Mathematical Modelling of Physical Systems [1st ed. 2018] 978-3-319-94757-0, 978-3-319-94758-7

Table of contents :
Front Matter ....Pages i-xvii
Thermostructure (Michel Cessenat)....Pages 1-149
Classical Mechanics (Michel Cessenat)....Pages 151-334
Fluid Mechanics Modelling (Michel Cessenat)....Pages 335-405
Behavior Laws (Michel Cessenat)....Pages 407-469
Back Matter ....Pages 471-505

Citation preview

Michel Cessenat

Mathematical Modelling of Physical Systems

Mathematical Modelling of Physical Systems

Michel Cessenat

Mathematical Modelling of Physical Systems

123

Michel Cessenat Institut Henri Poincaré Société de Mathématiques Appliquées et Industrielles Paris Cedex 05, Paris, France

ISBN 978-3-319-94757-0 ISBN 978-3-319-94758-7 (eBook) https://doi.org/10.1007/978-3-319-94758-7 Library of Congress Control Number: 2018948683 Mathematics Subject Classification: 35Q99, 53C80, 49S05, 76A99, 80A05 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To R. Dautray and to the memory of J.L. Lions

Foreword

After the first physical modelling of a system, a mathematical modelling is necessary in order to complete and in part to validate the previous model, to specify the goals that can be achieved by control theory, and then to prepare the work of numerical analysis. One of the main points is to specify the functional framework in agreement with physical requirements, and to make clear the intrinsic character of physical elements, independent of specific charts or frames. A goal of this work is to emphasize the main role played by differential geometry and convex analysis in the understanding of physical, chemical, and mechanical notions. It is indeed the case for the notions of reversible and irreversible evolutions in thermodynamics associated with a geometrization of the space of thermodynamic variables by foliation of that space. Classical mechanics for the motion of various solids and fluid mechanics is a natural framework for differential geometry, and using covariant derivatives is a central point, especially for the basic mechanics equations. The fluid mechanics leads to associated mechanical and thermodynamic evolution that must be modelled by an oriented foliated bundle coupling time and entropy. Many properties of the behavioral laws of materials and fluids are linked to convexity. References to differential geometry and convex analysis are made. This book will be useful for graduate students and researchers. I wish to thank Dr. Olivier Cessenat for his help. I wish also to thank late Professor G. Tronel for his useful advice.

vii

Introduction

Studying a physical system in order to solve a given problem involves many steps: A physical model defines variables, relevant parameters, and the type of modelling: deterministic, random, microscopic, macroscopic at several scale levels. A mathematical model specifies the conditions, the relevant functional frame, and the mathematical structure involved and often leads to linking variables through a system of equations with partial derivatives. The internal coherence of the previous physical modelling must be verified. A mathematical investigation and solution of the problem: existence, uniqueness, and stability of the solution. A numerical model adapts the model to a discretized processing (such as finite element methods), with stable schemes, leading to an approximate numerical solution. Back to the first step, comparing the numerical results to the experienced results in order to validate the choice of model. Otherwise, we have to correct the hypotheses and begin the modelling again. We first have to specify what we call a physical system. It is evolving matter, possibly with chemical reactions, waves, that are likely to be observed. Then we specify the notion of model: it is a rational description of the system using relevant characteristics of the system, and some knowledge of mathematical notions. Modelling deals with systems other than physical systems, especially in economics, sociology, linguistics, and even philosophy or theology. What makes modelling relevant is the introduction of some distance between the object of study (which is in a way transcendent) and what is represented, leading us to reconsider the value of our study compared to reality and to acknowledge some modesty (we no longer state that we know the real meaning of the studied phenomenon). We can classify certain models according to their predictive efficacy; perfect modelling, however, does not exist. From our study, we draw the possibilities of prediction, with the notions of causality, principles, and even the possibilities of knowing the past from the present. Physics (and on a larger scale science) should not be considered monolithic, but rather a succession of more or less accurate models.

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x

Introduction

A model can be characterized by its domain of validity, its coherence (internal logic), its richness (fitting various situations, with a sufficiently acute description), its simplicity, its efficiency. In a sense, different models represent nature, in the same way that an atlas with different charts represents a manifold—the Earth, for instance. Our goal is a mathematical model of several physical systems. The following question is essential: from a model of a physical system, is it possible to obtain a well-posed problem in the sense of Hadamard, i.e., with a unique solution, stable with respect to small data variations (which then allows a numerical solution)? The answer is very often negative, and then it leads us either to extrapolate data or to make behavioral hypotheses (about what happens in the infinite, for instance, or extra analyticity conditions), or to treat the problem with incomplete data (for example, with the theory of sentinels or with probabilities). Furthermore a mathematical model must control the model at various levels: Control of experiments: in order to obtain numerical results of experiments, it is necessary to use some signal processing or image processing for which a previous mathematical study has been made. Then results are often obtained through algorithms using a special type of modelling in a standard or canonical situation. We have also to solve inverse problems in order to obtain data for the direct problem. Control of a physical model: we have to check the relevance and the internal logic of a physical model. By simulation, we have to remedy the cases of difficult, expensive, or impossible experiments. We have to give some guarantees for the chosen model. A difficult point is to know the behavioral laws of materials, the state equations, and moreover their evolution as they age. Replacing physical experiments by numerical simulations is often a principal challenge in numerical studies. We also have to include in the model the possibility of operating on the system through the relevant parameters in order to obtain a fixed goal, which is the object of control theory. We also emphasize other interests in a mathematical model. A main point is to lead studies via well-adapted methods to the deep nature of the system and to employ powerful methods in solving a problem. A point frequently recognized (but often poorly understood) is rigor, which leads to a better understanding of nature by specifying all the hypotheses regarding the functional frame, initial conditions, and conditions at infinity; by specifying ambiguous notions, for example of reversibility or irreversibility from the macroscopic or microscopic point of view; by eliminating paradox in a consistent modelling; by clearing essential notions, intrinsic notions—that is, those that are frameindependent—which lead to the existence of mathematical structures and basic principles. Simplification of a model and numerical treatment. Returning to the modelling of a physical system, we have often to go from a microscopic model to a macroscopic one. We have to verify mathematically that it has been correctly formulated, with

Introduction

xi

the correct type of approximation, and we have to retrace our steps if the chosen model proves to be inadequate. If the given problem cannot be solved by computers, for instance due to very different scales, we can be led to solve a homogenization problem or a singular perturbation problem, or to change the problem into a nearby problem, possibly a nonlinear one (as in the approach of the Schrödinger equation with many variables by the Hartree–Fock equation). We must emphasize how sophisticated the mathematical structures for the modelling of physical systems are: a knowledge of matrix algebra is insufficient for obtaining a good understanding of the model. The difficulty in the mathematical modelling lies in the fact that we have to understand many languages—including those of the physician, chemist, engineer, and numerician—as well as a large domain of mathematics. The mathematical treatment implies a certain abstraction, which may be difficult for the reader. Studies of simple concrete examples facilitate comprehension. The main tools of the present modelling are differential geometry, convex analysis, and functional analysis. Notions of structures, manifolds, bundles, foliations, Lie groups, and Lie algebras occur naturally. A main notion of differential geometry is that of an integral manifold of a field of vector spaces, with the Frobenius theorem. Fundamental structures include contact structures for thermodynamics and notions of distance and covariant derivative on manifolds with Riemannian structures for usual mechanics; and then symplectic structures for Hamiltonian mechanics of conservative systems. A difficulty in using differential geometry lies in the smoothness conditions, which are not always satisfied. The use of convex analysis makes it possible to find substitutes, since convexity notions, extrema, are natural in physical systems, especially in thermodynamics and in the behavioral laws of materials. Of course, the functional frame is to be specified from the modelling of how certain quantities arise, such as the energy of the system. This allows us to give a meaning to equations, initial conditions, and boundary conditions. In this book, my goal is to give a mathematical model of certain physical systems from a physical model. My goal is not to solve equations, and I don’t offer any numerical treatment. Of course, there are many possible mathematical models from many physical models, specifically in thermodynamics with the entropy notion that may be defined from statistical thermodynamics, or also from optimal transport, or from Dirac structures. I only give, at the end, some references on points that are outside the scope of this work regarding a specific mathematical model. Of course, I in no way pretend that the given references are exhaustive. I only hope that they may be useful for students. The starting point of this study comes from fluid mechanics. A deep understanding requires intrinsic notions of differential geometry. This allows a natural definition of a physical quantity from the point of view of the fluid particle and the local thermodynamics. The first chapter is devoted to thermodynamics, with different points of view: global (modelling at zero order, where the volume appears), local, which will allow the functional thermodynamic representative of the thermodynamic state at every point of the system.

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Introduction

First we specify the main tools to be used. In the standard space F in which the thermodynamic variables are represented, which is of odd dimension, a basic differential form θ defines a contact structure. Then the state equation defines a foliation of F with n-dimensional leaves in the (2n + 1)-dimensional space F as maximal integral manifolds for θ . Evolution of the system along the leaves is reversible, while transverse evolution is irreversible, and it can occur in only one way, as required by the second law of thermodynamics. Convex analysis plays a basic role especially for stability. Many examples and generalizations are given: ideal gases, real gases with the van der Waals equation, systems with many gases and chemical reactions. Relative to functional thermodynamics, we emphasize the relation between the parabolic maximum principle and the second law of thermodynamics. We also consider thermal radiation with statistical thermodynamics, then electromagnetism in thermodynamics. The second chapter is devoted to usual mechanics with fields of forces that do not necessarily derive from a potential. With some simple examples, we see the necessity of the structure of Riemannian manifolds for the modelling of mechanical systems. Newton fundamental law of mechanics is then due to the covariant derivative in the space of differential forms. Special examples are solids, where the structure of Lie groups and Lie algebras is essential. In Chapter 3, the kinetic modelling of a fluid is given first; the structure of a Riemannian manifold is necessary, which allows us to define the strain tensor and the strain rate tensor from the velocity field. If the domain of the fluid is moving, the configuration space must include the time, and it is then equipped with the structure of a foliated Riemannian manifold. We are led to use Euler or Lagrange coordinates. Then the modelling of the fluid mechanics with thermodynamics leads to a fibre product bundle of the tangent space on spacetime for the kinetics, with a thermodynamic bundle. This leads to a foliation of the product bundle, due to time and the irreversible thermodynamic motion. Evolution of the given fluid must be obtained through two linked fields: the velocity field and the field giving the thermodynamic state of the fluid particle. These fields must satisfy the equations of fluid mechanics (Euler, Navier–Stokes, ENS) and the thermodynamic equations. These equations are obtained through a basic differential form on spacetime in connection with the principle of virtual power. We recall the usual hypotheses on the fluid, and we consider the natural functional frameworks for these equations. Chapter 4 gives behavior laws for fluids. Generally, they are Newtonian fluids. Generalizations are given as Bingham fluids. Systematic applications of convex analysis, following [Mor1], allow us to relate forces and velocities, and to give solutions of stationary problems in fluid mechanics. The goal of the appendix is essentially to recall some fundamental notions in differential geometry for independent reading.

Contents

1

Thermostructure .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Definitions. Thermoreversibility . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Local, Global, Semiglobal Modelling.. .. . . . . . . . . . . . . . . . . . . . 1.1.2 Mathematical Framework: Differential Geometry . . . . . . . . . 1.1.3 Some Parametrizations with Contact Structures .. . . . . . . . . . . 1.1.4 Relations Between the Basic Differential Forms . . . . . . . . . . . 1.1.5 Modelling of Work and Heat as Differential Forms .. . . . . . . 1.1.6 State Equations in a 5-Dimensional System .. . . . . . . . . . . . . . . 1.1.7 Reversible and Irreversible Evolutions ... . . . . . . . . . . . . . . . . . . . 1.2 Convexity Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Mathematical Framework: Convex Analysis . . . . . . . . . . . . . . . 1.2.2 Convexity in Thermodynamics .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Stability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Thermosystems with One Degree of Freedom . . .. . . . . . . . . . . . . . . . . . . . 1.4 Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 State Equations .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Specific Entropic Foliation for an Ideal Gas . . . . . . . . . . . . . . . . 1.4.3 Determination of Thermodynamic Functions . . . . . . . . . . . . . . 1.4.4 Carnot Cycle for an Ideal Gas . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Fluid with the van der Waals Equation . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Study of the van der Waals Equation.. . .. . . . . . . . . . . . . . . . . . . . 1.5.2 Loss of Convexity with van der Waals . .. . . . . . . . . . . . . . . . . . . . 1.5.3 Integrability and Convexity for van der Waals . . . . . . . . . . . . . 1.6 Thermosystems with Different Constituents . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.1 Local Thermodynamic Modelling.. . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.2 Global Thermodynamic Modelling . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.3 Going from Global to Local Thermodynamics .. . . . . . . . . . . . 1.6.4 From n Independent Constituents to a Mixture.. . . . . . . . . . . . 1.6.5 Semiglobal Thermodynamics . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.6.6 System of Ideal Gases, Global Thermodynamics . . . . . . . . . .

1 1 1 3 10 11 12 13 14 19 19 22 26 29 30 30 34 39 41 45 45 54 68 71 71 78 84 88 90 93

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1.6.7 System of Ideal Gases, Local Thermodynamics.. . . . . . . . . . . 1.6.8 Real Solutions .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7 Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.7.1 The Case of One Chemical Reaction. . . .. . . . . . . . . . . . . . . . . . . . 1.7.2 System of r Independent Chemical Reactions.. . . . . . . . . . . . . 1.8 Thermodynamic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.1 Thermodynamics Depending on Space .. . . . . . . . . . . . . . . . . . . . 1.8.2 Diffusion Process and Flux . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.8.3 Local Balance and Flux for a Moving Fluid .. . . . . . . . . . . . . . . 1.8.4 Connecting Local and Global Differential Forms . . . . . . . . . . 1.8.5 Functional Convex Analysis . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.9 The Use of the Free Helmholtz Energy .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.10 Thermoelectromagnetism . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.10.1 Constitutive Electromagnetic Relations.. . . . . . . . . . . . . . . . . . . . 1.10.2 Differential Forms in Thermoelectromagnetism . . . . . . . . . . . 1.10.3 Functional Thermoelectromagnetism . . .. . . . . . . . . . . . . . . . . . . . 1.11 Thermal Radiation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.11.1 Planck Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.11.2 Probability Measure and Expectation Values .. . . . . . . . . . . . . . 1.11.3 Global Thermodynamics . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.11.4 Local Thermodynamics .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

99 102 106 107 117 122 122 124 131 133 135 136 139 140 140 142 143 143 144 147 148

2 Classical Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Time Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Two Main Modelling Types .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Reminder of Mechanics with an Example .. . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Mechanics of Newton, Lagrange, and Hamilton .. . . . . . . . . . . . . . . . . . . . 2.4.1 Metric Map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 First-Order Equation .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 Second-Order Equation .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.4 Hamiltonian Vector Fields . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.5 Time-Dependent Force and Hamiltonian . . . . . . . . . . . . . . . . . . . 2.4.6 Fundamental Equation of Mechanics . . .. . . . . . . . . . . . . . . . . . . . 2.4.7 Canonical Forms Decompositions . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.8 Virtual Power Principle . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.9 Special Relativity and the Lorentz Metric . . . . . . . . . . . . . . . . . . 2.4.10 Evolution with an Electromagnetic Field . . . . . . . . . . . . . . . . . . . 2.5 Paths, Curves, and Geodesics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.1 On the Curve Parametrization . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.2 Exponentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.3 Wave Function of a Geodesic Field . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.4 Foliation and Eikonal Equation . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.5 Extremum Among Path Integrals . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.6 Euler–Cartan Equation and Some Other Extrema .. . . . . . . . .

151 151 152 154 157 160 161 162 165 167 174 178 182 184 203 216 216 219 220 222 223 228

Contents

2.6

2.7

2.8

2.9

2.10 2.11

2.12

2.13

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Some Simple Examples in Mechanics .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 Mechanics Problem on a Circle . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 Two Material Points with a Link . . . . . . . .. . . . . . . . . . . . . . . . . . . . Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.1 Mechanics Modelling of a Rigid Body .. . . . . . . . . . . . . . . . . . . . 2.7.2 Lie Group and Lie Algebra of Displacements .. . . . . . . . . . . . . 2.7.3 Adjoint Representation of D(3) . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.4 Coadjoint Representation of D(3) . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.5 Velocity Field of a Rigid Body .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.6 Inertial Riemannian Metrics . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.7 Modelling of Forces. Wrenches . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.8 Covariant Derivative of the Velocity . . . .. . . . . . . . . . . . . . . . . . . . 2.7.9 Euler Equation for a Rigid Body . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7.10 Some Angular Parametrizations of SO(3).. . . . . . . . . . . . . . . . . 2.7.11 Potential Energy for Forces with Potential . . . . . . . . . . . . . . . . . Relative Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.1 Relative Motion of Euclidean Frames . .. . . . . . . . . . . . . . . . . . . . 2.8.2 Composition of Velocities . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.3 Composition of Accelerations .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.8.4 Relative Evolution of a Material Point ... . . . . . . . . . . . . . . . . . . . Motion of Two Rigid Bodies .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.1 General Free Case. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.2 Rigid Bodies with an Axial Link .. . . . . . .. . . . . . . . . . . . . . . . . . . . 2.9.3 An Application with Nonholonomic Constraints .. . . . . . . . . . Incompressible Fluid in a Fixed Domain .. . . . . . . .. . . . . . . . . . . . . . . . . . . . Action of a Lie Group on a Manifold .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11.1 Action of G by Differential Operators . .. . . . . . . . . . . . . . . . . . . . 2.11.2 Lie Algebra of Functions . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11.3 Noether Theorem . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.11.4 Momentum Map, Poisson Action of a Lie Group . . . . . . . . . . Appendix 1. Lie Group and Lie Algebra . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.12.1 Definitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.12.2 Adjoint Representation of a Lie Group .. . . . . . . . . . . . . . . . . . . . 2.12.3 Coadjoint Representation .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.12.4 Splitting of the Tangent and Bitangent Bundles .. . . . . . . . . . . Appendix 2. Covariant Derivative . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.13.1 Connection, Horizontal Lift . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.13.2 Covariant Derivative . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.13.3 Extension of the Covariant Derivative . .. . . . . . . . . . . . . . . . . . . . 2.13.4 Riemannian Covariant Derivative . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.13.5 Connection on a Principal Fibre Bundle . . . . . . . . . . . . . . . . . . . . 2.13.6 Christoffel Symbols, Riemann–Christoffel Tensor .. . . . . . . . 2.13.7 Time-Covariant Derivative on a Riemannian Group . . . . . . .

232 232 236 245 245 247 250 253 253 258 267 269 271 276 287 290 290 291 293 294 296 296 297 301 303 305 306 307 309 310 315 315 316 317 318 320 320 324 325 327 330 331 332

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3 Fluid Mechanics Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Kinetic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Integral Flow and Time Foliation . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Euler and Lagrange Coordinates . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Euler and Lagrange Metrics . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.4 Associated Connection Form and Covariant Derivative . . . 3.1.5 Metric and Strain for a Continuum Medium.. . . . . . . . . . . . . . . 3.1.6 Rate of Strain Tensor Field for a Fluid . .. . . . . . . . . . . . . . . . . . . . 3.1.7 Strain Rate with Covariant Derivative . .. . . . . . . . . . . . . . . . . . . . 3.1.8 Trace Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.9 Functional Study of Existence of a Flow . . . . . . . . . . . . . . . . . . . 3.2 Thermodynamic Modelling . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction. Fibre Product of Bundles .. . . . . . . . . . . . . . . . . . . . 3.2.2 Some Reminders . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 Mass Conservation .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.4 Differential Structure: Foliated Fibre Bundles.. . . . . . . . . . . . . 3.2.5 Work and Heat on M˜ . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.6 Some Remarks and Extensions .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Mechanics Modelling.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 Euler–Navier–Stokes (ENS) Covariant Equations.. . . . . . . . . 3.3.2 Divergence Equations .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Modelling Hypotheses.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 Conditions on the Fluid Domain . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 Hypotheses on the Fluid Behavior . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.3 Mechanical Conditions on the Boundary . . . . . . . . . . . . . . . . . . . 3.4.4 Thermodynamic Boundary Conditions .. . . . . . . . . . . . . . . . . . . . 3.5 “Natural” Functional Framework . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Global Conditions on the Specific Mass . . . . . . . . . . . . . . . . . . . . 3.5.2 Framework for the Physical Model . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.3 A Priori Estimates from the Previous Hypotheses . . . . . . . . . 3.5.4 “Natural” Framework for the Thermodynamic Field.. . . . . . 3.5.5 Consequences of This Thermodynamic Framework . . . . . . . 3.5.6 Evolution with a Surface Discontinuity .. . . . . . . . . . . . . . . . . . . .

335 335 335 338 340 341 345 347 349 354 362 365 365 366 368 370 372 377 379 379 382 383 384 385 386 387 388 388 389 393 395 395 396

4 Behavior Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Some Review of Convex Analysis . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Generalized Fenchel Theorem.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Force–Velocity Relations in Convex Analysis . . .. . . . . . . . . . . . . . . . . . . . 4.3 Stress–Strain Relations in Mechanics . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Stress–Strain Relations for Newtonian Fluids .. . . . . . . . . . . . . 4.3.2 Stress–Strain Relations for Bingham Fluids . . . . . . . . . . . . . . . . 4.3.3 Stress–Strain Relations for Tensor Fields .. . . . . . . . . . . . . . . . . .

407 408 408 412 414 414 416 418 420

Contents

4.4

xvii

Relations Between Force and Velocity Fields . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Relations f, v with Homogeneous Dirichlet Conditions .. . 4.4.3 Dual Space of H 1 (M)3 . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.4 Relations f, v with Boundary Conditions . . . . . . . . . . . . . . . . . . 4.4.5 Mixed Type Condition .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.6 Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.7 Slipping Condition on the Boundary .. . .. . . . . . . . . . . . . . . . . . . . Relations σ,  with Trace Constraint.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Force–Velocity with Constraint on the Divergence . . . . . . . . . . . . . . . . . . 4.6.1 General Case for a Newtonian Fluid . . . .. . . . . . . . . . . . . . . . . . . . 4.6.2 Relation f, v with Constraint, Dirichlet Condition .. . . . . . . . 4.6.3 Relation f, v with Constraint, Neumann Condition .. . . . . . . 4.6.4 Stream with Constraint . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6.5 Application, Viscous Force on an Obstacle .. . . . . . . . . . . . . . . . 4.6.6 Superficial Tension .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

421 421 423 425 431 433 436 444 445 449 449 453 456 460 465 467

Appendix A .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.1 Some Notation and Definitions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2 Integral Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.1 Integrable k-Field . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.2 Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.3 Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.2.4 Applications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.3 Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4 Contact Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4.1 Definitions. Contact Structure .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4.2 Projective Contact Manifold .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4.3 Darboux Theorem for Contact Structures.. . . . . . . . . . . . . . . . . . A.4.4 Symplectification of a Contact Manifold (M, P ). . . . . . . . . . . A.4.5 Integral Submanifolds and Hamiltonian . . . . . . . . . . . . . . . . . . . . A.4.6 Contact Manifold of Jets. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A.4.7 Legendre Involution, Legendre Transformation .. . . . . . . . . . . A.4.8 Characteristics of a Contact Structure.. .. . . . . . . . . . . . . . . . . . . . A.4.9 Cauchy Problems, First-Order Partial Differential Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

471 471 474 474 475 478 480 482 484 484 485 486 488 489 491 492 494

4.5 4.6

496

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 499 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 503

Chapter 1

Thermostructure

Thermodynamics deals with quantities such as pressure, temperature, internal energy (which will be called energy for short if there is no misunderstanding); thus we can speak of a thermodynamic modelling of such physical systems. At the start, thermodynamics deals with global modelling at a macroscopic level (modelling at zero order), dealing with only global quantities such as the volume of the domain of the system (without its shape), and only one temperature and one pressure. There then appeared a semiglobal model for a system with two different parts, each part with a global model (volume, temperature, pressure). This leads to many difficult questions about the relations between the two parts and the evolution to an equilibrium. A local point of view and a macroscopic one (which we generally adopt in this book) is a finer sort of modelling, leading to the study of continuous media. At each point of the domain of the system and at each time, we have to specify thermodynamic variables and their relations. At a microscopic level, thermodynamic notions appear through statistical mechanics, which allow a (partial) understanding of the macroscopic thermodynamics. Here we do not use statistical mechanics; our goal is to specify some basic notions such as the possibility of reversible or irreversible motions, thanks to differential geometry and the notion of maximal integral manifold for a specific differential form. This leads to an oriented foliation, giving a geometrization of thermodynamics.

1.1 Definitions. Thermoreversibility 1.1.1 Local, Global, Semiglobal Modelling In a local model of a system with only one constituent in a connected bounded domain  in R 3 there are five state variables: pressure P , (absolute) temperature © Springer International Publishing AG, part of Springer Nature 2018 M. Cessenat, Mathematical Modelling of Physical Systems, https://doi.org/10.1007/978-3-319-94758-7_1

1

2

1 Thermostructure

T , specific mass ρ (or τ = 1/ρ), internal energy (per unit mass) e, and also a fundamental quantity that is not an observable (its direct measure is impossible), the entropy s. For systems with one constituent, the chosen state variables are (τ, e, s, P , T ). For systems with many constituents, there are supplementary state variables: specific masses (or their concentrations), chemical potentials of each constituent ρk and μk . Of course, absolute temperature T , specific masses ρk (and thus τk ) are positive; pressure P is always positive for gases. All these variables are not independent: there are relations called state equations, linked to the physical (or chemical) nature of the medium, so that for a oneconstituent system, we have only three independent variables, such as (τ, e, s). We note that entropy, not being observable, does not appear in the state equations. Of course all these variables are dependent of the position x ∈ , which will be studied in the end of this chapter. In order to make a global model of a system with one constituent, in a connected bounded domain  in R 3 , of volume V , we must have a homogeneous system with a pressure P and a temperature T . The state variables are, with P and T , the mass M (or the mole number N), the volume V , the total (internal) energy E, the chemical potential μ, and the total entropy S. In a homogeneous system with many constituents, the masses Mk (or the mole numbers Nk ) with the chemical potential μk of each constituent are also thermodynamic variables. The variables T , V , Mk (or Nk ) are positive (with the pressure P for gases). Thermodynamic variables are linked by state equations as in the local model, so that we have to choose independent variables. The possibility of changing the volume is modeled by a family (λ ), λ ∈ R + , of connected domains that depend on a parameter λ (for example by dilation), so that thermodynamic variables are λ-dependent. Extensive variables are homogeneous of order 1 in λ, while intensive variables are homogeneous of order 0 and thus do not depend on λ. We have Vλ = λV , Mλ = λM, Eλ = λE, Sλ = λS, but T , P , μ are λindependent. The action on the system in global thermodynamic modelling lies in the notion of exterior, which corresponds to boundary conditions in a local model. An isolated system is a system without exchange of mass or energy with the exterior; for a closed system, its total mass does not change, while an open system can exchange mass and energy. The exterior can impose a temperature T to the system (the exterior is said to be a reservoir) or a pressure P . The notion of irreversible variation of certain quantities is linked, notably in [Pri-Kon], to a variation that is imposed by the exterior. (In local thermodynamics, this choice is not possible, and the main goal of this chapter is to specify the notion of irreversible variation.) Exterior conditions change the evolution of the system by perturbation of its homogeneous character, so that we want to know the perturbation effect on the system when it recovers its homogeneity. The main goal of global thermodynamics is to specify the equilibrium states and the possibility of changing those states. The notion of stability of an equilibrium state is essential, and it depends on the convexity of certain thermodynamic functions.

1.1 Definitions. Thermoreversibility

3

Semiglobal thermodynamic modelling. We often have many interacting systems with a global model for each. To each system we associate its variables (Vi , Ei , Si , Ti , Pi ), i ∈ I . We have to specify joint variables (for example the temperature), and the possible exchanges from one system to another. A main problem is to know which equilibrium state is reached for the total system, starting from an equilibrium state for each subsystem. In certain cases we can improve the semiglobal thermodynamic model by increasing the number of variables. For example, in the case of two systems with a part of the boundary in common, and with the same temperature, the modelling of each system is given by (Vi , Ei , Si , Pi , T ), and the modelling of the total system is given by the variables (V1 , V2 , E, S, P1 , P2 , T ), with A the measure of the surface , and γ the surface tension, but with V = V1 + V2 constant. We call a physical system with a thermodynamic model a thermosystem. The notions of work and heat imply the possibilities of evolution of the thermosystem, which leads us to define reversible and irreversible evolutions.

1.1.2 Mathematical Framework: Differential Geometry Contact structure and foliation. The space of odd dimension of the thermodynamic variables of a system has the structure of a differentiable manifold with a differential form that defines a contact structure. We give a prototype of the frame of our study without specifying the thermodynamic variables. Let F = M × R × N be a subspace of R n × R × R n , with M and N two open (convex) subsets of R n . Let (x, u, p) be a point of F, with x i , i = 1, . . . , n, and pk , k = 1, . . . , n, the respective components of x ∈ M and p ∈ N . Now let θ be the differential form on F defined by θ(x,u,p) = du − p.dx = du −



pi dx i ,

(1.1)

i

and let G(x,u,p) = ker θ(x,u,p), the space of the tangent vectors Z to F such that < θ(x,u,p), Z > = 0. This is a subspace of the tangent space to F at (x, u, p); the map (x, u, p) ∈ F → G(x,u,p) defines a contact structure on F that is a subbundle of the tangent bundle T (F) of codimension 1. The integral manifolds of this structure, which are the manifolds  ⊂ F such that T(x,u,p)() ⊂ G(x,u,p), ∀(x, u, p) ∈ , are at most n-dimensional.

4

1 Thermostructure

Now let be a differentiable map1 from M×R onto N . With the canonical basis ˜ = I × of R n , is identified with n real functions ( i ). We again define the map ˜ from M × R into F such that (x, u) = (x, u, (x, u))∀(x, u) ∈ M × R. Then let  = {(x, u, (x, u)), ∀(x, u) ∈ M × R},

(1.2)

which is a submanifold of F. Let θ be the induced differential form θ on  , and we again define a subbundle Q of T ( ) by Q x,u,p = ker θx,u,p ⊂ T(x,u,p) ( ),

p = (x, u), (x, u, p) ∈  .

(1.3)

˜ is the differential form θ = ˜ ∗ (θ ) The pullback of the differential form θ by on M × R, which is also given by ˜ ∗ (θ ))(x,u) = du −



i (x, u)dx i .

(1.4)

Let ˜∗ G x,u = ker( (θ ))(x,u) ⊂ T(x,u) (M × R);

(1.5)

this defines a map (x, u) → G x,u , which is a subbundle W of T (M × R). Now the question is whether W is integrable, i.e., whether ∀(x, u) ∈ M × R, there is a submanifold V ϕ of M × R such that

T(x,u (V ϕ ) = GFx,u , ∀(x, u) ∈ V ϕ . A necessary and sufficient condition that W be integrable is that the (exterior) ˜ ∗ (θ ) = 0, thus [Bour.var, 9.3.8] that satisfy, for all i, j ∈ differential be null: d [1, n], ∂ j ∂ j ∂ i ∂ i i = j . + + ∂x i ∂u ∂x j ∂u

(1.6)

A map ϕ from an open set U ⊂ M into R is an integral of if and only if ∂ϕ = i (x, ϕ(x)) ∀x ∈ U, i ∈ [1, n], ∂x i ˜ ∗ (θ )) = 0, that is, which is equivalent to ϕ ∗ ( dϕ −

1 This



i (x, ϕ(x))dx i = 0.

map will be given in thermodynamics by the state equations.

(1.7)

1.1 Definitions. Thermoreversibility

5

A main special case is that in which (x, u) is independent2 of u: ˆ from M onto N , so that Hypothesis 1 is defined through a diffeomorphism3 ˆ (x, u) = (x), ∀(x, u) ∈ M × R. ˆ simply by , then the integrability condition is given by the Schwarz Denoting conditions: Hypothesis 2 ∂ j ∂ i − = 0, i ∂x ∂x j

∀i, j = 1, . . . , n,

(1.8)

and ϕ is an integral of if and only if dϕ −



i (x)dx i = 0.

If Hypothesis 2 is satisfied, then ϕ is said to be an integral of , and W (or ) is said to be integrable, and there is a foliation, denoted4 by (M×R, W ), of the space M × R, whose leaves are the maximal connected submanifolds Vα of M × R, with , ∀(x, u) ∈ V , and the partition of M × R into leaves is “locally Tx,u (Vα ) = Wx,u α is 1, and trivial.” Moreover, Hypotheses 1 and 2 imply that the codimension of Wx,u if ϕ is an integral of , then ϕu0 = ϕ + u0 with a constant u0 is again an integral of . Thus the family of leaves may be named by the subscript α = u0 ∈ R, and the leaves of (M × R, W ) are (at least locally) the graphs G(ϕu0 ) of functions ϕu0 . Then this also implies that Q is integrable, and thus that ( , Q ) is a foliation of  , whose leaves are the jets of functions ϕu0 , that is, J (ϕu0 ) = {(x, ϕu0 (x), p),

x ∈ M, p = (pj ), pj =

∂ϕu0 }. ∂x j

These two foliations of M × R and  are isomorphic, and u0 is a variable transverse to the foliations. We note that we can change the variables ((x j ), u) into ((x j ), u0 ) by ζ (x, u0 ) = (x, u0 + ϕ(x)) = (x, u). Then if f is a function of (x, u) ∈ M × R, we have f (x, u) = f (x, u0 + ϕ(x)) = F (x, u0 ), whence F = f ◦ ζ , and thus ∂F ∂f , = ∂u0 ∂u

∂F ∂f ∂ϕ ∂f . = + j ∂x j ∂x j ∂x ∂u

Thus the action of ζ on the tangent space T (M × R) is given by 2 This

is the case in thermodynamics when u is the entropy, since the state equations do not depend on the entropy. 3 Or more generally a differential map. 4 According to the notation of [Connes].

6

1 Thermostructure

ζ∗ (

∂ ∂ ∂ϕ ∂ , )= + j ∂x j ∂x j ∂x ∂u

ζ∗ (

∂ ∂ ) = ( ). ∂u0 ∂u

Then the n tangent vector fields to M × R, ∂ ∂ ∂ ∂ϕ ∂ = j + j , + pj j ∂x ∂u ∂x ∂x ∂u

Xj =

(1.9)

satisfy the following properties: < θ , Xj > = 0,

[Xi , Xj ] = 0,

∀i, j = 1, . . . , n.

(1.10)

Indeed, we have [Xi , Xj ] = 0 thanks to (1.8). Then < θ , Xj > = −pj +pj = 0, and thus Xj is tangent to the leaves of the foliation (M × R, W ), whereas for the transverse variable, < θ , ∂u∂ 0 > = 1. Then we can define the projectors in the space T (M × R): Ph =



Xi ⊗ dx i ,

Pv =

∂ ⊗ θ P hi . ∂u

(1.11)

We easily verify the properties P h ◦ P h = P h , P v ◦ P v = P v , P h ◦ P v = P v ◦ P h = 0,

P h + P v = I.

If X ∈ T (M × R), then P h X is its projection on the tangent space to the leaves, and P v X is its transverse projection. By duality we have the projectors in the space T ∗ (M × R): (P h )∗ =



dx i ⊗ Xi ,

(P v )∗ = θ ⊗

∂ . ∂u

(1.12)

Then if df is the differential of a function f on M × R, then d r f = (P h )∗ (df ), respectively d irr f = (P v )∗ (df ), is the projection on the dual spaces, so that we have df = d r f + d irr f . Product and splitting of foliations. Now with Hypothesis 1, we assume the following: Hypothesis 3 The diffeomorphism splits into two diffeomorphisms 1 and 2 , that is, there are open sets Mi ⊂ R ni , i = 1, 2, with M = M1 × M2 , n = n1 + n2 , and N i ⊂ R ni , i = 1, 2, with N = N 1 × N 2 , so that i is a diffeomorphism from Mi onto N i , and = ( 1 , 2 ). i i i i Now let θ(x i ,ui ,p i ) = du − p .dx , i = 1, 2, be the two basic differential forms on i

Mi × R × N i , and let W , i = 1, 2 be subbundles of T (Mi × R) corresponding to i .

1.1 Definitions. Thermoreversibility

7 1

2

A necessary and sufficient condition that W be integrable is that W , W both be integrable, thus that 1 , 2 both satisfy the Schwarz conditions 2. If these i conditions are satisfied, then we obtain two foliations (Mi × R, W ) with leaves Li i . From these two foliations, we can define the product of the spaces (M1 × u0

R) × M2 × R and the product of the leaves L1 1 × L2 2 . Then we obtain the foliation u0

(M × R, W ) with leaves Lu0 thanks to the map

u0

((x 1 , u1 , p1 ), (x 2 , u2 , p2 )) → ((x 1 , x 2 ), u1 + u2 , (p1 , p2 )) with ui = ui0 + ϕ i (x i ), u = u1 + u2 = (u10 + u20 ) + (ϕ 1 (x 1 ) + ϕ 2 (x 2 )), if ϕ i is an integral of i , and thus the function ϕ defined on M by ϕ(x) = ϕ(x 1 , x 2 ) = (ϕ 1 (x 1 ) + ϕ 2 (x 2 )) is an integral of , giving by its graph a leaf of the foliation (M × R, W ), and by its jet a leaf of the foliation ( , Q ). ¶ Convexity condition. Furthermore, we assume Hypothesis 4 below. ∂

Hypothesis 4 The Jacobian matrix J ( ) = ( ∂x ij ) is positive definite. Then Hypothesis 4 implies that the Hessian matrix H (ϕ) of ϕ is positive definite, and thus that ϕ is a real convex function on M. Of course we have the same property for ϕu0 , ∀u0 ∈ R, and thus all the leaves of the foliation (M ×R, W ) (respectively of ( , Q )) are the graphs (respectively the jets) of convex functions. Often we will assume that the opposite of the Jacobian matrix is positive definite, and thus we have to change “convex” to “concave.” ˆ is invertible, we can also Remark 1 Conjugate foliation. Since by Hypothesis 1, −1 ˆ use its inverse p ∈ N → x = (p) ∈ M, and thus define a new foliation, “conjugate” to the above foliation in the sense of convex analysis, with the graphs (or the jets) of convex functions, which is obtained from the previous foliation by Legendre transformation, as will be seen below. This new foliation may be obtained directly from the differential form ∗ ∗ ∗ θ(p,u ∗ ,x) = du − x.dp = du −



xi dpi ,

(1.13)

i

with u∗ = p.x − u.

¶

Remark 2 Notion of “irreversible differential.” Let Y = (Fα ) be a foliation of a manifold M, and π a submersion M → W in a manifold W , so that Y = Mπ (i.e., the leaves can be defined by π −1 (w), w ∈ W ). Then for all x ∈ M, there

8

1 Thermostructure

exists a submanifold Wx of M such that the tangent space Tx M in x ∈ M splits into the tangent space to the leaf Tx (M, Y ) and the tangent space Tx (Wx ) to Wx (see [Bour.var][9.2.9]): Tx M = Tx (M, Y ) ⊕ Tx (Wx ). Passing to the dual spaces, we have Tx∗ M = Tx∗ (M, Y ) ⊕ Tx∗ (Wx ). This corresponds to the splitting of differential forms; for example, if g is a differentiable function of M into R, then dg is split in a reversible part drev g(x) (such that < drev g(x), w > = 0, ∀w ∈ Tx (Wx )) and an irreversible part (such that we have < dir g(x), w >= 0 for all tangents w to the leaf that contains x):5 dg(x) = drev g(x) + dir g(x). The notation of irreversible differential, typically for entropy, is used by [Pri-Kon] in a frame of global thermodynamics where all that is coming from the exterior is said to be irreversible. ¶ Remark 3 Contact Transformations and Foliation Morphisms Let us call (see [Arn1][App. 4]) a diffeomorphism that preserves the contact structure of a manifold F a contact transformation in F with a contact structure. The set of contact transformations forms a group G. In the thermodynamic frame with a basic differential form θ giving the contact structure, every diffeomorphism φ of F such that φ ∗ (θ ) is proportional to θ is a contact transformation. For given state equations defining a manifold  in F , we denote by G the set of contact transformations leaving the manifold  stable. If Q is integrable, then ( , Q ) is a foliation, and elements of G transform the foliation into itself. They are morphisms of the foliated structure, a leaf Lα being transformed into another leaf. A simple example is that of translations of u, gu0 (x, u, p) = (x, u + u0, p), with Hypothesis 1. ¶ Remark 4 The Darboux theorem for the contact structure. Through the Darboux theorem6( [Arn1][App. 4]), the space F with N = R n can be identified with the space Qn = J 1 (M, R) of jets of real functions on M of class C 1 , defined by 7 j = (x; u; p) ∈ Qn ⇔ ∃g ∈ C 1 (M, R) such that j = (x; g(x); dg(x)).

5 It

is not clear in the general case whether drev g and dir g are exact differentials. reader can skip this remark on a first reading. 7 Using notation near that of mechanics. 6 The

1.1 Definitions. Thermoreversibility

9

The space J 1 (M, R) can be seen as a phase space (or cotangent bundle) “enlarged by R,” that is, (T ∗ M) × R. The space T ∗ M being naturally equipped with the canonical differential form α = p.dx = pk dxk , the following differential form is defined on Qn : θ = du − p.dx = du −



pk dxk .

Let f ∈ C 1 (M, R) be a real function on M. Then Jx (f ) = (x; f (x); df (x)) is called the jet of f at x, and p = df (x) ∈ Tx∗ M is a linear form on Tx M: y →<  ∂f (x) = ∂k f (x), with df (x), y > = ∂k f (x)y k ; we identify p with (pk ), pk = ∂x k k = 1 at n. LetJ (f ) be the jet of f (respectively G(f ) its graph) defined by def

J (f ) = {(x, f (x), df (x)), x ∈ M},

def

G(f ) = {(x, f (x)), x ∈ M}.

Thus J (f ) (respectively G(f )) is a submanifold of J 1 (M, R) (respectively of M × R). The canonical projection π from J 1 (M, R) onto J 0 (M, R) = M × R, π(x; u; p) = (x; u) transforms the jet of f into its graph: π(J (f )) = G(f ). If f is of class C k , k ≥ 1, then the map x ∈ M → Jx (f ) ∈ J 1 (M, R), also denoted by J (f ), is of class C k−1 . In the special case in which f is of class C 2 , J (f ) is an embedding in J 1 (M, R), and its image is a submanifold of class C 1 of J 1 (M, R). More generally, we can start from the phase space T ∗ (M × R), and from its projective space Qn = P T ∗ (M×R).8 The phase space T ∗ (M×R) being naturally equipped with the canonical differential form θc = p.dx + pn+1 dxn+1 , the choice of a map Un with pn+1 = 1 for its projective space gives the differential form p.dx − du = −θ with u = −xn+1 . We thus identify (x; u; p) ∈ J 1 (M, R) with ((x, u), (p, 1)) ∈ Un ⊂ P T ∗ (M × R). The space of jets of differentiable functions J 1 (M, R) comes naturally from the projective space Qn . ¶ In the case of M = R n , the Legendre involution L : (x, u, p) → (p, U, x) with U = p.x−u (see (1.40) below) transforms the basic differential form θ = du−p.dx into L∗ (θ ) = dU −x.dp, and we have L∗ (θ )+θ = 0. Thus the Legendre involution transforms a maximal integral manifold of θ , called a Legendre manifold, into a maximal integral manifold, and transforms (locally generally) the jet J (f ) into a manifold L(J (f )) when f ∈ C 1 (R n , R)).9 Note that the chosen parametrization of a Euclidean space by R n is due to the choice of units.

that it is the quotient space of T ∗ (M × R) by the equivalence relation (x1 , . . . , xn+1 ; p1 , . . . , pn+1 ) ∼ (x1 , . . . , xn+1 ; μp1 , . . . , μpn+1 ), μ ∈ R ∗ . 9 But a priori, L(J (f )) is not globally the jet of a function F . 8 Recall

10

1 Thermostructure

1.1.3 Some Parametrizations with Contact Structures We give some examples of frequently used parametrizations of the space F for a system with one constituent. In the first parametrization, notions of work and heat directly appear; the second is associated with reversible or irreversible notions of evolution. Other parametrizations can be very interesting to use in some particular situations. Below we specify the variables (x1 , x2 ; u ; p1 , p2 ) in different parametrizations, with basic differential forms. The domain of (absolute) temperatures and that of specific masses is R + = ]0, +∞[. The domains of the other state variables such as the pressure depend on the state equations.10 Parametrization(τ, s; e ; −P , T ), with energy as the main variable. The space F is identified with F˜ = (R + × R) × R × (R × R + ),

with M = R + × R,

(1.14)

and is equipped with the differential form θ = θ e = de + P dτ − T ds.

(1.15)

This differential form is never degenerate, except for P = 0. ¶ Parametrization(τ, e; s ; PT , T1 ), with entropy as the main variable. As in (1.14), the space F is identified with F˜ == (R + × R) × R × (R × R + )

with M = R + × R.

(1.16)

The space F is equipped with the differential form θ s = ds −

1 P dτ − de. T T

(1.17)

This choice of parametrization is essential from the state equations, as we will see below. It is fairly natural from the point of view of units, with s without dimension, the units of (p1 , p2 ) being the inverses of those of (x1 , x2 ). Here the dual variable of the energy is the inverse of the temperature (the Boltzmann constant kB being taken equal to 1), which appears naturally in many physical expressions. In the case of gas, energy is positive, and then the space F is F = (R + )2 × R × (R + )2 .

¶

Parametrization (P , s; h ; τ, T ) with enthalpy h = e + P τ as main variable, with M = R + × R. The differential form is then given by

10 In many models of systems (especially at the level of molecules), energy is bounded from below, but there is no universal bound.

1.1 Definitions. Thermoreversibility

11

θ h = dh − τ dP − T ds.

(1.18)

We also give the following parametrizations. Parametrization (τ, T ; f ; −P , −s), with f = e − T s the free (Helmholtz) energy as main variable, with M = R + × R + . The associated differential form is then given by θ f = df + P dτ + s dT .

(1.19)

Parametrization (P , T ; g ; τ, −s) with g the free Gibbs energy as main variable g = e − T s + P τ = f + P τ = h − T s, also called free enthalpy, or thermodynamic potential, with M = R + × R + . The associated differential form is then given by θ g = dg − τ dP + s dT .

(1.20)

Parametrization (ρ, s˜ ; e˜ ; μ, ˜ T ) with energy “per unit volume” as main variable, e˜ = ρe with s˜ = ρs, also called pseudoentropy. Multiplying (1.15) by ρ, we obtain the differential form θ˜ e = ρθ e = d e˜ − T d s˜ − μdρ, ˜

(1.21)

with μ˜ = g = P τ + e − T s = ρ1 (P + e˜ − T s˜ ), called the chemical potential. We can also keep entropy s as the main variable and define the differential form θ˜ s = ρT ds + hdρ − d e, ˜ with h = e + P τ =

1 (e˜ + P ). ρ

¶

1.1.4 Relations Between the Basic Differential Forms Let he,s be the map he,s (τ, s; e; −P , T ) = (τ, e; s;

P 1 , ). T T

We verify that the pullback of θ s by he,s is such that h∗e,s θ s = T1 θ e . Let he,f (respectively he,g , he,h , he,e˜ ) be the map such that if j (τ, s; e; −P , T ), then he,f (j ) = (τ, T ; f ; −P , −s), he,h (j ) = (P , s; h; τ, T ),

=

(respectively he,g (j ) = (P , T ; g; τ, −s),

he,e˜ (j ) = (ρ, s˜ ; e˜; μ, ˜ T )),

(1.22)

12

1 Thermostructure

θ e is the inverse differential form of θ f (respectively θ g ) by he,f (respectively he,g ), θ e = h∗e,f θ f = h∗e,g θ g = h∗e,h θ h ,

ρθ e = h∗e,e˜ θ e˜ .

(1.23)

The map he,g is a Legendre involution (up to the sign of certain variables), since L((τ, s; e; −P , T ) = (−P , T ; −g; τ, s). If  is a manifold such that the restriction of θ s to  is null, then the restriction of e θ to  is also null. Since θ e −θ g = d(g−e−P τ +T s), the choice g = e+P τ −T s implies that θ e | = 0 is equivalent to θ g | = 0. The given examples in the case of local modelling can be transposed to global modelling of closed systems with one constituent. The case of systems with many constituents will be studied below. Remark 5 It is possible to make a model of a thermosystem with one constituent by the four variables τ, e, P , T , hence by a manifold V of even  dimension, and a symplectic structure thanks to the 2-form ω, so that ω = dpj ∧ dxj , with x1 = τ, x2 = e, p1 = P /T , p2 = 1/T . The search for the Lagrange manifold, i.e., the maximal integral manifolds of ω, leads to a foliation of the manifold V .11 Let θ 0 be the differential form defined on V = (R + )4 by θ0 =

1 1 Q P θ = dτ + de = p1 dx1 + p2 dx2 . T T T

Then θ 0 is induced by the canonical form of the phase space 4 = T ∗ ((R + )2 ). Without the state equation, this space has a symplectic structure thanks to the 2differential form ω = dθ 0 . We can define a contact structure “above the symplectic structure” that corresponds to the previous contact structure. Following Arnold, we “contactize” the space , and on this new space the canonical differential form θ s = ds − θ 0 is defined. ¶

1.1.5 Modelling of Work and Heat as Differential Forms The modelling of the basic notions of work and heat in the space F with one constituent will be by the following differential forms.12 Definition 1 Work and heat for a closed system with one constituent. In global thermodynamics, the modelling of work is the differential form θW = −P dV (for

11 The “phase” associated with a Lagrange manifold (following [Ler2]) can be identified with the entropy of the system. 12 A first difficulty in the presentations of “abstract thermostatics” (see [Germ][RIII]) is in the mathematical definition of the notion of heat. We give it implicitly in F according to the first law of “thermodynamics.” Of course we would have to specify these notions in more general cases.

1.1 Definitions. Thermoreversibility

13

a volume “variation” dV of the system), and the modelling of heat (received by the system) is such that θ Q = dE + P dV (E being the (internal) energy of the system). In local thermodynamics, we have two levels: The first level corresponds to specific internal energy (i.e., per unit mass). The notions of work (of pressure forces) and of heat are represented by the differential forms θ W = p1 dx1 = −P dτ,

θ Q = du − p1 dx1 = de + P dτ,

so that θ W + θ Q = de.

(1.24)

The second level corresponds to the weighted specific internal energy (or per unit volume). The notions of work and heat are represented by θ˜ W = ρθ W = −ρP dτ,

θ˜ Q = ρθ Q = ρ(de + P dτ ),

(1.25)

so that θ˜ Q + θ˜ W = ρde is not an exact differential. With e˜ = ρe, we have d e˜ = θ˜ W + θ˜ Q + θ mat ,

with θ mat = edρ = eτ ˜ dρ.

(1.26)

The differential form θ mat corresponds to a variation of matter. The relation (1.26) may be viewed as a generalization of the first law. An often used notation in physics for these differential forms is δW , δQ. From the algebraic point of view, these differential forms take what is received by the system into account. In the first level, the unit of the specific internal energy, thus of the differential forms of work and heat, is [de] = [e] = L2 T −2 , whereas at the second level, the unit is [d e˜] = [e] ˜ = ML−1 T −2 , so that by integration over space, we have the unit of the global energy [dE] = [E] = ML2 T −2 = L3 [e]. ˜

1.1.6 State Equations in a 5-Dimensional System Between the thermodynamic variables of a system there are relations called state equations, each with its domain of validity, especially the domains of temperature and pressure. Thus the set of possible states of a system is a submanifold  of the set F. Here the dimension of , that is, the degree of freedom, must be 2 when F is 5-dimensional. Therefore, we must have two independent state equations, 1 , 2 ; a (smooth) function j , j = 1, 2, on F is associated with a state equation j (x1 , x2 ; u; p1, p2 ) = 0,

(1.27)

14

1 Thermostructure

which defines a hypersurface j =  j as the set of (x, u, p) = (x1 , x2 ; u; p1, p2 ), satisfying (1.27). When j is of class C 2 , and d j (x, u, p) = 0, ∀(x, u, p) ∈ F, so that j (x, u, p) = 0, then the hypersurface j is a fourdimensional submanifold of F. More generally, let  or simply  be the manifold of the states of the system, denoting the state equations. Now we have two main remarks. (i) Recall that entropy does not appear in the state equations; this implies the preferential choice u = s as the main variable. (ii) The choice of the spaces M and N used at the beginning of the chapter is related to the choice of the differential form θ and to the state equations if they can be solved as p = (x). Then the coordinate system must be adapted to the state equations. Condition (ii) is satisfied for an ideal gas, also called a polytropic perfect gas. This will be developed below.

1.1.7 Reversible and Irreversible Evolutions Evolution and Foliation A main problem in thermodynamics is to characterize reversible evolutions, and the solution is based on a foliation of  (which defines a structure of a foliated manifold (see [Bour.var][9.3.2, 9.2]), which is a family of maximal integral manifolds of the kernel of the differential form θ . When F is of dimension 5, these maximal integral manifolds are of dimension at most n = 2 (whence the terminology of thermosystem with two degrees of freedom). Recall that an integral manifold V of θ is such that if Tj (V ) is the tangent space of V at the point j ∈ V , then < θ, ξ > = 0,

∀ξ ∈ Tj (V )

(we write θ |V = 0).

(1.28)

We admit the following evolution postulate. Hypothesis 5 The evolution of a state of a system is given by a path in  , i.e., there is an (open) interval J in R and γ ∈ C k (J,  ), (k ∈ N), such that the state of the system is given by = {γ (t), t ∈ [t0 , t1 ]}, ⊂  for every interval [t0 , t1 ] ⊂ J . We will assume that the path is of class C 1 (or at the least of class C 1 piecewise). Now we specify the basic notion of reversible evolution. Let be an oriented curve in , and γ (t) a parametric representation of , i.e., γ is (locally) an embedding of R in  (see [Mall][ch IV.1.1]). Recall that every other parametric representation γ˜ of is such that γ˜ = γ ◦ h with h a (smooth) increasing function in R.

1.1 Definitions. Thermoreversibility

15

Reversible Evolution Definition 2 θ -reversible evolution (or thermo-reversible). An evolution γ in  is said to be θ -reversible if γ is an integral curve of θ . With this hypothesis, (1.28) is equivalent to   T ds| − θ Q  = 0.

(1.29)



This is the first part of the second principle relative to reversible evolutions.13 Let X be a vector field on , and γ an integral curve of X (with J an interval of the lifetime of γ ): dγ (t) = Xγ (t ) ∈ Tγ (t ) ⊂ Tγ (t ), dt

∀t ∈ J = ]t0 , t1 [ .

The θ -reversibility of the evolution is equivalent to < θ, Xγ (t ) > = −

dτ ds de −P +T = 0, dt dt dt

∀t ∈ J ;

(1.30)

a vector field X generates an integral flow that is θ -reversible if < θ, X > = 0. Definition 3 Adiabatic evolution. An evolution γ in  is said to be adiabatic if γ is an integral curve of θ Q : θ Q | = de| + P dτ | = 0.

(1.31)

An evolution γ that is both adiabatic and reversible is isentropic: ds| = 0, or also < ds, Xγ (t ) > =

ds = 0, dt

t ∈ J.

(1.32)

The restriction of the differential form of the heat θ Q to an integral manifold V of θ is such that   θ Q  = T ds|V . (1.33) V

Therefore, V can be split into integral manifolds of θ Q (of dimension 1), which gives an adiabatic foliation of V . The principle of Carathéodory: “There exist adiabatically inaccessible states in every neighborhood of a given state” is a consequence of an adiabatic foliation of 13 The notion of reversibility in physics can be defined in many different

ways, which are not always in agreement; see [Lan-Lif1][ch. II.11, 20], for example linked to the absence of heat.

16

1 Thermostructure

the space  (or of V ). But a foliation of  by a family (Fα ) of integral manifolds of θ s satisfying the state equations is more important than that of θ Q , because it gives a geometric point of view of the notion of irreversibility, as expounded below.

θ -Irreversible Evolution When (1.29) is not satisfied, the evolution is θ -irreversible. But it is not a characterization: The second law of thermodynamics imposes that an irreversible evolution (γ (t)) (for all t ∈ J ) must be such that (denoting by Xγ (t ) its generator) < θ s , Xγ (t ) > = ds −

1 Q θ , Xγ (t )  > 0, T

∀t ∈ J.

(1.34)

Following the second law, a necessary condition of existence for an evolution γ in  is that its generator Xγ (t ) satisfy, ∀t ∈ J , < θ s , Xγ (t ) > = < ds −

P dτ 1 de 1 Q ds θ ,X >= − − ≥ 0. T dt T dt T dt

(1.35)

Thus we have < θ Q , Xγ (t ) > ≤ T

ds . dt

The intrinsic character of the second law is expressed by the fact that the contact structure is oriented. We see the main place of the entropy relative to these notions of irreversibility. Definition 4 θ -admissible evolution. An evolution in , (γ (t)), t ∈ J, is said to be thermo-admissible (θ -admissible) if its vector field Xγ (t ) satisfies (1.35) for all t ∈ J . Likewise, the oriented path in  and the vector Xγ (t ) , t ∈ J , are said to be θ -admissible. Every oriented θ -admissible path in  with origin A and end B in  must be such that   1 Q 1 Q s(B) − s(A) − θ ≥ 0 thus θ ≤ s(B) − s(A), (1.36) T T   or also θ Q ≤ T ds. Every cyclic evolution14 in F that is θ -admissible is reversible, which implies that irreversible cycles in F are impossible. The definition of θ -reversibility differs from that of “usual” reversibility, which is the possibility of going back from the final state to the initial state by the inverse path (which is the case if evolution is given by a group). 14 That

is, there exists t0 ∈ R such that γ (t + t0 ) = γ (t), ∀t ∈ R.

1.1 Definitions. Thermoreversibility

17

A θ -admissible evolution (γ (t))t ∈R in  can be cyclic in  only if it is reversible. Entropic Foliation and the Second Law There are state equations such that the heat differential form θ Q is, up to the factor 1 1 Q 0 T , an exact differential form, T θ = θ = dsrev , and thus θ s = ds − θ 0 = ds − dsrev = ds0 . The variation along the time of s0 expressing an irreversible evolution is only positive and transverse to the foliation. Every leaf is referred to15 by the variable s0 , and thus denoted by Fs0 . The free Helmholtz energy and the free Gibbs energy are therefore defined on each leaf Fs0 by f = (e − T srev ) − T s0 , g = (e + P τ − T srev ) − T s0 . Examples Consider the case of an ideal gas with two independent state equations 1 and 2 ; the space of variables is therefore reduced to a manifold  of dimension 3 (with entropy) on which the differential form θ s = ds − θ 0 = ds0 is exact. It is also the case of mixtures of ideal gases, and also fluids with the van der Waals equation. When the differential form θ s is given by (1.17), we can find necessary conditions on the state equations in order that θ s be exact, thus such that θ s = ds0 . We assume that the state equations are of the following form: e = f (T , P , τ ),

P = g(T , τ )

with differentiable functions f, g. Then e and P are functions of the two independent variables T , τ , or β = T1 , τ, so that we can write e = −ψ(β, τ ), PT = φ(β, τ ) with differentiable functions ψ, φ. In order that θ s be exact, it is necessary that the ∂φ Schwarz condition be satisfied, that is, that ∂ψ ∂τ = ∂β . We can obtain the condition on the previous functions f, g by writing that the exterior derivative dθ s is null, so that d( PT ) ∧ dτ + d( T1 ) ∧ de = 0. This gives the relation g−T

∂g ∂f ∂f ∂g + + = 0. ∂T ∂τ ∂P ∂τ

(Of course if e and PT are functions respectively of only T and τ , this relation is satisfied.) Thus there are fairly general state equations giving sense to such a foliation with entropy s0 .

15 The

map j = (τ, e; s; P , T ) ∈  → s0 ∈ R is a submersion.

18

1 Thermostructure

Projectors We can specify the evolution in the 3-dimensional space M × R, thanks to projectors, as with (1.11), (1.12). Let X1 and X2 be vectors in Te,τ,s ( ) defined as follows: X1 =

1 ∂ P ∂ ∂ ∂ + , X2 = + . ∂e T ∂s ∂τ T ∂s

We have < θ, X1 > = 0, < θ, X2 > = 0, with [X1 , X2 ] = 0. Then (X1 , X2 ) is a basis of the space Hj = He,τ,s = ker θj tangent to the leaf ∂ Fs0 in  , and (X1 , X2 , ∂s ) is a basis of the space Te,τ,s ( ). We can define the projectors in this space as Pv =

∂ ⊗ θ, ∂s

P h = X1 ⊗ de + X2 ⊗ dτ = I − P v ,

with I the identity in T ( ). Thus P h is the projector on Hj , and P v is the projector ∂ on Vj = R( ∂s )e,τ . An evolution (ϕx (t)) in  is reversible if P v(

dϕx ds 1 de P dτ ) = 0, ∀t ∈ R, i.e., − − = 0. dt dt T dt T dt

Relative to the differential forms with t I = I ∗ the identity in T ∗ ( ), we have t

Pv = θ ⊗

∂ , ∂s

t

Ph = I∗ − tPv = I∗ − θ ⊗

∂ = de ⊗ X1 + dτ ⊗ X2 . ∂s

Thus t P v and t P h are the projectors in T ∗ ( ) onto the subspaces H ∗ and V ∗ such that Hj∗ = {ω ∈ Tj∗ ( ), ω|Vj = 0},

Vj∗ = {ω ∈ Tj∗ ( ), ω|Hj = 0}.

Therefore, if du is the differential of a function u on  , we note that d irr u = t P v du, d r u = t P h du, are respectively the irreversible and reversible parts of du. Thus d irr e = d irr τ = 0 and d irr s = t P v ds = θ = ds0 , t P h θ = 0,

d r s = t P h ds = ds − θ =

θQ . T

1.2 Convexity Properties

19

1.2 Convexity Properties Convexity is essential in local and global thermodynamics for stability questions. Let us recall some notions in convex analysis and specify some conventional notation.

1.2.1 Mathematical Framework: Convex Analysis The convexity of a real function u on a vector space (of finite dimension) F is equivalent to the convexity of its epigraph Gu = {(x, z), x ∈ F, z ∈ R, z ≥ u(x)} ⊂ F × R. We recall the following main properties (see Chap. 4, and [Aub, Eke-Tem]). Given a function u on a vector space (of finite dimension) F with values in R ∪ {+∞}, we define the conjugate function u∗ of u, the map of the dual F ∗ of F into R ∪ {+∞}, by u∗ (p) = sup (p.x − u(x)), ∀ p ∈ F ∗ .

(1.37)

x∈F

The function u∗ is convex and lower semicontinuous (lsc). The map u → u∗ is a generalization of the Legendre transformation (also called the Fenchel transformation). When the function u is convex lsc, then we have u(x) = sup (p.x − u∗ (p)), ∀ x ∈ F, p∈F ∗

and thus u is also the conjugate function of u∗ . The domain of u∗ (denoted by Dom u∗ ) is the set of p ∈ F ∗ such that u∗ (p) is finite. The subdifferential of u at x0 (u convex or not; see [Eke-Tem][ch. I.5]) is the set, denoted by ∂u(x0 ), of elements p ∈ F ∗ such that u(x0 ) is finite, and < x − x0 , p > +u(x0) ≤ u(x), ∀ x ∈ F,

(1.38)

which formally reads < δx, p > ≤ δu(x0 ). We say that u is subdifferentiable at x0 if ∂u(x0 ) is not empty (there is a straight line below the graph of u that contains the point (x0 , u(x0 ))). When u is convex and differentiable at x0 , then u is subdifferentiable at x0 , and ∂u(x0) is identified with the derivative Du(x0 ). Note that a function that is differentiable at x0 , but not convex, is not necessarily subdifferentiable at x0 . We have the following main property [Aub][ch. I.4.2, Prop. 4.2; p. 56]: u(x) + u∗ (p) = p.x, ⇐⇒ p ∈ ∂u(x).

(1.39)

20

1 Thermostructure

Let us point out the difference between u∗ (p) and the “usual”  Legendre  transform of a not necessarily convex differentiable function u(x): let x0i , i ∈ I be the set of solutions of the equation u (x0 ) = p with given p; zi = p.x0i − u(x0i ) is a multivalued function of p, zi = u∗i (p) (that is, the “usual” Legendre transform), whereas u∗ (p) = supI zi = supI u∗i (p). With the hypothesis that the map x ∈ F → p = ∇u(x) = u (x) ∈ F is a homeomorphism (i.e., bijective and bicontinuous), we define the “usual” Legendre transform of u by U (p) = p.(∇u)−1 (p) − u((∇u)−1 (p)),

(1.40)

and we have ∇U = (∇u)−1 . Minimization properties relative to a convex function. Let u be a convex function on F = R 2 (or F = R + × R) of class C 2 , so that Du(x) = 0 for all x ∈ F . The definition (1.37) can be seen as a minimization problem on the function up (x) = u(x) − x.p with given p = Du(x0 ): inf (u(x) − x.p) = − sup (x.p − u(x)) = −u∗ (p);

x∈F

x∈F

hence up (x0 ) = u(x0 ) − x0 .p = −u∗ (p) if x0 is a minimum (see, for example, [Dieud3][ch. 16.5.12] and [Ciar][ch. II.7.4, p. 156]), since the function up is such that Dup (x0 ) = 0 and its Hessian H (up ) = H (u) is positive definite. Now let f be a concave function; thus f˜ = −f is convex. We define the conjugate function f∗ of f by f∗ (p) = inf(p.x − f (x)) = − sup(−p.x + f (x)) = −f˜∗ (−p). x

(1.41)

x

Then f∗ is a concave function, with f (x) + f∗ (p) = p.x,

−p ∈ ∂(−f )(x).

If f is smooth, we can differentiate, so that df − p.dx = −(df∗ (p) − x.dp), and ∂f ∂f∗ j pj = ∂x j is equivalent to x = ∂pj . Properties of the Legendre (or Fenchel) transformation L associated with the Legendre involution L and the relationship with the basic differential form θ . Let L be the Legendre transformation that transforms a real function u on F (respectively its graph G(u)) into its conjugate function u∗ (respectively its graph G(u∗ )). If u is a real convex continuous function on F with only one subgradient p at every point x ∈ F (i.e., ∂u(x) = p, and thus p = u (x))),16 then let us call, by extension, the set J (u) = {(x, u(x), p), p = u (x), x ∈ F, u(x) ∈ R, } 16 With the

derivative in the sense of the Gateaux differential or if u ∈ C 1 (F, R) in the usual sense.

1.2 Convexity Properties

21

the jet of u. Then by the Legendre (or Fenchel) transformation, we transform the jet of u onto the jet of u∗ (if the conjugate function is smooth, i.e., C 1 (F, R), at least Gateaux differentiable), and we also denote this transformation by L. Then the following diagram is commutative: L

(x, u(x), p(x)) ∈ J (u) −→ (p, u∗ (p), x(p)) ∈ J (u∗ ) ↓π ↓π (x, u(x)) ∈ G(u)

L

−→

(p, u∗ (p)) ∈ G(u∗ )

In this diagram we identify F with F ∗ . The projection π from F = F × R × F onto F × R, so that π(x, v, p) = (x, v), allows us to identify J (u) with G(u) and J (u∗ ) with G(u∗ ), and we assume that u and u∗ are of class C 1 . We recall that the Legendre involution L is defined on F by L(x, u, p) = (p, U, x), U = p.x − u. Then the restriction of L to J (u) may be identified with the Legendre transformation L, but with L(J (u)) = L(J (u)) = J (u∗ ). If the function u is convex, then we also have L(J (u∗ )) = L(J (u∗ )) = J (u∗∗ ) = J (u). In order that J (u) be a submanifold of F of class C 1 , we must have a priori u ∈ C 2 (F, R). Then it is the same for J (u∗ ). If the function u ∈ C 1 (F, R) is strictly convex, then the map u ∈ F → p = u (x) ∈ F is a homeomorphism, and thus on these functions, the Legendre transformation L is identified with the “usual” Legendre transformation according to (1.40):(x, u, p) → (p, U (p), x) with x = ∇U (p) = (∇u)−1 (p), and we also have that u∗ = U ∈ C 1 (F, R) and u∗ is strictly convex.17 The pullback of the differential form θ = du − p.dx on F by the Legendre involution L is θ ∗ = L∗ (θ ) = dU − x.dp, and we have θ + θ ∗ = du − p.dx + dU − x.dp = d(u + U − p.x) = 0, and thus L∗ (θ ) = θ ∗ = −θ on F. We can also take the differential of (1.39) if u and u∗ are convex of class C 2 ; with θ ∗ = du∗ − xdp, we obtain du − pdx + du∗ − xdp = θ + θ ∗ = 0. Thus if J (u) is an integral manifold for the contact structure due to θ , then J (u∗ ) is an integral manifold for that structure due to θ ∗ , and conversely. Now if there are state equations such that p = (x) is a function of x (or conversely), with a diffeomorphism satisfying the Schwarz conditions, then the differential form p.dx is exact, and we note that durev = p.dx, whence 17 If u is a Gateaux-differentiable strictly convex function, we have [Eke-Tem][ch. I, Prop. 5.4] u(y) > u(x)+ < u (x), y − x >, ∀x = y, and < y − x, u (y) − u (x) > > 0, ∀x = y. Thus u (y) = u (x) implies y = x.

22

1 Thermostructure

θ = du − durev = du0 on F × R, and θ ∗ = −du0 on F × R. If J (uλ ), λ = u0 ∈ R, −1 is a foliation for θ of  , then J (u∗λ ) is a foliation for θ , and the Legendre involution L is an isomorphism for the corresponding jets’ foliations. To change a thermodynamic function into another we have often to use partial Legendre involutions and transformations. Let L1 and L2 be the partial Legendre involutions L1 (x; u; p) = (p1 , x2 ; p1 x1 −u; x1 , −p2 ),

L2 (x; u; p) = (x1 , p2 ; p2 x2 −u; −p1, x2 ).

Let I be the mapping I(x; u; p) = (x; −u; −p). Thus IJ (φ) = J (−φ), ∀φ. We have L = L1 ◦ I ◦ L2 . The jets of the functions e, f, g satisfy the relations J (f ) = I ◦ L2 (J (e)),

J (g) = I ◦ L(J (e)) = I ◦ L1 (J (f )).

¶

1.2.2 Convexity in Thermodynamics In this section we are concerned only with the simple case of the five variables (τ, e, s, P , T ). Thereafter we will consider more general situations. In the framework of local thermodynamics, we postulate that the map (due to the state equations) satisfies Hypothesis 1 and Hypothesis 2, thus giving a (unique) foliation of M × R with leaves (Ls0 ), with previously used notation in thermodynamics. Moreover, we assume that satisfies Hypothesis 4, so that the integral ϕ = −s of is a convex function of (τ, e). Then the entropy s(τ, e) is a concave function of (τ, e); the energy e(τ, s) is a convex function of (τ, s). First we give properties associated with the subdifferential of convex functions, especially with sˆ = −s: (−

1 P , − ) ∈ ∂ sˆ (τ, e), T T

which formally reads − PT δτ −

1 T δe

(−P , T ) ∈ ∂e(τ, s),

≤ δ sˆ (τ, e) and thus

1 P δτ + δe ≥ δs(τ, e), T T

−P δτ + T δs ≤ δe(τ, s),

which means the control of entropy in reversible evolutions. If the function s → e(τ, s) is invertible for all τ , which allows us to define a function s(τ, e), and conversely, the two hypotheses (convexity of e, concavity of s) are equivalent.

1.2 Convexity Properties

23

Indeed, if e(τ, s) is convex, then its graph lies globally on one side of every tangent plane, and thus that is also true for the graph of the function s(τ, e), which implies that this function is either concave or convex. Now ∂e ∂s = T > 0. Let us denote by eτ and sτ the functions (with fixed τ ) eτ (s) = e(τ, s) and sτ (e) = s(τ, e). ∂s ∂ 2 e ∂s 2 Since eτ ◦sτ = I , we have by taking partial derivatives that ∂e ∂s ∂e = 1 and ∂s 2 ( ∂e ) + ∂e ∂ 2 s ∂s ∂e2

2

= 0, which implies ∂∂es2 < 0, and hence s(τ, e) is concave. The converse is straightforward. ¶ The convexity of sˆ = −s (if it is of class C 2 ) is a consequence of the fact that its Hessian matrix H (ˆs ) = −H (s) = −J ( ) is positive definite, with H (s) = ∂ P (T ) = Since ∂e equivalent to

∂ 1 ∂τ ( T ),

∂ P ( ) ≤ 0, ∂τ T

∂2s ∂2s ∂τ 2 ∂τ ∂e ∂2s ∂2s ∂e∂τ ∂e2



 =

∂ P ∂τ ( T ) ∂ 1 ∂τ ( T )

∂ P ∂e ( T ) ∂ 1 ∂e ( T )

.

the positive definite character of the matrix H (ˆs ) is

∂ 1 ( ) ≤ 0, ∂e T

(

∂ 1 2 ∂ P ∂ 1 ( )) − ( ) ( ) ≤ 0. ∂τ T ∂τ T ∂e T

Now (H (s)) = 0 is equivalent to the map (τ, e) → (P , T ) being an embedding. This is a consequence of the relation (H (s)) = −

1 ∂P ∂T ∂P ∂T − ]. [ T 3 ∂τ ∂e ∂e ∂τ

Convexity properties of other thermodynamic functions. The Gibbs and Helmholtz functions have the following properties of convexity: (i) The Gibbs free energy g(P , T ) is a concave function of (P , T ). (ii) The Helmholtz free energy f (τ, T ) is a convex function of τ and a concave function of T . Proof of (i) from the convexity of the energy. We use the parametrization based on energy; hence x = (τ, s). We extend the definition of e to the entire space R 2 by e(x) = +∞ if x ∈ R + × R. The domain of this function is Dom e = R + × R. Then we define the conjugate function e∗ of e by e∗ (p) = sup (p.x − e(x)); thus e∗ (−P , T ) = x∈R 2

sup (−P τ + T s − e(τ, s)). (τ,s)∈R 2

(1.42) The Gibbs free energy is then defined by def

g(P , T ) = −e∗ (p) = −e∗ (−P , T ).

(1.43)

24

1 Thermostructure

Hence a priori, the opposite of the function e∗ , conjugate to e, is an extension of the “usual” Gibbs free energy (up to the change of p = (−P , T ) into (P , T )); therefore, g(P , T ) is a concave function of (P , T ). Indeed, the relation (1.39) with p = (−P , T ) ∈ ∂e(x) and x = (τ, s) ∈ R + × R implies e(x) + e∗(p) = p.x. Thus e(τ, s) − g(P , T ) = −P τ + T s.

(1.44)

Conversely, e is also the conjugate function of e∗ (see [Aub][ch. I.3.2, Thm. 3.1]): e(x) + e∗(p) = p.x ⇐⇒ p ∈ ∂e(x),

e∗∗ (x) + e∗(p) = p.x ⇐⇒ x ∈ ∂e∗ (p).

When e is a differentiable function on R + × R, we have that p = (−P , T ) ∈ ∂e ∂e(τ, s) implies −P = ∂τ , T = ∂e ∂s . In the case of a gas, with P > 0, e(τ, s) is a decreasing function of τ and an increasing function of s. Then Dom e∗ = R − ×R + . Therefore, we have e(τ, s) =

sup

(g(P , T ) − P τ + T s).

(1.45)

2 (P ,T )∈R+

The following minimization problem with (−P , T ) ∈ ∂e(τ0 , s0 ) corresponds to the definition (1.42): inf(e(τ, s) + P τ − T s) = e(τ0 , s0 ) + τ0 P − s0 T = −e∗ (−P , T ) = g(P , T ). τ,s

Proof of (ii). We can define the free energy f (τ, T ) from e or from g thanks to the relations f (τ, T ) = inf (e(τ, s) − T s) = sup (−P τ + g(P , T ). s

(1.46)

P

Setting eτ (s) = e(τ, s), gT (P ) = g(P , T ), we can identify −f (τ, T ) with the conjugate functions f (τ, T ) = −eτ∗ (T ) = (−gT )∗ (τ ); with T ∈ ∂eτ (s), we have f (τ, T ) = e(τ, s) − T s = g(P , T ) − P τ. Thus the free energy is a convex function of τ and a concave function of T . ¶ From the entropy s(τ, e), a concave function of (τ, e), let us define, similarly to the energy, its conjugate function s∗ according to (1.41), with x = (τ, e), and p = ( PT , T1 ) by s∗ (p) =

inf

x∈R + ×R

(p.x − s(x)) = −(−s)∗ (−p).

(1.47)

Then s∗ is a concave function, so that s(x) + s∗ (p) = p.x, with −p ∈ ∂(−s)(x), which gives

1.2 Convexity Properties

25

s∗ (

1 P 1 , ) = g(P , T ). T T T

(1.48)

Indeed, with ( PT , T1 ) = Ds(τ0 , e0 ), the definition (1.47) corresponds to the following maximum problem: sup (s(τ, e)−τ τ,e

1 1 P 1 P P 1 −e ) = s(τ0 , e0 )−τ0 −e0 = −s∗ ( , ) = − g(P , T ). T T T T T T T

An important consequence of the convexity of the energy is the following. Speed of sound. We can define a “module of velocity” (the speed of sound) by the positive quantity  c=

since



∂2e ∂τ 2 s

∂P ∂ρ

1/2 s

  2 1/2 ∂P 1/2 ∂ e =τ − =τ , ∂τ s ∂τ 2 s

≥ 0, by the convexity of e.

(1.49)

¶

For what concerns the energy per unit volume e(˜ ˜ s , ρ) (see the parametrization based on energy per unit volume), we have the following main property, linked to the concavity of entropy. The energy per unit volume e(˜ ˜ s , ρ) is a convex function of (˜s , ρ) if and only if the energy per unit mass e(s, τ ) is a convex function of (s, τ ). This follows directly from the following lemma (see [God-Rav][ch. II.1, p. 102]). Lemma 1 Let C0 be a convex cone in R p and π : R + × C0 → R a function (of C 2 class). The function f : R + × C0 → R defined by f (μ, v) = μg( μ1 , μv ) is convex if and only if g is convex. It is then interesting to specify the conjugate function e˜∗ (T , μ) ˜ of e(˜ ˜ s , ρ). When (T , μ) ˜ ∈ ∂ e(˜ ˜ s , ρ), we have, following (1.39), e˜∗ (T , μ) ˜ = −e(˜ ˜ s , ρ) + T s˜ + μρ. ˜ If we recall (see (1.21)) that the chemical potential is such that ρ μ˜ + T s˜ − e˜ = P , we obtain that the pressure P is just the desired conjugate function ˜ P = P (T , μ) ˜ = e˜∗ (T , μ).

(1.50)

Thus P (T , μ) ˜ is a convex function of (T , μ). ˜ Another consequence of (1.48) is that the Gibbs free energy is a concave function of (P , T ), since s∗ is concave. In the framework of global thermodynamics, with Hypothesis 4 we have that the entropy S(V , E) is a concave function of (V , E); the energy E(V , S) is a convex function of (V , S). We will see later applications of these properties.

26

1 Thermostructure

1.2.3 Stability The meaning of the word stability will be specified below. Perturbation of a fixed state. Here we consider the five variables τ, e, s, P , T . Let us first recall the Taylor–Young formula (see [p. 144]Ciarlet) in this frame. For every increase (δτ, δs) of (τ, s), respectively (δτ, δe) of (τ, e), the increase of e(τ, s), respectively of s(τ, e), δe = e(τ + δτ, s + δs) − e(τ, s), δs = s(τ + δτ, e + δe) − s(τ, e) (if these functions are of class C 2 ) is given by 1 δe = De(τ, s)(δτ, δs) + H (e)(τ, s)((δτ, δs), (δτ, δs)) + |(δτ, δs)|2 )(|(δτ, δs)|) 2 1 δs = Ds(τ, e)(δτ, δe) + H (s)(τ, e)((δτ, δe), (δτ, δe)) + |(δτ, δe)|2 )s(|(δτ, δe)|) 2 with lim (|(δτ, δs)|) = 0 when |(δτ, δs)| → 0, and similarly by exchanging s for e. We can see the perturbation property in the following way, with the entropy. Using the “conjugate” function s∗ of s, see definition (1.47), with given (P , T ), s∗ (

1 P 1 P , ) = inf( τ + e − s(τ, e)), τ,e T T T T

then if (τm , em ) realize the infimum, we have for every (τ  , e ) that (

P  1 P 1 P 1 τ + e − s(τ  , e ) ≥ ( τm + em − s(τm , em ) = s∗ ( , ), T T T T T T

or with δτ = τ  − τm , δe = e − em , δs = s(τ  , e ) − s(τm , em ) = s  − sm , we obtain (

1 P δτ + δe) − δs ≥ 0, T T

that is, with the derivative Ds of s, Ds(τ, e)(δτ, δe) = ( PT δτ + T1 δe), and its Hessian H (s), using the Taylor–Young formula, we have 1 Ds(τ, e)(δτ, δe) − δs ≈ − H (s)(τ, e)((δτ, δe), (δτ, δe)) ≥ 0. 2 With given (P , T ), the change of (τ, e) is controlled by the conjugate s∗ of s. ¶ Stability and evolution. Here we consider the general case with the space F. Let (x(t)) (with 0 ≤ t ≤ a) be an evolution in M starting from x 0 , which implies a reversible evolution of the system (x(t), u(x(t)), p(t)), with p(t)) = ∂u ∂x (x(t)) and u(x) a strictly convex function of class C 2 , with Hessian H (u) a positive definite

1.2 Convexity Properties

27

matrix. In a neighborhood of (x 0 , u(x 0 )), we have H (u)(x(t)) ≥ αI, with I the unit matrix and α > 0. The evolution equation of p is then dp dx = H (u).( ), dt dt

whence

dx dp dx dx dx dx . = .H (u(t))( ) ≥ α . . dt dt dt dt dt dt

dx By the Cauchy–Schwarz inequality, we deduce | dp dt | ≥ α| dt |, which proves that the evolution of x is “controlled” by that of p. Conversely, the conjugate function u∗ (p) is then strictly convex, H (u∗ ) = (H (u))−1 is also a positive definite matrix, and thus locally the evolution of p(t) is controlled by that of x(t). Example with x = (τ, e) or x = (τ, s). When the energy is strictly convex and of class C 1 (that is, the case with ideal gases), then for a reversible evolution of the system, the evolution of (τ, s) is controlled by that of (P , T ) (and conversely), and similarly, the evolution of (τ, e) is controlled by that of ( PT , T1 ) (and conversely), since the map x → p is a homeomorphism. Irreversibility of physical evolution at second order, according to the second law. Let F0 = M × R, M ⊂ R n , and let be the map x → p = (x), due to the state equations, which is assumed to be integrable for the differential form ∗ (θ ) =  i ds − i dx , denoted simply by θ . Now let x˜ 0 = (x 0 , s 0 ), x˜ 1 = (x 1 , s 1 ), be two states in F0 . Let γ (t) with t ∈ [t0 , t1 ] be a physical path in F0 from x 0 , s 0 to x 1 , s 1 . We assume that γ is of class C 2 . We denote by δs0 the jump of irreversible entropy dx ds between the two states. Then with γ (t) = (xt , st ), γ˙ = dγ dt = ( dt , dt ),





δs0 =

t1

θ= γ

< θγ (t ), γ˙ (t) > dt.

t0

Now let θ˜ be the map from T (F0 ) into R defined from the differential form θ by < θ˜ , (x, ˜ v) ˜ > = < θx˜ , v˜ >,

∀v˜ ∈ Tx˜ (F0 ),

˜ 18 so that we can define the differential form d θ. Then let γˆ = (γˆ (t)) = (x˜t , v˜t ), with t ∈ [t0 , t1 ], be a path in T (F0 ), with d γˆ 19 ˙ ˙ dt = (x˜ t , v˜ t ), above the path (γ (t)); with obvious notation, we have  γ˙

d θ˜ = < θx˜ 1 , v˜ 1 > − < θx˜ 0 , v˜ 0 > .

We assume that < θx˜ 0 , v˜ 0 > = 0. Then we have  δs0 =

18 Do 19 Of

γ˙

d θ˜ =



t1

< d θ˜ , (x˙˜t , v˙˜t ) > dt.

t0

not confuse this with the exterior differential! course, if γˆ is the derivative of the path γ (t), we must have v(t) ˜ = x˙˜t .

28

1 Thermostructure

From the second law, such an evolution is physically possible if δs0 ≥ 0, so that we must have, for every physically allowed path (γˆt ), ˜ ˙ ˙ < dx(t ˜ ) θ , (x˜ t , v˜t ) > ≥ 0,

∀t ∈ [t0 , t1 ].

The calculation of this quantity gives20 ˙ ˙ ˙ < dx(t ˜ ) θ˜ , (x˜ t , v˜t ) > = < ds, v˜t > −



< d i , x˙t >< dx i , vt > .

(1.51)

Thus d θ˜ = ds −



d i ⊗ dx i = ds −

 ∂ i ∂x j

dx j ⊗ dx i .

When x˙t = vt , then (1.51) is ˙ ˜ ˙ ˙ < dx(t ˜ ) θ , (x˜ t , v˜t ) > = < ds, v˜t > −

 ∂ i dx j dx i . ∂x j dt dt

We can also write, with v˜t = (vt , vts ) and the components (vti ) of vt , < v˙ts , ds > −



vti < x˙t , d i >≥ 0.

With the Hessian of s, we finally obtain v˙ts − < vt , H (s)x˙t > ≥ 0.

(1.52)

Recall that the opposite of the Hessian of s is (locally) positive definite when (−s) is strictly convex, so that < vt , H (−s)x˙t > = 12 Lx˙t < vt , H (−s)vt > is the Lie derivative of the quadratic form 12 < vt , H (−s)vt >. Let us emphasize that this relation connects the concavity of the entropy to the concept of irreversible movement. If the evolution is reversible, we must have (with x˙t = v(t)) < ds, v˙˜t > − < vt , H (s)vt > = 0,

∀t ∈ [t0 , t1 ],

and thus < ds, v˙˜t > = < vt , H (s)vt > ≤ 0,

∀t.

that if we consider s0 a function of x˜ given by s0 (x) ˜ = s + srev (x), then we have <  0 ), (x˙˜t , v˙˜t ) > = D 2 s0 (x˙˜t , v˜t ) + Ds0 (v˙˜t ). d(ds

20 Note

1.3 Thermosystems with One Degree of Freedom

29

1.3 Thermosystems with One Degree of Freedom Thermosystems are generally derived from simplified models in the case of solids and incompressible fluids. Boundary problems of fluid mechanics with transmission conditions are often associated with different degrees of freedom from each part of the boundary, for example a compressible fluid in contact with a rigid wall. Here we consider a system modeled by the space F = Q1 (see Remark 4) identifiable with R 3 , and the standard contact form θ : θ = − du + p1 dx1 .

(1.53)

We insist on the fact that there is no integral manifold of dimension 2 in R 3 (the system is sometimes said to be not integrable). Here the maximal integral manifolds are of dimension 1. The choice of such a family will be made thanks to a state equation. First example: “barotropic” fluids. Here the variables are τ, e, P , and the chosen parametrization is x1 = τ, u = e, p1 = −P .

(1.54)

The space F is identifiable with F˜ = R + × R × R. Here the basic form is expressed by θ = −de − P dτ = −θ Q . We assume that these parameters satisfy the following state equation p1 x1 + (γa − 1)u = 0,

thus P τ − (γa − 1)e = 0

(1.55)

in R 3 (see below (1.65)), which defines the surface (γa ). The restriction of the form θ to (γa ) is θ |(γa ) = −de−(γa −1)e

dτ = −e[d log e+(γa −1)d log τ ] = −ed log(eτ (γa −1) ). τ

Thus 1e is an integrating factor for θ |(γa ) , and we can write,21 with a constant cv , s (γa −1) , and therefore eτ (γa −1) = exp s . Hence θ | (γa ) = 0 implies cv = log(eτ cv s that cv = log e0 , e0 > 0, is constant. The integral curves of θ = 0 in (γa ), −γa +1

uu0 (x1 ) = u0 x1

,

thus e(τ ) = e0 τ −γa +1 ,

(1.56)

are adiabatic. This family of integral curves makes a foliation of (γa ), with s as transverse variable.

21 Similarly

to the relation (1.86) below.

30

1 Thermostructure

∂e Then the pressure p is given by P = − ∂τ = e0 (γa − 1)τ −γa ; hence p is a function of τ , P = gs (τ ), which corresponds to the notion of barotropic fluid. But here the usual notions of thermodynamics (notably the second law) are not applicable, since here the entropy is not a priori a variable. Second example: incompressible fluids. The density of incompressible fluids is constant, and we have only to consider the variables (s, e, T ). With the parametrization based on entropy, let

x2 = e, u = s, p2 = 1/T ,

(1.57)

with the associated space F = R + × R × R + . The basic 1-form is θ s = du − p2 dx2 = ds − (1/T )de.

(1.58)

We assume that these variables satisfy the state equation e − cv T = 0. The integral curves of the equation θ = 0 with this state equation are given by the family of (concave) functions s = s0 + cv log e = s0 + cv log cv + cv log T ,

(1.59)

which defines a foliation (Lcs0v ) of the manifold (cv ); the variation of the parameter (of entropy) s0 expresses an irreversible evolution. More generally, θ s is an exact differential if and only if e is a function of T (or T is a function of e). Third example. Here the basic 1-form is θ s = du − p1 dx1 = ds −

P dτ, T

(1.60)

and the state equation is that of perfect gases: PT = r τ1 . We obtain a new foliation, which together with the foliation of the second example and according to Hypothesis 3 provides the foliation relative to the ideal gas.

1.4 Ideal Gas 1.4.1 State Equations We will study θ -reversible evolutions and find integral manifolds of θ in , thanks to the theory of characteristics, which can be done by explicit calculation in the present case. This allows a good understanding of the theory. Here we use a parametrization based on energy (which is not the best choice, as we shall see later).

1.4 Ideal Gas

31

In a local model, we call the following relation between P , T , and τ = 1/ρ the ideal gas law: P τ = rT ,

(1.61)

with r = R0 /m0 , where m0 is the molar mass of the gas, and R0 is the “ideal gas constant”; R0 = N0 kB , N0 is the Avogadro number N0 = 6.023.1023, and kB is the Boltzmann constant, kB = 1.38.10−23 joules/kelvin; hence R0 = 8.314 joules/kelvin). Thus r is a positive constant that depends on the gas. In a global model, the ideal gas law is P V = NR0 T ,

(1.62)

with N the number of moles of the (pure) gas of the system, and V is the volume of the domain filled by the gas. Thus R = NR0 is a positive constant if the total mass of the gas is constant. When we have a mixture of gases, let mi be the molar mass, Mi the mass, Ni the number of moles  (hence Ni= Mi /mi ) of each gas. Then we have the law (1.62) with N = Ni . If M = Mi is the total mass of the gas, we have the relation Mr = NR0 (with the requirement that the fluid be “homogeneous”). Such a law has a limited domain of validity, notably thanks to a possible change of phase. In the local situation, with parametrization based on energy (τ, s, e), we can write the state equation (1.61) in the form pg (x, u, p) = (x1 , x2 ; u ; p1 , p2 ) = p1 x1 + r p2 = 0,

(1.63)

which defines a manifold  = (r). Note that neither u nor x2 appears in this equation. Determination of characteristics. From the state equation and the differential form θ = p.dx − du, we can define characteristic vectors ξ that are tangent vectors to  and for which there exist constants aξ , bξ such that i(ξ )dθ = aξ d + bξ θ, i(ξ )θ = 0,

i(ξ )d = 0.

Thus i(ξ )dθ is a differential form, a linear combination of the forms d and θ . We can choose aξ = −1, since if ξ is a characteristic vector, then so is λξ, ∀λ ∈ R. We ∂ > = 0, and therefore bξ = − ∂ have < i(ξ )dθ, ∂u ∂u , whence i(ξ )dθ = −d − The partial derivatives of pg = are

∂ θ. ∂u

32

1 Thermostructure

∂ = x1 = τ, ∂p1

∂ = r, ∂p2

∂ = 0, ∂u

∂ = p1 = −P , ∂x1

∂ = 0, ∂x2

which gives the characteristic vector ξ = ( p , p. p , −( x + p u )) = (x1 , r ; 0 ; − p1, 0) = (τ, r; 0; P , 0). The differential equation of a characteristic curve is dτ = τ, dt

ds =r; dt

de =0; dt

d(−P ) = P, dt

dT = 0. dt

Integration of this system gives, with various constants with subscript 0 and with the notation exp to indicate the exponential function, τ (t) = τ0 exp t, s(t) = rt + s0 , e(t) = e0 , P (t) = P0 exp(−t), T (t) = T0 . We see that the trajectory is an isotherm and that the entropy s is proportional to the evolution parameter t. We verify that P (t)τ (t) = P0 τ0 = rT = constant. The search for integral manifolds of θ in (r) can be accomplished through the usual method (see [Arn2]), here using the “initial” manifold u0 = e0 = cv T0 with cv a positive constant called the specific heat.22 This is equivalent to joining to the state equation (1.63) the so-called polytropic gas equation (which may be considered a first integral) pol (x, u, p) = u − cv p2 = 0, whence e − cv T = 0,

or also

E − NCv T = 0,

(1.64)

in the framework of global thermodynamics, with Cv = m0 cv , and m0 the molar mass. The Gibbs free energy g is such that g = e + P τ − T s = g0 − rT0 t, and the rate of heat deposit is constant: < θ Q,

de dτ d >= +P = P τ = rT = rT0 . dt dt dt

Remark 6 A variation. Instead of (1.61), we consider the state equation P τ − (γa − 1)e = 0

(1.65)

with constants cp = cv + r,

γa = cp /cv = 1 + (r/cv ) > 1,

thus cv =

1 R0 . m0 γ a − 1

(1.66)

22 The notation of the subscript v is used according to the usual notation: in the present case of ideal gases, cv is a constant independent of temperature and pressure.

1.4 Ideal Gas

33

The constant γa , called the adiabatic index, depends on the type of gas: for a monoatomic gas, γa = 5/3, while for a diatomic gas, one has γa = 7/5. With the parametrization based on energy, we also write (1.65) as (x1 , x2 ; u ; p1 , p2 ) = p1 x1 + (γa − 1)u = −P τ + (γa − 1)e = 0. This defines a manifold  = (γa ) in F. Let us determine the characteristics associated with . The partial derivatives of are ∂ = x1 , ∂p1

∂ = 0, ∂p2

∂ = (γa − 1), ∂u

∂ = p1 , ∂x1

∂ = 0, ∂x2

which gives the characteristic vector, up to a multiplicative coefficient, ξ = (ξx , ξu , ξp ) = ( p , p. p , −( x +p u ) = (x1 , 0 ; p1 x1 ; −γa p1 , −(γa −1)p2 ). The equation of characteristic curves is given by dx1 = x1 , dt

dx2 =0; dt

du = p1 x1 ; dt

dp1 = −γa p1 , dt

dp2 = −(γa −1)p2 . dt

Integration of this system gives, with various constants with subscript 0, x1 (t) = x10 exp t, x2 (t) = x20, u(t) = u0 exp(−(γa − 1)t), p1 (t) = p10 exp(−γa t), p2 (t) = p20 exp(−(γa − 1)t).

(1.67)

Indeed, we have du/dt = p10 x10 exp(−(γa − 1)t), and hence u(t) = u0 exp(−(γa − 1)t),

u0 = −

p10 x10 . γa − 1

(1.68)

With previous thermodynamic variables, we have τ (t) = τ0 exp t, s(t) = s0 , e(t) = e0 exp(−(γa − 1)t), P (t) = P0 exp(−γa t), T (t) = T0 exp(−(γa − 1)t). These characteristic curves are isentropic curves (or reversible adiabatic curves). We have only to choose an initial manifold. This may be done in many ways. With previous initial conditions u0 = cv p20 , i.e., e0 = cv T0 ,

(1.69)

cv being a positive constant, (1.67) implies u(t) = cv p2 (t), i.e., e(t) = cv T (t),

∀t ∈ R.

34

1 Thermostructure

Notably, P (t)τ γa (t) = P (0)τ γa (0) is constant. We note that the rate of work deposit is < θW ,

dτ d >= P = P τ = rT = rT0 exp(−(γa − 1)t). dt dt

We see that the present choice of the polytropic equation instead of the ideal gas equation exchanges adiabatic curves with isotherms in the evolution along characteristics. ¶

1.4.2 Specific Entropic Foliation for an Ideal Gas Entropy for an Ideal Gas in Local Thermodynamics For systems of ideal gases in the framework of local thermodynamics, we admit that the variables (τ, e, s, −P , T ) satisfy two independent state equations defining a submanifold of F: (i) the state equation (1.61), which defines a submanifold (r) of codimension 1 in F; (ii) the state equation of a polytropic gas (1.64), which defines a submanifold (cv ): e = cv T , with cv = r/(γa − 1) > 0, also of codimension 1 in F. These state equations define a map : (τ, e) ∈ R + × R + → (

r cv P 1 , ) = ( , ) ∈ R+ × R+ , T T τ e

which is a diffeomorphism and is integrable. Moreover, it satisfies < (τ, e), (τ, e) > = r + cv . Remark 7 On the necessity of a state equation linking e and T . With the state equation of (1.61) alone, the differential form θ s becomes θ |r = ds −

1 1 P dτ − de = d(s − r log τ ) − de. T T T

This is a differential form in R 3 with s  = s −r log τ, T , e, thus with three variables, and we cannot have integral manifolds of dimension 2. But along the integral manifolds, ds  − T1 de must be an exact form, implying dT ∧ de = 0; hence the energy must be a function of T : e = f (T ). Then we have a new state equation f , which allows also (up to a smoothness condition) a foliation of (r) ∩ (f ) with parameter s0 , whose time variation indicates an irreversible evolution. When f is a linear function, we have the previous situation with e = f (T ) = cv T . ¶

1.4 Ideal Gas

35

The intersection of the two manifolds (r), (cv ) is a submanifold  (also denoted by (r, cv )) of codimension 2 in F, thus of dimension 3:  = (r) ∩ (cv ) = (r, cv ). Then the restriction of the differential form θ 0 = θ 0 | =

1 Q Tθ

(1.70)

to  is

1 r cv P dτ + de = dτ + de T T τ e

(1.71)

= d(r log τ + cv log e); thus it is an exact differential θ 0 = dsrev with s(rev)(τ, e) = cv log e + r log τ.

(1.72)

Hence θ s = ds − dsrev = ds0 , with s0 constant on the integral manifolds of θ s | . Therefore, the entropy is determined with respect to (τ, e) by s(τ, e) = s0 + r log τ + cv log e.

(1.73)

Entropy for an Ideal Gas in Global Thermodynamics (i) Closed systems. In the framework of global thermodynamics, for a closed system with N moles of an ideal gas, with variables (V , E, S, −P , T ), the state equations are given by (1.62), (1.64), and thus N P = R0 , T V

1 N = Cv , T E

(1.74)

and the basic differential form is θ S = dS −

P 1 dE − dV = dS − θ 0 . T T

(1.75)

Hence we have θ 0 = N(R0

1 1 dV + Cv dE) = Nd(R0 log V + Cv log E). V E

Since N is constant, it follows that with Srev = N(R0 log V + Cv log E), θ 0 is an exact differential, θ 0 = dSrev . Thus θ S = dS − dSrev = dS0 , with S(V , E) = S0 + N(R0 log V + Cv log E).

(1.76)

36

1 Thermostructure

We can also express the entropy with the variables P , T , or with ( PT , T1 ), so that we have S(V , E) = S( −1 (( PT , T1 ) = S∗ (( PT , T1 ) with the “Legendre transform” S∗ of S: S(P , T ) = S∗ ((

P 1 P 1 , ) = S˜0 + N(−R0 log − Cv log ), T T T T

with

S˜0 = S0 + NR0 log(NR0 ) + NCv log(NCv ), (1.77) with S0 and S˜0 constants on the integral leaves of θ S , with N constant. (ii) Open systems. N too is a variable, so that the modelling of the system is based on the variables (V , E, N), S, ( PT , T1 , − Tμ ) with μ the chemical potential. The basic differential form is S = dS − PT dV − T1 dE + Tμ dN. The state equations have to define a map , so that (V , E, N) = ( 1 (V , E, N), 2 (V , E, N), 3 (V , E, N)) = (

μ P 1 , , − ), T T T

with (1.74). A first condition is that the map be integrable. This implies R0 ∂ 3 Cv ∂ 3 ∂ 1 ∂ 2 3 ∂V = ∂N = V , ∂E = ∂N = E . Thus must be given, up to an arbitrary function f (N), by 3 (V , E, N) = R0 log V + Cv log E + f (N). If the map is positively homogeneous, (λV , λE, λN) = (V , E, N), ∀λ > 0, we obtain that 3 is given (up to a constant) by 3 (V , E, N) = R0 log

V E + Cv log . N N

We obtain the same result if we require that the Jacobian matrix J ( ) of be negative, i.e., that the quadratic form associated with J ( ) be negative (degenerate): (x, J ( )x) ≤ 0, ∀x ∈ R 3 . Thus the map is given by μ P 1 N N V E , , − ) = (R0 , Cv , R0 log + Cv log ). T T T V E N N (1.78) Note that is not bijective, and thus that it is not a diffeomorphism. Now the basic differential form θ S becomes (V , E, N) = (

θ S = d[S − N(R0 log

V E + Cv log ) − (R0 + Cv )N]. N N

Thus the entropy is obtained, up to a constant S0 , by S − N(R0 log

V E + Cv log ) − (R0 + Cv )N = S0 N N

1.4 Ideal Gas

37

(compare with the case of a closed system (1.76)) or S − S0 V E = R0 log + Cv log − (R0 + Cv ). N N N

(1.79)

We see that the entropy is a positively homogeneous function of degree 1, that is, S(λV , λE, λN) = λS(V , E, N), ∀λ > 0, and that it is a concave function, since its Hessian H (S) = −J ( ) is negative. E V V E V E , τ = N . Then S(V , E, N) = S(N( N , N , 1)) = NS( N , N , 1). Now let e = N S0 V E Thus with s(τ, e) = S( N , N , 1) = S(τ, e, 1), and s0 = N , (1.79) becomes s(τ, e) − s0 = R0 log τ + Cv log e, that is, the entropy in local thermodynamics (1.73), up to a change of unit: in (1.73), V V τ corresponds to M = m1 N with m the molar mass of the gas. Specific Entropic Foliation of an Ideal Gas We return to the frame of local thermodynamics. The integral manifolds of θ in  are the leaves s 0 , s0 ∈ R (of dimension 2), s 0 is the set of elements j = (e, τ ; s; T ; P ) ∈ F satisfying (1.73) and the state equations (1.64), (1.61). Let S0 be the map j ∈ (r, cv ) → s0 ∈ R; S0 is a submersion from  to R. The family of integral manifolds (s 0 , s0 ∈ R) is a foliation of  , thus a partition of  . The sum set of this family is denoted by S 0 = ∪s 0 . The pair (S 0 ,  ) is called a foliated manifold (see [Bour.var, 9.2]). We give a thermodynamic interpretation of this foliation: a reversible evolution in  is in a leaf; an irreversible evolution in  is transverse to the leaves, thus with a variation of the “constant” s0 in (1.73). For an irreversible evolution in  , with velocity vector X, (1.34) reads 1 ds0 θ s | , X =  θ e | , X = ds0 , X = > 0; T dt

(1.80)

the reversibility of evolution in  is specified by23 ds0 |γ = 0, i.e., ds0 , X = 0.

23 There

(1.81)

is a similar point of view in [Lan-Lif1], but we have to note that s is to be changed into s0 ; the difference appears in a Carnot cycle, which is composed of adiabatics and isotherms. Following [Lan-Lif1], evolution along isotherms of the system is reversible only using an outside system.

38

1 Thermostructure

Morphisms of the thermodynamic foliation with a plane foliation of R 3 . Let x1 = log e, x2 = log τ, x3 = s, and the map ϕ : (e, τ, s) → (x1 , x2 , x3 ) from F = R + × R + × R onto R 3 . Then the basic differential form θ s is the pullback θ s = ϕ ∗ θˆ by ϕ of the differential form θˆ = dx3 − cv dx1 − rdx2 , whose integral manifolds are the family of planes Fx0 , x0 ∈ R, defined by x3 − cv x1 − rx2 = x0 . The map π such that π(x1 , x2 , x3 ) = x0 is a submersion defines a foliation of Rπ3 , with the leaves Fx0 . Notice that θˆ = π ∗ dx0 , so that θ s = ϕ ∗ π ∗ dx0 . The foliation of Rπ3 may be oriented with increasing x0 , which corresponds to increasing s0 of S 0 , and each leaf s 0 separates  into two parts. Moreover, we see that we can take the variables (x1 , x2 , x0 ) as a new coordinate system in R 3 , and thus (e, τ, s0 ) as a new coordinate system in  , instead of (e, τ, s). We can sum up these properties by saying that the map ϕ is a morphism of the oriented foliation S 0 onto the oriented foliation Rπ3 . Remark 8 We lay emphasis on the following property: for every θ -admissible  oriented path (from A to B) in  , the integral θ s = T1 θ e depends only on the starting leaf (thus of A) and on the arrival leaf (thus of B). Therefore,  θ s = s0 (B) − s0 (A) ≥ 0.

This specifies (1.36); indeed, we have also 



s0 (B) − s0 (A) =

θs =

(ds −

1 Q θ ) = s(B) − s(A) − T



1 Q θ . T

¶

Contact transformations and morphisms of the foliation for an ideal gas Here we give examples of contact transformations associated with a change of units. Let G0 be the abelian group of transformations, with two (positive) parameters κ1 , κ2 and a real parameter κ3 , defined (in the parametrization with entropy as the main variable) by gκ1 ,κ2 ,κ3 (x1 , x2 ; s; p1 , p2 ) = (κ1 x1 , κ2 x2 ; s − κ3 ; κ1−1 p1 , κ2−1 p2 ).

(1.82)

The differential form θ s is invariant under these transformations. Then the state equations of an ideal gas p1 x1 = r,

p2 x2 = cv ,

are invariant under the group G0 . If κ3 = κg = cv log κ1 + r log κ2 , then we have a group G0 that is the product of two abelian groups, with gκ1 ,κ2 = gκ1 ,1 .g1,κ2 . The family (gκ1 ,1 ), with κ1 > 0 is a group of isotherm transformations, and the family (g1,κ2 ), κ2 > 0 is a one-parameter group of incompressible transformations, i.e., conserving τ (hence ρ = 1/τ ).

1.4 Ideal Gas

39

The leaves Lα = s 0 are stable by G0 . The action of the group G0 on the entropy function is given with g = gκ1 ,κ2 by R(g)(s) = s ◦g−κg ; thus R(g)(s)(x1 , x2 ) = s(κ1 x1 , κ2 x2 )−(r log κ1 +cv log κ2 ), with x1 , x2 ) = (τ, e) corresponding to the action of g on the graph of s. Thus each leaf Lα is transformed into itself. A special subgroup of G0 is the group G0,0 with one parameter defined with μ ∈ R + by gμ (x1 , x2 , s, p1 , p2 ) = (μ3 x1 , μ2 x2 , s − (3r + 2cv ) log μ, μ−3 p1 , μ−2 p2 ). (1.83) This transformation preserves the differential form θ , keeping the space (r, γa ) invariant and leaving stable the leaves Lα . ¶ The other differential forms with respect to θ s , i.e., θ e , θ f , θ h , θ g , θ˜ s , restricted to (r, cv ), are not exact, which shows the interest in the parametrization with entropy s as the basic variable. ¶ Convexity properties. In the frame of ideal gases, the entropy s(τ, e) is a concave function of (τ, e), whence (−s(τ, e)) (and thus s0 = s −s(τ, e)) is a convex function of (τ, e). Indeed, we have s − s0 = cv log e + r log τ. Thus cv ∂s = , ∂e e

∂ 2s cv = − 2, ∂e2 e

∂s r = , ∂τ τ

∂ 2s r = − 2, ∂τ 2 τ

∂ 2s = 0. ∂e∂τ

Hence the Hessian matrix H (s) is given by H (s) =

∂2s ∂2s ∂τ 2 ∂τ ∂e ∂2s ∂2s ∂e∂τ ∂e2



=−

r τ2

0

0 cv e2

.

Thus −H (s) = H (−s) is a positive definite matrix. Then the energy e is a strictly convex function of (τ, s), and the speed c of sound is given by c = (γa (γa − 1) e)1/2 = (γa (γa − 1) )1/2 e1/2 = (rγa T )1/2

¶.

1.4.3 Determination of Thermodynamic Functions Determination of the Energy We can obtain the energy thanks to (1.73) as a function of entropy and of τ . We can also determine the energy by the method of characteristics, solving the system (with (1.63) and (1.64))

40

1 Thermostructure

(i) x1

∂u ∂u +r = 0, ∂x1 ∂x2

(ii) u − cv

∂u = 0, ∂x2

(1.84)

also with τ = x1 , s = x2 . The general solution of (1.84) (denoting by u0 a constant) is −r/cv

u(x1 , x2 ) = u0 x1

exp (x2 /cv ).

(1.85)

From (1.64), u, and thus also u0 , is positive. Let u0 = exp − cs0v . With the above thermodynamic notation, we have eα (τ, s) = τ −r/cv exp((s − s0 )/cv ) = τ −γa +1 exp((s − s0 )/cv ).

(1.86)

The expressions for pressure and temperature as functions of τ and s are deduced immediately, either by derivation or by the state equations. Remark 9 “Equation of adiabatic curves.” With (1.86), we note that −γa

p1 = ∂u/∂x1 = (−γa + 1)x1

exp(cv−1 (x2 − s0 )),

whence the relation P τ γa = (γa − 1) exp((s − s0 )/cv ), which is the “equation of adiabatic curves” as seen in Remark 6.

(1.87) ¶

Determination of the Gibbs Free Energy and the Chemical Potential With ϕ(v) = v − v log v, ∀v > 0, the free energy f (τ, T ) is then given by f (τ, T ) = −rT log τ + ϕ(cv T ) − s0 T . The Gibbs free energy g is obtained from the expression for e (see (1.86)), and the state equations by g(P , T ) = rT log P + ϕ(rT ) + ϕ(cv T ) − s0 T , or also with s˜ 0 = −ϕ(cv ) − ϕ(r) + s0 by g(P , T ) = rT log

P − cv T log T − s˜ 0 T . T

(1.88)

Then, thanks to the state equations of the ideal gas, the chemical potential μ˜ (see (1.21)) is given as a function of (e, ˜ ρ) by

1.4 Ideal Gas

41

g r μ˜ e r e˜ ˜ ρ) = − = cv log − r log + s˜ 0 , − (e, − r log + s˜ 0 = cv log T T cv τ cv ρ τ whence −

μ˜ (e, ˜ ρ) = cv log e˜ − (r + cv ) log ρ + C, T

with C = −cv − r + s0 .

(1.89)

˜ For an ideal gas, with the new variables ρ, e, ˜ we have a new integrable map μ˜ 1 ρ ˜ (ρ, e) ˜ = (− , ) = (cv log e˜ − (r + cv ) log ρ + C, cv ), T T e˜ with the new state equations given by e˜ = cv ρT and (1.89), so that we have (as in (1.21), with θ˜ s = ρθ s ) θ˜ s = d s˜ +

1 μ˜ ˜ dρ − d e˜ = d s˜ − (ρ, e).(ρ, ˜ e), ˜ T T

(1.90)

giving s˜ = −(r + cv ) log ρ + cv log e˜ + s˜0 , with s˜0 a constant, so that we have a new entropic foliation (L˜ s˜ 0 ) (of the space  ˜ ), associated with the weighted entropy s˜. ¶

1.4.4 Carnot Cycle for an Ideal Gas Let a system made up of an ideal gas, i.e., the energy E of the system (in global thermodynamics) depends linearly on the temperature T by E = Cv T , with Cv such that24 Cv = Mcv = Vρcv , M being the total mass of the system. Let m0 be the molar mass of the gas, N the number of moles, hence M = m0 N. We assume that the system is closed (i.e., N is a constant), and that the two state equations (1.64) and (1.62) are satisfied, with the differential form (1.75). We obtain a family of maximal integral manifolds in  by the jets of the entropy S given by (1.76). We then have an interpretation similar to the local frame of the evolutions (reversible or not) by the foliation due to S0 . The action of the exterior on the system may be made by imposing a given temperature T (the exterior is then said to be a reservoir) or a given pressure P ,

we choose to give the unit of local energy to the temperature, then [e] = L2 T −2 , Cv has units of mass, but cv is without unit.

24 If

42

1 Thermostructure

or also with given exchanges of work or heat. If there is no exchange of heat, the evolution of the system is said to be adiabatic, that is, < θ Q,

dE dV d >= −P = 0. dt dt dt

Note also that a more realistic model must take a change of phase into account, from water to vapor, for example, using the van der Waals equation, which we study below. The function of the closed system uses two “reservoirs” representative of the action of the exterior, one with temperature T1 , the other with temperature T2 , with T1 > T2 . If the evolution of the system during a Carnot cycle is reversible (with a real cycle in the manifold R0 ,Cv ), it is given by a path γ in a leaf Ls0 . If the evolution of the system is irreversible, it is given by an oriented path γ˜ in the manifold R0 ,Cv (of extremities α, β), which is not a cycle with variables (S, V , P , T , E), but is a cycle γ in projection with variables (V , P , T , E) (with an evolution of S0 , which is a variable transverse to the leaves FS0 ). Then we have 

 S(β) − S(α) =

γ˜

dS =

γ˜

dS0 ≥ 0,

  since γ˜ dSrev = γ dSrev = 0, whence the positivity of S(β) − S(α) while following γ˜ over time according to the orientation of the path. Whether the evolution is   reversible or not, we set balance laws with integrals on γ : γ dE and γ dSrev = γ T1 θ Q , both equal to zero. Thus we have 





dE = γ

θQ +

θ W = Q − W = 0,

γ

γ

where W is the total work provided by the system along the cycle, and thus the provided work is equal to the quantity of received heat Q. The cycle γ is composed of two isotherms and two adiabatic curves,25 and can be made in one sense or the other. Here we choose a cycle that produces work. The cycle contains the following parts of the path (see Figure 1.1): (1) γA,B isotherm relaxation by contact with the hot reservoir to the temperature T1 . The system receives a heat quantity QAB = Q1 > 0 (which determines the point B), and provides work WA,B , the volume of the system going from VA to VB , with VB > VA . Then the energy of the system is constant E1 = Cv T1 , and the quantity of provided work WA,B is equal to the quantity of received heat: 



WA,B =

P dV = γA,B

25 Hence

 dE + P dV =

γA,B

of characteristic curves, as seen above.

θ Q = Q1 > 0. γA,B

1.4 Ideal Gas

43

Using the ideal gas law, the provided work is expressed by 



WA,B =

P dV = γA,B

VB

NR0 T1 VA

1 VB dV = NR0 T1 log . V VA

(2) γB,C going from temperature T1 to temperature T2 < T1 by adiabatic relaxation, the system being “thermically isolated”: hence θ Q = 0 on γB,C , which corresponds to T V γa −1 = constant, hence TT21 = ( VVBC )γa −1 . There is no exchange of heat, the energy E decreases, and the system provides the work: 



WB,C =

P dV = −

dE = −[E(C)−E(B)] = −NCv (T2 −T1 ) > 0.

γB,C

γB,C

(3) γC,D , isotherm compression at T2 , contact with the cold reservoir at temperature T2 : We have to provide work WC,D to the system (this work determines the point D), whose volume decreases (from VC to VD ), and the system provides a heat quantity −QC,D = Q2 > 0 equal to −WC,D , where 



WC,D =

P dV = γC,D

VD

NR0 T2 VC

VD 1 = −Q2 < 0. dV = NR0 T2 log V VC

(4) γD,A going from temperature T2 to temperature T1 > T2 by adiabatic compression, the system being “thermically isolated,” so that θ Q = 0 on γD,A . There is no exchange of heat, we provide work to the system −WD,A > 0, and

Fig. 1.1 Carnot cycle, P (V ).

44

1 Thermostructure

the energy E increases and returns to its initial value. Then the work is expressed by 

T1 T2

= ( VVDA )γa −1 , and



WD,A =

P dV = − γD,A

dE = E(D)−E(A) = −NCv (T1 −T2 ) = −WB,C . γD,A

The balance law of work W provided by the system during the Carnot cycle is26  W =

P dV = NR0 T1 log γ

Moreover, since θ 0 =

1 E Tθ

 γ

thus

Q1 T1

−

Q2 T2

VB VD + NR0 T2 log . VA VC

is a total differential, we have Q1 Q2 1 Q − = 0; θ = T T1 T2

= 0, whence the equality

VB VA

W = Q1 − Q2 = NR0 (T1 − T2 ) log

=

VC VD ,

and thus

VB PA = NR0 (T1 − T2 ) log . VA PB

The yield ηrev = η is the ratio between the provided work W (corresponding to the area of the domain inside the cycle of the diagram P , V ) and the received heat Q1 , thus expressed by η=

W Q1 − Q2 Q2 T2 = =1− =1− . Q1 Q1 Q1 T1

Remark 10 On the smoothness of the evolution. Let t → γ (t) be a parametric representation of the Carnot cycle that gives the evolution of the system over time. The speed of evolution depends on the speed of bringing heat to the system in the part γA,B , then work in the part γC,D . The rate of bringing heat is constant if we follow the evolution along the characteristic isotherm curve. Furthermore, the speed while going along the Carnot cycle is linked to the possibility that the system is homogeneous in global thermodynamics. Since the derivative ∂P ∂V is not continuous at the points A, B, C, D, the function 1 γ can be of class C only if its derivative is zero at times tA , tB , tC , tD such that γ (tA ) = A, . . . , γ (tD ) = D, that is, exchanging isotherm and adiabatic. Indeed, the dP dP dV fact that dV is not continuous at these points and the relation dP dt = dV dt require dV dP dT dE dt = 0, thus dt = 0, and from the ideal gas law dt = 0, we have dt = 0 and dS also dt = 0. ¶ 26 Since the cycle γ = ∂ is the boundary of a domain , this work is also equal to the area  W =  dP dV .

1.5 Fluid with the van der Waals Equation

45

1.5 Fluid with the van der Waals Equation A “change of phase” corresponds to a change of physical structure of a system such as the change of a solid state to a liquid state, or liquid to gas, and conversely. Furthermore, several solid states or liquid states can exist. Such changes may be described in several ways with thermodynamic variables. We consider the simple case of a system with one constituent (water, for example), modeled by five variables (τ, e, s, P , T ). From experiment we know that at given T and P , several phases can coexist, with different properties, notably with different values of (τ, e, s). The state equation of van der Waals allows a modelling of the liquid–gas change of phase to a certain extent, but a priori not the solid–liquid one; hence the validity domain of that state equation consists of temperatures greater that a temperature T1 . This model has a nonphysical domain with a lack of convexity for certain thermodynamic functions that will be corrected in a second step. Similarly to the situation of an ideal gas, we can join another state equation linking energy, temperature, and specific mass.

1.5.1 Study of the van der Waals Equation In a global model of a “pure” fluid (with molar mass m), the van der Waals state equation is (P + a0 (

N 2 ) )(V − Nb) − NR0 T = 0, V

(1.91)

where N is the number of moles of the fluid, V the volume of the domain of fluid, and a0 , b are positive constants; a0 is said to be the “cohesion coefficient,” and b is a coefficient proportional to the size of the molecules. Naturally we must have T > 0,

V > Nb,

P + a0 (

N 2 ) > 0. V

A local model is obtained by letting M = Nm,

M/V = ρ,

and τ 0 = b/m,

hence N/V = ρ/m = 1/(τ m),

r = R0 /m,

a = a0 /m2 ;

the van der Waals equation is (P +

a )(τ − τ 0 ) − rT = 0, τ2

(1.92)

46

1 Thermostructure

or also (P , T , τ ) = 0, with (P , T , τ ) = P − P (τ, T ),

with P (τ, T ) =

rT a − 2. 0 τ −τ τ

(1.93)

In the three-dimensional space of (P , T , τ ), that equation defines a manifold, with conditions T > 0,

τ > τ 0,

P+

a > 0. τ2

(1.94)

Since the absolute temperature is positive, this implies τ > τ 0 , hence ρ < ρ0 , with ρ0 such that ρ0 = 1/τ 0 = m/b; this is the compressibility limit of the material. We can also give a formulation without units which eliminates the various constants: Tc =

8 a0 , 27 bR0

8 a , Tc = 27 rτ 0

Pc =

a0 , 27b2

τc =

1 a Pc = , 27 (τ 0 )2

3b , m

Vc /N = 3b,

or (1.95)

τc = 3τ , 0

which leads to the reduced van der Waals equation for (1.91) with a0 = 3, b = 1 8 1 8 0 3 , R0 = 3 , and for (1.92) with a = 3, τ = 3 , r = 3 . For fixed (P , T ), the van der Waals equation is of third order with respect to τ , which admits τc as a triple root for (P , T ) = (Pc, Tc ); Pc and Tc are said to be the critical pressure and critical temperature. The pressure is positive when (τ − τ 0 ) τa2 < rT . In order for this to be true for a 27 all τ > τ 0 , we must have T > T+ , with T+ = 4rτ 0 = 32 Tc . It is not obvious that pressure must necessarily be positive for a liquid state, but that may be a limit of validity of modelling fluids by the van der Waals equation. Study of the function P (τ, T ) at fixed temperature, simply denoted by P (τ ). The derivative of the pressure with respect to τ (with T constant) is expressed from van der Waals equation by P  (τ ) = −

(τ − τ 0 )2 1 2a 2a rT + ψ(τ )], with ψ(τ ) = rT + = [− . (τ − τ 0 )2 τ 3 (τ − τ 0 )2 2a τ3

Now ψ is an increasing function of τ on the interval ]τ 0 , τc ], and decreasing on rTc 4  [τc , +∞[, with a maximum at τ = τc such that ψ(τc ) = 27τ 0 = 2a . Thus P (τc ) = r 8 a [T − T ], with rTc = 27 . τ 0 . The corresponding pressure Pc such that Pc = 4(τ 0 )2 c P (τc , Tc ) = 8τ1 0 rTc is the critical pressure. 0 The condition ( ∂P ∂τ )T (τ ) < 0, ∀τ > τ , giving the decreasing of P (τ ) with τ (at fixed temperature) is therefore satisfied when T > Tc .

1.5 Fluid with the van der Waals Equation

47

When T < Tc , there exist two numbers τm (T ), τM (T ) such that P  (τm (T )) = P  (τM (T )) = 0,

with τ 0 < τm (T ) < τc < τM (T ),

(1.96)

and such that P  (τ ) > 0 in the interval ]τm (T ), τM (T )[, and with P  (τ ) ≤ 0 outside. Thus P (τ ) is increasing in the interval ]τm (T ), τM (T )[, and decreasing outside. Denoting τm (T ) and τM (T ) by τm and τM , we have P (τm , T ) =

rT a 2a(τm − τ 0 ) a a − = − 2 = 3 (τm − 2τ 0 ); τm − τ 0 τm2 τm3 τm τm

(1.97)

thus P (τm , T ) ≤ 0 for τm ≤ 2τ 0 . Let Pm (T ) = P (τm , T ), PM (T ) = P (τM , T ). For fixed (T , P ), T < Tc Pm (T ) < P < PM (T ), P > 0, there exist three values τj , j = 1, 2, 3, for τ , with τ 0 < τ1 < τ2 < τ3 , 3 , the set of triples solutions of the van der Waals equation (1.92). In the space R+ (2) (P , T , τ ) satisfying (1.92) is a 2-dimensional manifold  with a fold such that the projection π on the plane (P , T ) is not invertible for (P , T ), so that T < Tc , Pm (T ) < P < PM (T ), but is invertible outside. We note that for τ = τm , we have ∂ ∂P (τm , T ) = − (τm , T ) = 0 ∂τ ∂τ (2) 3 at (and for τ = τM ), so that the tangent plane to the manifold  ⊂ R+ (P (τm , T ), T , τm ) (or (P (τM , T ), T , τM )) is parallel to the axis Oτ , thus vertical (2) with respect to the plane (P , T ). (Of course, it is the same if  is taken as a submanifold of the space R 5 , since the derivatives along the supplementary variables are also zero.) It will be useful to specify the lifts of π in order to distinguish the liquid and gas domains. A first physical interpretation of different parts of the domain D such that D = {(P , T , τ ), P > 0, T > T1 , τ > τ 0 } (if T1 > 0 is a limit of validity of the modelling of the system) is the following: in where T > T , representative of a state where liquid and gas (i) the domain Dlv c (vapor) are not segregated. c (with T (ii) the complementary domain Dlv 1 ≤ T ≤ Tc ) having several subdomains:

• a subdomain Dl , with τ “small” (notably τ < τm (T )), hence with high specific mass, representative of the liquid state;

48

1 Thermostructure

• v, with τ “high” (notably τ > τM (T ), hence for specific mass small, representative of the vapor state; • an intermediate subdomain Dm , corresponding to a mixing liquid vapor, which will be specified below. ¶ A first question is to know the condition for θ 0 = T1 θ Q = T1 de + PT dτ to be an exact differential form, with the van der Waals equation (1.93). We have θ0 =

r 1 a a 1 1 de + dτ − dτ = d(e + ) + rd log (τ − τ 0 ). T (τ − τ 0 ) T τ2 T τ

Hence θ 0 is exact if e + aτ is a function of T . With this condition, we can write θ 0 = dsrev , and thus θ s = ds − dsrev = ds0 , which allows us to characterize the irreversible evolutions easily by the mere variation of s0 . Similarly to the case of ideal gases, we are led to adopt the local (respectively global) relation between energy and temperature as a new state equation: e+

a = cv T , τ

respectively E + a0

N2 = Cv NT , V

with cv (respectively Cv = mcv ) constant. Then this defines a map from M = {(τ, e), τ > τ 0 , e + P 1 {( T , T ), T > 0, P + τa0 > 0} by (τ, e) = ( 1 (τ, e), 2 (τ, e)) = (

(1.98)

a τ

> 0}, into N =

P 1 a cv cv r − 2 ). , )=( a, T T τ − τ0 τ e + τ e + aτ (1.99)

This map is integrable, since we have ∂ 2 a cv a 1 ∂ 1 = = 2 = . ∂e ∂τ τ (e + τa )2 cv (τ T )2 Of course we directly verify this property below by obtaining the integral function for . Let  be the set of (τ, e, s, PT , T1 ) (or simply (τ, e, P , T )) satisfying the state equations (1.98), (1.93), with (1.94). The restriction of the differential form θ 0 = P 1 T dτ + T de to the manifold  is such that θ 0 | = d (r log (τ − τ 0 ) + cv log (cv T )) = dsrev .

(1.100)

1.5 Fluid with the van der Waals Equation

49

Entropy Determination Entropy in local thermodynamics. The relation (1.100) gives the expression of entropy directly by s(τ, T ) = r log (τ − τ 0 ) + cv log (cv T ) + s0 ,

(1.101)

which implies the expression of entropy as a function of τ and of the energy s(τ, e) = r log(τ − τ 0 ) + cv log (e +

a ) + s0 . τ

(1.102)

Weighted entropy in local thermodynamics and chemical potential. From (1.102) we have s˜ (ρ, e) ˜ = ρs(τ, e) = rρ log(

e˜ 1 − τ 0 ) + cv ρ log( + aρ). ρ ρ

(1.103)

Then the chemical potential is given by −

∂ s˜ 1 −rτ 0 ρ 2aρ 2 e˜ μ˜ (ρ, e) ˜ = = r log( − τ 0 ) + cv log( + aρ) + + c . v T ∂ρ ρ ρ 1 − τ 0ρ e˜ + aρ 2 (1.104)

˜ such that (ρ, ˜ Then we obtain the integrable map e) ˜ = (− Tμ˜ , − T1 ) for the weighted entropy, the relation (1.104) being a state equation supplying the van der Waals equation (1.92). Entropy in global thermodynamics. Using (1.98) and (1.91) (and the homogeneity property), we obtain by integration S(V , E, N) = N[Cv log (

N E V + a0 ) + R0 log( − b) + s˜0 ]. N V N

(1.105)

We can deduce this relation from (1.105): let S = s¯ N, E = eN, ¯ V = τ¯ N, with the molar mass m and the relations s¯ = ms,

e¯ = me, τ¯ = mτ,

whereby the van der Waals equations (1.91) and (1.98) read P+

T a0 , = R0 τ¯ − b τ¯ 2

The differential form θ S = dS −

1 T

dE −

P T

e¯ +

a0 = Cv T . τ¯

dV +

μ T

(1.106)

dN for an open system is

50

1 Thermostructure

P μ 1 d(eN) ¯ − d(τ¯ N) + dN T T T 1 P e¯ P τ¯ μ = N(d s¯ − d e¯ − d τ¯ ) + (¯s − − + ) dN. T T T T T

θ S = d(¯s N) −

Hence if g¯ =

G N

(1.107)

= e¯ + P τ¯ − T s¯ = μ, we have θs =

1 S 1 P θ = d s¯ − d e¯ − d τ¯ . N T T

The above equation, with (1.106), leads to θ s = ds−Cv d log Cv T −R0 d log(τ¯ −b), and the entropy s reads s = Cv log Cv T + R0 log(τ¯ − b) + s¯0 , s(τ¯ , e) ¯ = Cv log(e¯ +

thus

a0 ) + R0 log(τ¯ − b) + s¯0 , τ¯

(1.108)

whence (1.103). ¶ Study of the manifold  in local thermodynamics. From the state equations (1.98), (1.93), we can replace variables (τ, e) by variables (P , T ) thanks to the map : (τ, e) ∈ M → (P , T ) defined by the relations P (τ, e) =

r a a 1 e+ − 2, 0 cv (τ − τ ) τ τ

a 1 T (τ, e) = (e + ). cv τ

(1.109)

Let M = {(τ, e), τ > τ 0 ,

e+

a > 0}. τ

(1.110)

Then the image set is such that (M) ⊂ {(P , T ), P > − τa0 , T > 0}. Now the question is to know whether is an embedding. With partial derivatives at fixed τ or e, let =

a 2a ∂P ∂T 1 ∂P ∂T 1 P + + − =− . 0 2 ∂τ ∂e ∂e ∂τ cv τ − τ τ cv τ 3

When P + τa2 = 2a τ −τ , we have  = 0. Using the van der Waals equation, this τ3 defines the curve CmM (over R 2 ) as 0

P =

a (τ − 2τ 0 ), τ3

rT =

2a (τ − τ 0 )2 . τ3

(1.111)

1.5 Fluid with the van der Waals Equation

51

These relations are also obtained by writing ∂P ∂τ |T = 0; hence the curve CmM is that of minima and maxima of P with fixed T . This curve also is the set of points where  (τ, e) is not injective, hence where is not an embedding. We obtain the inverse image of the curve CmM by thanks to the expression of the energy e=−

acv 2a a + (τ − τ 0 )2 , τ r τ3

which gives also the curve C¯mM in  , which is the graph of the function . The set DmM of points of  that are “inside” this curve, that is, DmM ={(τ, e, P , T ),

T < Tc ,

τm (T ) < τ < τM (T ),

Pm (T ) < P < PM (T ), em (T ) < e < eM (T )}

(1.112)

(with Pm (T ), PM (T ), τm (T ), τM (T ) expressed by (1.111), (1.94)), has no physical meaning.27 Some properties of the curve CmM . We have 2a ∂P = 4 (−τ + τc ), ∂τ τ

∂(rT ) 2a = 4 (−τ + τc )(τ − τ 0 ). ∂τ τ

These two derivatives are null at τ = τc , at the critical point (Pc , rTc ) ∈ CmM . Moreover, these two derivatives are positive on the interval ]τ 0 , τc ] and negative on [τc , +∞[, so that P and T are increasing functions on ]τ 0 , τc ], decreasing on ]τc , +∞[, and they reach their respective maxima Pc , Tc , for τ = τc . Now taking τ¯ = τ − τ 0 , we have from (1.111), α=

τ¯ 2 rT =2 P τ¯ − τ 0

thus 2τ¯ 2 − α τ¯ + ατ 0 = 0;

hence τ¯ must be a solution for fixed T , P , and thus fixed α, of a second-order c equation that has no real solution when 0 < α < 8τ 0 = rT Pc . Thus we have rT P

≥

rTc Pc , whereby P

≤ Pc

T Tc , and since T

≤ Tc , we verify that P ≤ Pc . Moreover,

) 0 at every point (rT (τ1 ), P (τ1 )) ∈ CmM , we have ∂(rT ∂P = (τ − τ ) > 0, so that T is an increasing function of P . ¶ The determination of the energy may be made in several ways, in order to show how the state equation (1.98) occurs.

ˆ =  \DmM , which represents the set of thermodynamic states, is a the manifold  manifold with boundary. But here it is not question of stability of states.

27 Thus

52

1 Thermostructure

Determination of the Energy First method. The van der Waals equation (1.92) is transformed into a partial ∂e derivative equation of the energy e(τ, s) using the relations P = − ∂τ , T = ∂e ∂s : (−

∂e a ∂e + 2 )(τ − τ 0 ) − r = 0. ∂τ τ ∂s

(1.113)

The solution of this equation is expressed with an arbitrary (but differentiable) function φ by s a e(τ, s) = φ( − log(τ − τ 0 )) − . r τ

(1.114)

This is verified with the change of function e¯ = e + aτ . This implies ∂e 1 s = φ  ( − log(τ − τ 0 )), ∂s r r 1 s a ∂e = φ  ( − log(τ − τ 0 )) − 2 . P =− 0 ∂τ τ −τ r τ T =

(1.115)

But the temperature T must be positive, which implies that φ  must be a positive 0 . function (hence φ must be increasing); if furthermore P > 0, then φ  > a τ −τ τ2 In other respects, the relation between energy and temperature e+

a = cv T τ

(1.116)

is a first integral for the van der Waals equation. This implies the equation e − cv

∂e a + = 0, ∂s τ

whose solution is expressed with another new arbitrary function ψ(τ ) by e(τ, s) +

a s = ψ(τ ) exp , τ cv

whence cv T = ψ(τ ) exp csv , and the comparison with (1.115) implies (with a constant s0 ) φ(λ) = C exp(α0 λ), with C = exp −

s0 , cv

α0 =

r = γa − 1, cv

with γa the adiabatic index; see (1.86). Then (1.115) and (1.114) give

(1.117)

1.5 Fluid with the van der Waals Equation

53

e(τ, s) = (τ − τ 0 )−α0 exp

s − s0 a − , cv τ

1 s − s0 (τ − τ 0 )−α0 exp , cv cv s − s0 a P = α0 (τ − τ 0 )−α0 −1 exp − 2. cv τ

T =

(1.118)

Characteristic method. As in the case of a perfect gas, let x1 = τ,

x2 = s,

u = e,

p1 = −P ,

p2 = T .

The van der Waals equation is def

(x1 , x2 ; u; p1 , p2 ) = (p1 −

a )(x1 − τ 0 ) + rp2 = 0. x12

The characteristic vector ξ is expressed by ξ = ( p ; p. p ; −( x + p u )) = (x1 − τ 0 , r; p. p ; − x1 , 0), with p. p = p1 (x1 − τ 0 ) + p2 r =

a (x1 − τ 0 ), x12

x1 = p1 −

d a [ (x1 − τ 0 )]. dx1 x12

The characteristic equations are dx1 = (x1 − τ 0 ), dt

dx2 = r, dt

du a = 2 (x1 − τ 0 ), dt x1

dp1 d a = −p1 + [ (x1 − τ 0 )], dt dx1 x12

dp2 = 0. dt

Returning to the thermodynamic variables and denoting by the subscript 0 the initial conditions, we obtain e+

a a = (e + )0 , τ τ

P+

τ − τ 0 = et (τ − τ 0 )0 , Moreover, we have involving p1 may be e+

a (x − τ 0 ) x12 1 d written e−t dt (et p1 )

du dt

=

a a = (e + )0 , τ τ

a a = e−t (P + 2 )0 , 2 τ τ

s = rt + s0 , = =

P+

T = T0 .

a dx1 d a = − dt ( x1 ), and x12 dt d e−t dt (et a2 ), which gives x 1

a a = e−t (P + 2 )0 . 2 τ τ

the equation

54

1 Thermostructure

Taking (e + aτ )0 = cv T0 , we obtain the state equation e + τa = cv T . Returning to the ideal gas. We denote by the subscript v the thermodynamic variables of the van der Waals equation, (τv , Pv , Tv ), and we also denote by ev , sv the variables corresponding to e, s. Let P = Pv +

a , τv2

τ = τv − τ 0 ,

T = Tv .

The van der Waals equation is thereby transformed into the equation of an ideal gas. The differential form θ ev = (T ds − P dτ − de)v becomes, changing the variables with subscript v by the new variables, θ ev = T ds − (P −

a )dτ − dev = T ds − P dτ − de = θ e , (τ + τ 0 )2

with e = ev +

a a = ev + . 0 τ +τ τv

¶

Remark 11 Some positivity conditions. Let us recall that the energy e is not necessarily positive for a liquid or a solid, but e is bounded from below. Here we have e ≥ cv T − τa0 = cv (T − 27 8 α0 Tc ). (The energy is then positive only for cv T ≥ τa ∀τ ≥ τ 0 , thus cv T ≥ τa0 .) 0 Finally, the condition ( ∂P ∂τ )s ≤ 0, ∀τ ≥ τ , which implies the positivity of the ∂2e 1 (τ −τ 0 )2 ≥ 0, hence for rT 2a > γa ∂τ 2 τ3 1 1 rTc 1 ψ(τ ) = ; hence T > c γa γa 2a γ a Tc .

speed of sound, is satisfied if is obtained for all τ if

rT 2a

>

=

1 γa ψ(τ ).

This

¶

1.5.2 Loss of Convexity with van der Waals ˆ . For this study We did not study the question of stability of states of the manifold  we are brought to see that the van der Waals equation does not imply the convexity of thermodynamic functions, especially of the opposite of entropy. We have seen in Hypothesis 4 requirements such that a state equation defines such convex functions. In the parametrization based on entropy, the van der Waals equation is (p1 , p2 , x1 ) = p1 + with x1 > τ 0 . Hence we have 1 = 3 =

∂ ∂p1

a r p2 − = 0, x1 − τ 0 x12

= 1, and

P  (τ ) ∂ r 2a 1 = − . = − ∂x1 (τ − τ 0 )2 τ3 T T

1.5 Fluid with the van der Waals Equation

55



Thus 1 3 = − P T(τ ) . Therefore, if T < Tc and if τ ∈]τ l (T ), τ v (T )[, then P  (τ ) > 0 and 1 3 < 0. We deduce from Hypothesis 4 that the entropy s(τ, e) defined by (1.101) cannot be concave for all τ > τ 0 , and the energy e(τ, s) cannot be convex for all τ > τ 0 . The loss of convexity is understood as a loss of stability corresponding to a change of phase.

Convexification of the Opposite of Entropy We turn the function s˜ such that s˜ (τ, e) = −s(τ, e) defined on the set M (see (1.110)) into a convex function s˜ ∗∗ , which is the Fenchel biconjugate of s˜ (according to convex analysis). More precisely, the epigraph of s˜ ∗∗ is the convex envelope28 of the epigraph of s˜ (τ, e). (i) Calculation of the conjugate function of s˜. Extending the domain of definition of s˜ to R × R by s˜ (τ, e) = +∞ for (τ, e) outside M, we define the conjugate function of s˜ by (see (1.37)) (without taking s0 into account, which is of no interest here), s˜∗ (p1 , p2 ) = sup(p1 τ + p2 e − s˜ (τ, e)) τ,e

= sup[p1 τ + p2 e + r log (τ − τ 0 ) + cv log (e + τ,e

a )]. τ

When p1 > 0, as when p2 > 0, we have s˜∗ (p1 , p2 ) = +∞. When p1 ≤ 0, the sup with respect to e is reached for e + aτ = − pcv2 . Thus s˜ ∗ (p1 , p2 ) = sup[p1 τ + r log (τ − τ 0 ) − τ

a cv p2 + cv log (− ) − cv ]. τ p2

The supremum with respect to τ is reached for p1 +

a r cv −a r + = p1 + + p2 2 = 0. e + aτ τ 2 τ − τ0 τ − τ0 τ

The dual variables p1 , p2 correspond to variables P and T through p1 = −

P 1 , p2 = − , T T

whence

p1 = P. p2

(1.119)

Therefore τ must be a solution of the van der Waals equation.

28 Recall (see [Aub][ch. I.3.4, p. 41]) that if co(f ) is the convex envelope of a function f , we have co(f ) ≤ f and f ∗ ≤ (co(f ))∗ , thus co(f ) = (co(f ))∗∗ ≤ f ∗∗ .

56

1 Thermostructure

In order to express the infimum, using the variables p1 and p2 as in (1.119), let ψP ,T (τ ) =

1 a a (p1 τ − p2 + r log(τ − τ 0 )) = P τ − − rT log(τ − τ 0 ). p2 τ τ (1.120)

Thus s˜ ∗ (p1 , p2 ) = p2 inf[ψP ,T (τ )] + cv log (− τ

cv ) − cv . p2

Calculation of the infimum of ψP ,T (τ ). The derivative of the function ψP ,T is such that dψP ,T a 1 (τ ) = P + 2 − rT = P − P (τ, T ) = (P , T , τ ). dτ τ τ − τ0

(1.121)

Hence the function ψP ,T is a primitive of the van der Waals equation. The minimum of ψP ,T (τ ) is such that P − P (τ, T ) = 0. (i) If T ≥ Tc , then there exists one value τ = τm such that P − P (τ, T ) = 0. Then s˜∗ is given by s˜∗ (−

1 1 P , − ) = − ψP ,T (τm ) + cv log (cv T ) − cv . T T T

(1.122)

(ii) If T < Tc and P > 0, then there exist three extrema τ1 < τ2 < τ3 of the function ψP ,T (τ ) for fixed T ; τ1 and τ3 are minima, whereas τ2 is a maximum. Note that ψP ,T (τ ) → +∞ when τ → τ 0 and when τ → +∞. Hence the infimum of ψP ,T (τ ) is expressed by min{ψP ,T (τ1 ), ψP ,T (τ3 )}. Let τm be the value of τ at this minimum (if there is one minimum). In what follows, τ1 and τ3 will be denoted by τl and τv , corresponding to liquid and vapor. We have  ψP ,T (τv ) − ψP ,T (τl ) =

τv

(P − P (τ, T ))dτ.

(1.123)

τl

Let PS (T ) be the value of P , so that 

τv

(P − P (τ, T )) dτ = 0, hence PS (T ) =

τl

1 τv − τl



τv

P (τ, T ) dτ, τl

with PS (T ) = P (τv , T ) = P (τl , T ). (1.124)

1.5 Fluid with the van der Waals Equation

57

Fig. 1.2 Van der Waals curves, P (τ ).

If P (τ, T ) > PS (T ), then τm = τl , and if P (τ, T ) < PS (T ), then τm = τv . The position of PS (T ) corresponds to the equality of area of each loop (see Figure 1.2). For P = PS (T ), we have ψP ,T (τv ) = ψP ,T (τl ), and the infimum of ψP ,T is reached by τl and τv . This implies that the function s˜ ∗ is continuous across the (saturation) curve (but with a jump of the derivatives) S = {(PS (T ), T ), PS (T ) ≥ 0, 0 ≤ T ≤ Tc }. Let τl , τv be the extreme roots of the equation PS (T ) − P (τ, T ) = 0 (of course they depend on T ). This allows us to define the saturation domain D˜ S with variables (τ, T ): DS = {(τ, T ), 0 < T < Tc , τl < τ < τv , P (τ, T ) > 0}.

(1.125)

With el = cv T − a τ1l , ev = cv T − a τ1v (of course depending on T ) we also define the saturation domain DS with variables (τ, e) by DS = {(τ, e), τl < τ < τv ,

el < e < ev }.

(1.126)

(ii) Calculation of the biconjugate of s˜ . Then with ϕ(cv ) = cv − cv log cv , the biconjugate of s˜ is expressed by

58

1 Thermostructure

s˜∗∗ (τ, e) = sup (p1 τ + p2 e − s˜∗ (p1 , p2 )) p1 ,p2

= sup [p1 τ + p2 e − (p1 τm + r log(τm − τ 0 ) − p1 ,p2

a p2 ) τm

+ cv log(−p2 )] + ϕ(cv ) = sup ζτ,τm (p1 , p2 ) + ϕ(cv ),

with

p1 ,p2

ζτ,τm (p1 , p2 ) = [p1 (τ − τm ) − r log(τm − τ 0 ) + p2 (e +

a ) + cv log(−p2 )]. τm

We recall that τm depends on p1 and p2 through p1 +

a r + p2 2 = 0. τm − τ 0 τm

(1.127)

By taking the derivative of ζτ,τm with respect to p2 , then p1 , we obtain e+

a ∂τm cv r a + + (−p1 − − p2 2 ) = 0, τm p2 τm − τ 0 τm ∂p2 (τ − τm ) + (−p1 −

a ∂τm r − p2 2 ) = 0. 0 τm − τ τm ∂p1

With (1.127), we have e+

cv a + = 0, τm p2

τ − τm = 0,

(1.128)

but fixing (τ, e) determines the temperature T using (1.116). If (τ, T ) is outside DS , and if P (τ, T ) ≥ PS (T ) (respectively P (τ, T ) ≤ PS (T )), then τm = τl = τ, (respectively τm = τv = τ ),

e+

cv a =− , τ p2

1 P whence p2 = − , p1 = − . T T Hence if (τ, T ) is outside the saturation domain DS , that is, if (τ, e) is outside the domain DS , see (1.126), then the biconjugate of s˜ is identical to s˜: s˜∗∗ (τ, e) = s˜ (τ, e).

(1.129)

Now we have to study the case (τ, T ) ∈ D˜ S , whence (τ, e) ∈ DS . Then the identification of (1.128) is impossible, and ((τ, e), s˜ (τ, e)) cannot belong to the convex envelope of the epigraph of s˜. Let P = PS (T ). We recall that

1.5 Fluid with the van der Waals Equation

59

ψP ,T (τv ) − ψP ,T (τl ) = 0. Let ξ ∈]0, 1[ be such that τ = ξ τl + (1 − ξ )τv ∈ [τl , τv ]. At (τl , el ) and (τv , ev ), we have (

∂ s˜ ∂ s˜ P 1 , ) = (− , − ) = (p1 , p2 ), ∂τ ∂e T T

and since P (τl , T ) = P (τv , T ) = PS (T ), the values of (p1 , p2 ) at (τl , el ) and (τv , ev ) are the same. Thus the tangent planes to the graph of s˜ (τ, e) at (τl , el ) and (τv , ev ) are identifiable, and the convex envelope of the epigraph of s˜ (τ, e) is made up of the envelope of these tangent planes and the convex part of this epigraph (with (τ, e) outside DS ). Thus in DS , s˜ ∗∗ (τ, e) = ξ s˜ (τl , el ) + (1 − ξ ) s˜ (τv , ev ), with τ = ξ τl + (1 − ξ )τv ,

e = ξ el + (1 − ξ )ev ,

(1.130)

that is, (τ, e) = ξ(τl , el ) + (1 − ξ )(τv , ev ) ∈ DS .

(1.131)

We can also derive (1.130) using the usual convex relation, with 1 1 P P , − ) ∈ ∂ s˜ ∗∗ (τ, e), (τ, e) ∈ ∂ s˜ ∗ (− , − ) : T T T T 1 P 1 P s˜ ∗∗ (τ, e) + s˜ ∗ (− , − ) = − τ − e T T T T P 1 = − (ξ τl + (1 − ξ )τv ) − (ξ el + (1 − ξ )ev ) T T (−

P 1 P 1 τl − el ] + (1 − ξ )[− τv − ev ] T T T T 1 1 P P = ξ [˜s (τl , el ) + s˜∗ (− , − )] + (1 − ξ )[˜s (τv , ev ) + s˜ ∗ (− , − )] T T T T 1 P = ξ [˜s (τl , el ) + (1 − ξ )[˜s (τv , ev ) + s˜∗ (− , − ). T T

= ξ [−

State equation with energy in the saturation domain DS . In DS , we can take (ξ, T ) as basic state variables, and we have to prove that (τ, e) are still basic state variables in DS , and that the map (ξ, T ) → (τ, e) given by (1.131) is a diffeomorphism.29 Let

29 Note that (P , T ) are no longer basic state variables in the saturation domain, since then the pressure P depends on the variable T . Recall also that the set {(ξ, T ), ξ ∈ [0, 1]} is identifiable with the subdifferential ∂ s˜ ∗ (− PTS , − T1 ).

60

1 Thermostructure

[τ ]lv = τl − τv ,

[e]lv = el − ev .

Then from (1.130), we have ξ=

τ − τv e − ev = . [τ ]lv [e]lv

From the state equation (1.98), we have [e]lv = −a[ τ1 ]lv , and thus in DS , we also have the state equation e(τ, T ) − cv T =

a (τ − τl − τl ). τl τv

(1.132)

Entropy as a function of (τ, T ). As seen above, the entropy is the interpolant of the liquid vapor entropies at given temperature in the domain DS : s(τ, T ) = s(ξ τl + (1 − ξ )τv , T ) = ξ sl (τl , T ) + (1 − ξ )sv (τv , T ),

(1.133)

according to (1.130), with the variable T changed into e. We can also write s(τ, T ) = ξ r log(τl − τ 0 ) + (1 − ξ )r log(τv − τ 0 ) + cv log(cv T ) + s0 . Eliminating ξ by ξ = s(τ, T ) = r

τ −τv τl −τv ,

(1.134)

we obtain for τl ≤ τ ≤ τv , T < Tc ,

τl − τ 0 (τ − τv ) log + r log(τv − τ 0 ) + cv log(cv T ) + s0 . τl − τv τv − τ 0

(1.135)

Further Information About the Saturation Domain We sum up the relations on τl , τv . The saturation curve S that bounds the set of liquid–vapor equilibrium points in the space of (P , T ) (or (P , T , τ )) is defined by the set of values (P , T ) (respectively (P , T , τv ) and (P , T , τv )) such that  (PS (T ), T , τl ) = 0, (PS (T ), T , τv ) = 0,

τv

(PS (T ), T , τ ) dτ = 0.

τl

This last relation corresponds to the equality of areas and implicitly defines the function PS (T ) for T ≤ Tc . It can be written P (τv − τl ) − a(τv−1 − τl−1 ) − rT log

τv − τ 0 = 0. τl − τ 0

Therefore, with T < Tc given, the three variables P , τv , τl are determined from T by the three equations

1.5 Fluid with the van der Waals Equation

61

(P + aτl−2 )(τl − τ 0 ) − rT = 0, (P + aτv−2 )(τv − τ 0 ) − rT = 0,

(1.136)

τv − τ 0 a )(τv − τl ) − rT log = 0, (P + τl τv τl − τ 0 which define the saturation curve S by PS (T ), τv (T ), τl (T ). We can eliminate T (respectively P ) by the two first equations, which gives P =

  1 1 a 1 − τ 0( + ) , τv τl τl τv

τ0 1 1 τ0 rT = a( − 1)( − 1)( + ). τl τv τl τv

(1.137)

Now we specify some properties of τl , τv , and the saturation domain. First, we have, with the above notation, τ 0 < τl (T ) < τm (T ) < τc < τM (T ) < τv (T ),

0 < T < Tc .

(1.138)

To prove that τc is between the extrema τm and τM , we have only to check that P  (τc , T ) is positive. Since T < Tc , we obtain P  (τc , T ) = −

rT a rT 6 6 +2 3 =− + 3 > 0. 0 2 2 (τc − τ ) τc rTc τc τc

We also have (i) τl (P , T1 )) > τl (P , T2 ),

τv (P , T1 ) < τv (P , T2 )

(ii) τl (P1 , T ) < τl (P2 , T ),

if P1 > P2 > PS (T ), if PS (T ) > P1 > P2 ,

τv (P1 , T ) < τv (P2 , T ), (iii) PS (T1 ) > PS (T2 ),

if Tc > T1 > T2 > 0,

if Tc > T1 > T2 > 0, (1.139)

that is, τl (P , T ) is an increasing function of T , τv (P , T ) is a decreasing function of T at fixed P , Pc > P > 0; τl (P , T ) is a decreasing function of P , τv (P , T ) is an increasing function of P at fixed T , Tc > T > 0; PS (T ) is an increasing function of T . This is a simple consequence of the inequality P (τ, T1 ) > P (τ, T2 ) when T1 > T2 for (i), a balance of areas for (iii). Furthermore, τl = τl (PS (T ), T ) is an increasing function of T , and τv = τv (PS (T ), T ) is a decreasing function of T : ∂τl > 0, ∂T

∂τv < 0, ∂T

0 < T < Tc .

(1.140)

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1 Thermostructure

Proof τl is defined by the relation PS (T ) = P (τl , T ), which we derive with respect to T . Thus ∂P ∂τl ∂P dPS − − = 0, dT ∂τ ∂T ∂T Now

∂P ∂τ

therefore

∂P ∂τl dPS ∂P = − . ∂τ ∂T dT ∂T

< 0 in a neighborhood of τl . Besides, we have (see (1.151) below) 

τv

( τl

∂P dPS − ) dτ = dT ∂T



τv

( τl

1 dPS −r ) dτ = 0, dT τ − τ0

and thus dPS (τv − τl ) = dT



τv

r τl

1 ) dτ, τ − τ0

so that dPS ∂P 1 − = dT ∂T τv − τl



τv

r τl

1 1 ) dτ − r < 0, 0 τ −τ τ − τ0

since τl − τ 0 < τ − τ 0 with τl < τ < τv . As a consequence,

∂τl ∂T

> 0. ¶

Behavior of τl , τv as P and T tend to 0. The van der Waals equation (1.92) reads P τ 3 − (P τ 0 + rT )τ 2 + aτ − aτ 0 = 0.

(1.141)

Let X = τ1 . Then with respect to X, this equation is P − (P τ 0 + rT )X + aX2 − aτ 0 X3 = 0. When P = 0 and T = 0, it is reduced to aX2 (1 − τ 0 X) = 0, thus giving X = 0 and X = τ10 , hence τv = +∞, τl = τ 0 . Then we can verify that as P and T tend to 0, τv → +∞ and τv → τ 0 . But the modelling of a fluid by the van der Waals equation stops before T = 0, since there is no modelling of the transition in it from liquid to solid. Neighborhood of the critical point (Pc , Tc , τc ). From (1.136), we have, with [τ ]lv = τl − τv , the jump of τ across S (and similarly for τ1 , τ12 ): 1 1 (i) P [τ ]lv + a[ ]lv − aτ 0 [ 2 ]lv = 0, τ τ 1 (ii) − P [τ ]lv + a[ ]lv + rT [log(τ − τ 0 )]lv = 0. τ

(1.142)

We eliminate P by summing, then keeping the second relation of (1.137); we have

1.5 Fluid with the van der Waals Equation

63

1 1 (i) 2a[ ]lv − aτ 0 [ 2 ]lv + rT [log(τ − τ 0 )]lv = 0, τ τ τ0 τ0 1 1 (ii) rT = a( − 1)( − 1)( + ). τl τv τl τv

(1.143)

In a neighborhood of τc we have, using τc − τ 0 = 2τ 0 = 23 τc , log

τl − τ 0 τl − τc + 2τ 0 τl − τc τv − τc = log = log (1 + ) − log (1 + ). 0 0 0 τv − τ τv − τc + 2τ 2τ 2τ 0

Let τl = 1 − l , τc

τv = 1 + v , τc

T = 1 − T , Tc

P = 1 − P . Pc

(1.144)

Thus log

τl − τ 0 τl − τv [τ ]lv 3 ≈ ≈ ≈ − (l + v ). 0 0 0 τv − τ 2τ 2τ 2

With the relation rTc =

8 a 27 τ 0

=

8 a 9 τc ,

then (1.143)(i) becomes

  rT 1 τc 3 τc τc τc 2− ( + ) . ≈ rTc 4 τl τv 3 τl τv

(1.145)

Then using (1.144) in this equation and in (1.143)(ii), we have (i)

rT 3 1 1 ≈ 1 − (v − l ) + (v2 + l2 ) − l v , rTc 4 2 2

(ii)

rT 1 1 ≈ 1 − (v − l )2 − l v . rTc 4 4

(1.146)

Subtracting (ii) from (i), we obtain v − l ≈ (v2 + l2 ) − l v = l v + (v − l )2 , whence v − l ≈ l v . From the relation 4l v = (v − l )2 + (v + l )2 , we again obtain v − l ≈ (

v + l 2 ) . 2

(1.147)

64

1 Thermostructure

From (1.147) it is trivial that v (1 − l ) ≈ l , whence v ≈ l (1 + l ), meaning that v and l are infinitesimal of the same order. From (1.146) and (1.147) we obtain at once rT 1 ≈ 1 − (v + l ), rTc 4

thus T ≈

1 (v + l ). 4

(1.148)

Now from (1.142) and (1.144), we have, with (1.144), P ≈ 1 − (v − l )2 + v l ≈ 1 + v l , Pc

thus P ≈ v l ≈ 4T .

(1.149)

We can obtain this last relation directly from the van der Waals equation (1.141) with τ = 1 + , so that (1.141) is 1 P (1 + )3 − (P + 8T )(1 + )2 +  3 ≈ 0, 3 giving P − 13 (P + 8T ) ≈ 0, and thus P ≈ 4T . Now we should compare these results with experiments to see whether the van der Waals equation is a good model in a neighborhood of the critical point. Remark 12 Clausius–Clapeyron equation; heat of transition. The quantity of heat in the transition liquid–vapor at temperature T is given by the jump of enthalpy  Qlv =

d e˜ + PS (T )dτ = ev − el + PS (T )(τv − τl ) = hv − hl T

a τv − τ 0 = (τv − τl )( + PS (T )) = rT log > 0, τl τv τl − τ 0

(1.150)

where T is a path joining (el , τl , PS (T )) to (ev , τv , PS (T )). We have the relation Qlv = T (sv − sl ). Furthermore, writing the equality of areas for the function (P , T , τ ) with van der Waals, see (1.93), 

τv

(PS (T ), T , τ ) dτ = 0,

τl

then taking the derivative with respect to T , with (PS (T ), T , τl ) (PS (T ), T , τv ) = 0, we obtain 

τv τl

and thus

d (PS (T ), T , τ ) dτ = dT



τv

( τl

1 dPS −r ) dτ = 0, dT τ − τ0

=

(1.151)

1.5 Fluid with the van der Waals Equation

65

τv − τ 0 dPS − r log = 0, dT τl − τ 0

(1.152)

dPS Qlv 1 hv − hl = = , dT T (τv − τl ) T τv − τl

(1.153)

(τv − τl ) and therefore

which is the so-called Clausius–Clapeyron formula, which gives the slope of the curve PS (T ) as a function of the heat of transition. In a neighborhood of the critical point, these relations are Qlv ≈ rTc τv2τ−τ0 l . Then Qlv ≈ 4Tc [τ ]vl ,

and

r dPS ≈ 0 ≈ 4. dT 2τ

¶

Determination of Other Thermodynamic Functions Determination of the concave Gibbs free energy. From g(P , T ) = T s∗ ( PT , T1 ) (see (1.48)), we know that g(P , T ) is a concave function of (P , T ). Thus from the properties of s˜ ∗ , we know that it is continuous across the saturation curve S . Then we have the following results. (i) When T ≥ Tc , there exists one value τ such that P − P (τ, T ) = 0. Hence g(P , T ) is expressed by g(P , T ) = P τ −

a − rT log(τ − τ 0 ) + ϕ(cv T ) − T s0 . τ

(1.154)

(ii) When T < Tc , we recall that • (i) P > PS (T ), implies infτ >τ 0 ψP ,T (τ ) = ψP ,T (τl ), • (ii) P < PS (T ), implies infτ >τ 0 ψP ,T (τ ) = ψP ,T (τv ). Let ˜ , T , τl ), gl (P , T ) = g(P

gv (P , T ) = g(P ˜ , T , τv ).

(1.155)

Therefore, when liquid and vapor are in equilibrium, we have  gv (P , T ) − gl (P , T ) = (τv − τl )P −

τv

P (τ, T ) dτ = 0.

τl

When T < Tc , g(P , T ) is expressed • (i) by P > PS (T ), then g(P , T ) = gl (P , T ); • (ii) by P < PS (T ), then g(P , T ) = gv (P , T );

(1.156)

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1 Thermostructure

• (iii) by P = PS (T ), then g(P , T ) = gl (P , T ) = gv (P , T ), and for P = PS (T ), we have g(P , T ) = f (τl , T ) + P τl = f (τv , T ) + P τv .

¶

Remark 13 The Gibbs free energy g(P , T ) defined in this way, being a concave function of (P , T ), if −P is in the subdifferential30 ∂fT (τ ) of the function fT (τ ), then we have fT∗ (P ) + fT (τ ) = −P τ , and thus f (τ, T ) − g(P , T ) = −P τ for τ > τ 0 , −P ∈ ∂fT (τ ).

¶

(1.157)

Remark 14 Multivalued Gibbs free energy. The map (τ, s) → (P , T ) given by (1.118) is not invertible on the set of pairs {(P , T ), T < Tc ,

Pm (T ) < P < PM (T )},

with Pm (T ) = P (τm (T )), PM (T ) = P (τM (T )). Let sj and ej be the entropy and energy corresponding (by (1.118) and (1.101)) to the solution τj of the van der Waals equation, with j = 1, 2, 3, and τ1 < τ2 < τ3 . We define gj (P , T ) = g(P ˜ , T , τj ) = e(τj , sj ) + P τj − T sj (also called the Gibbs free energy); gj (P , T ) is a differentiable function of (P , T ) (on their domains) such that dgj = τj dP − sj dT , thus satisfying θ gj = 0. In fact, g2 (P , T ) has no physical sense, whereas gl (P , T ) = g1 (P , T ) (respectively gv (P , T ) = g3 (P , T )) extended continuously by g(P , T ) for {(P , T ), T < Tc , P < PS (T )},

respectively {(P , T ), T < Tc , P > PS (T )},

represents the Gibbs free energy for the liquid (respectively gas) state. A Gibbs free energy gˆ for the set of states is defined by g(P ˆ , T ) = g(P , T ) for T < Tc , P > PM (T ) and P < Pm (T ) = (gl (P , T ), gv (P , T )) for T < Tc , P ∈ [Pm (T ), PM (T )].

(1.158)

This is a multivalued function of (P , T ) on the domain representing metastable states, on Pm (T ) < P < PS (T ) for the liquid states, and PS (T ) < P < PM (T ) for the gas states. ¶ Remark 15 On the saturation curve, the equation θ g = dg − τ dP + sdT = 0 is to be replaced by31 30 Recall

that the function fT (τ ) is not convex. results from the formula (see [Aub][App. 17.6, p. 397]) ∂(g ◦ A)(x) = A∗ ∂g(Ax), with A a linear map; here A(P , T ) = (−P , T ).

31 This

1.5 Fluid with the van der Waals Equation

67

(−τ, s) ∈ ∂(−g)(P , T ), which is equivalent (see [Aub][ch. I.4.2, p. 56]) to the relation g(P , T ) = e˜(τ, s) − T s + P τ . Therefore, (−τl , sl ) ∈ ∂(−g)(P , T ), and (−τv , sv ) ∈ ∂(−g)(P , T ). ¶ With (1.121) and (1.93), the τ derivative of g(P ˜ , T , τ ) is g˜τ (P , T , τ ) =

dψP ,T ∂ g˜ (P , T , τ ) = (τ ) = P − P (τ, T ). ∂τ dτ

(1.159)

Convexification of the energy. We recall the relation of g with the conjugate functions of e(−P , T ), g(P , T ) = −e∗ (−P , T ). Then we obtain the convexification of the energy e˜(τ, s) = e∗∗ (τ, s) (also called the -regularization of the energy e(τ, s)), i.e., the epigraph of the function e(τ, ˜ s) is the convex envelope of the epigraph of the function e(τ, s). In the domain DS , e(τ, ˜ s) is obtained by interpolation: e(τ, ˜ s) = ξ e(τl , sl ) + (1 − ξ )e(τv , sv ),

(1.160)

with τ = ξ τl + (1 − ξ )τv and s(τ, T ) = s = ξ sl + (1 − ξ )sv ) (see (1.133)), which gives e(τ, ˜ s) =

s − sv sl − s e(τl , sl ) + e(τv , sv ). sl − sv sl − sv

Convexification of the free energy fT (τ ) with fixed T . Several methods are possible. We can determine f from e and s using the relation f = e − T s; thus we have the function f (τ, T ) (which is not convex in τ for all fixed T for the not convex functions e and s˜ = −s), given thanks to (1.101) and with ϕ(u) = u − u log u: f (τ, T ) = −

a − rT log(τ − τ 0 ) + ϕ(cv T ) − s0 T . τ

(1.161)

Then we can use the convexification of e and s˜ to obtain the convexification of fT (τ ) with fixed T . Now we can directly find this function. With given T < Tc , (τl , fT (τl ) and (τv , fT (τv )) are the connection points of the epigraph of fT with its convex envelope, which is the epigraph of a function f˜T . We verify that f˜T is defined for τ ∈ [τl , τv ] by interpolation, i.e., by f˜T (τ ) = ξfT (τl )+(1−ξ )fT (τv ) pour τ = ξ τl +(1−ξ )τv , 0 ≤ ξ ≤ 1,

(1.162)

and by f˜T (τ ) = fT (τ ) outside of this interval, the domain on which the function fT (τ ) is convex and smooth.32 Thus we replace the nonconvex part of the epigraph 32 f˜ (τ ) T

= fT∗∗ (τ ) is the -regularization of the function fT (τ ).

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1 Thermostructure

of fT by a straight line tangent at its two extremes to the graph of fT . Let (τl , yl ) and (τv , yv ) be two points with yl = fT (τl ) and yv = fT (τv ). The slope p of the straight line is such that p=

∂fT ∂fT (τl ) = (τv ) = −PS (T ), ∂τ ∂τ

and p =

yl − yv . τl − τv

Using the formulas (on the saturation curve) fT (τl ) + f˜T∗ (P ) = −P τl ,

−P ∈ ∂ f˜T (τl ),

fT (τv ) + f˜T∗ (P ) = −P τv ,

−P ∈ ∂ f˜T (τv ),

(1.163)

and multiplying the first relation by ξ and the second by 1 − ξ , we obtain, thanks to (1.162), in DS , f˜T (τ ) + f˜T∗ (P ) = −P τ.

(1.164)

1.5.3 Integrability and Convexity for van der Waals The modelling of the thermodynamic state of a system by the van der Waals equation (1.92) is not physically acceptable for stable states in the saturation domain DS , where liquid and vapor can coexist. Corresponding to the state variables (τ, e) ∈ DS , the pressure must be replaced as state variable by the variable ξ ∈ [0, 1], giving (τ, e) by (1.131). Let  be the map (ξ, T ) = (τ, e). Then the differential forms dτ and de are given by ∂τv ∂τl + (1 − ξ ) ], ∂T ∂T ∂ev ∂el de = [e]lv dξ + [ξ + (1 − ξ ) . ∂T ∂T

dτ = [τ ]lv dξ + [ξ

Then the basic differential form θ s = ds −

PS T dτ

−

1 T de

(1.165)

becomes

 ∗ (θ s ) = ds − α(T )dξ − β(ξ, T )dT , with PS 1 [τ ]lv + [e]lv , T T 1 ∂el PS ∂τv 1 ∂ev PS ∂τl + ) + (1 − ξ )( + ). β(ξ, T ) = ξ( T ∂T T ∂T T ∂T T ∂T

α(T ) =

(1.166)

1.5 Fluid with the van der Waals Equation

We have α(T ) =

Qlv T

69

= [s]lv , and PS ∂τl 1 ∂el ∂sl = + ) ∂T T ∂T T ∂T

with sl and similarly with sv , so that β(ξ, T ) = ξ

∂sv ∂sl ∂sv ∂sv ∂sl + (1 − ξ ) = ξ( − )+ . ∂T ∂T ∂T ∂T ∂T

Thus ∂α ∂β(ξ, T ) ∂sl ∂sv = =( − ), ∂T ∂ξ ∂T ∂T so that the differential form  ∗ (θ s ) is integrable on [0, 1] × [T1 , Tc ] (with the temperature T1 such that P (τ, T ) > 0, ∀T ∈ [T1 , Tc ]).

Foliation for van der Waals The family (Ls0 ), s0 ∈ R, of integral manifolds of the map (1.99) in the space  ⊂ F = M × R × N is made up of the jets of the entropy defined by (1.101), and is a foliation of the space  of the thermodynamic states satisfying equations (1.92) and (1.116) (as in the case of an ideal gas). The projection π of F on M × R transforms the family (Ls0 ) into the family (Gs0 ) of the graphs of the entropy, which gives a foliation of the space M × R. But they are not the graphs of a concave function. Then taking the stability of states into account, we have to replace the leaves ˜ s0 ) corresponding (Ls0 ) (respectively (Gs0 )) by the leaves (L˜ s0 ) (respectively (G ∗∗ to the jets (respectively graphs) of concave functions −˜s defined by (1.130) and (1.129), with s˜∗∗ the convexification of the opposite of the entropy. This is possible because the same value of s0 is on each part of the saturation domain DS . ˜ corresponding to the new Then (L˜ s0 ) constitutes a foliation of the space  ˜ , with state equations in the saturation domain DS . The foliation (L˜ s0 ) allows us to define reversible evolutions, and thus a reversible way from liquid state to vapor state or conversely, but the smoothness is not of class C 2 . The variation of the variable s0 indicates an irreversible evolution according to the second law of thermodynamics.

Phase Transition in Global or Local Thermodynamics Let N be the number of moles of a substance occupying a domain  of volume V , with Vl for the liquid state and Vv for the gas state, under the same conditions

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1 Thermostructure

of pressure and temperature. Thus the variable ξ such that V = ξ Vl + (1 − ξ )Vv indicates the percentage of volume occupied by the liquid state. In local thermodynamics, the variable ξ can represent a limit of percentage of volume occupied by the liquid state in a family of neighborhoods V (x) of a point x of the space, or a stage of progress of the change of phase, indicated in the diagram (P , τ ) by τ = ξ τl + (1 − ξ )τv , i.e., a point of the plateau. This variable ξ can come from a problem of homogenization if the local separation between liquid and vapor is due to the existence of numerous little bubbles of vapor in the liquid. Remark 16 Change of phase and chemical reaction. We compare a change of phase to a chemical reaction. We consider, as a typical change of phase, the transformation of water from the liquid state to vapor: H2 0 (liq)  H2 0 (vap). If here ξ is the step of advancement of the reaction, we have (see (1.280)) dξ =

dNv dNl = , −1 1

(1.167)

or also, on dividing by N = Nl + Nv , which is a constant, d ξ˜ = −

dNv dNl = , N N

with ξ˜ =

ξ ∈ [0, 1]. N

Therefore, the numbers of moles of liquid and of vapor, here denoted by Nl,ξ˜ and Nv,ξ˜ , are such that Nl,ξ˜ = (1 − ξ˜ )N,

Nv,ξ˜ = ξ˜ N.

(1.168)

The total volume of the fluid domain during the phase transition is expressed by V = (1 − λ)Vl + λVv ,

λ ∈ [0, 1],

or also by V = Vl,λ + Vv,λ , with Vl,λ = (1 − λ)Vl ,

Vv,λ = λVv .

(1.169)

Now we have dNl = nl dVl,λ ,

dNv = nv dVv,λ

with nl and nv independent of λ, nl (respectively nv ) being the larger (respectively smaller) root of the van der Waals equation at given P , T : (P = an2 )(1 − bn) − R0 T n = 0.

1.6 Thermosystems with Different Constituents

Then with λ = ξ˜ , with (1.168) and (1.169), we have

71 Nl,ξ˜ Vl,ξ˜

= nl ,

Nv,ξ˜ Vv,ξ˜

= nv . Thus

we see that the parameter λ corresponding to the value V of the volume of fluid in the liquid vapor transition can be identified with the stage of progress (divided by N) of the chemical reaction of the liquid–vapor transition. But the chemical kinetic equation for the speed of the reaction (similar to (1.289) below) does not seem to be valid; this speed is essentially related to the speed of bringing heat from liquid to vapor. ¶

1.6 Thermosystems with Different Constituents We deal with systems with several constituents Ci , i = 1, . . . , n, each characterized at the microscopic level by a molecule Mi (with a molar mass mi and with Ni moles in the volume V of the system). Each molecule can be dissociated into atoms and combined into new molecules thanks to chemical reactions, and then the constituents and their numbers are changed. There can be also changes of phase (such as the transition from a solid state to a liquid state), with new mean positions of molecules and new internal physical structures. In the chosen model, there is only one variable of temperature and at most only one variable of pressure.

1.6.1 Local Thermodynamic Modelling The local thermodynamic modelling of a system with n+1 degrees of freedom (with n constituents at most) can be done in several ways. With variables s, τ, e, T , P , there occur the mass fractions33 of the constituents (ck ), and the corresponding chemical potentials (μck ) (which are dual variables of the mass fractions), or also the specific masses (ρk ) of the constituents, and the new corresponding chemical ρ potentials (μ˜ k ). Modelling with Specific Masses or with Mass Fractions The set of variables (ρ, e, (ck )) (or (τ, e, (ck )), τ = 1/ρ) is the convex space M = ¯ n−1 , with R + =]0, +∞[, and with the simplex34 R+ × R × 

call the ratio ck = ρρk of the specific mass ρk of the k-constituent to the specific total mass ρ the mass fraction of the constituent. 34 Note that  ¯ n−1 is a manifold with boundary. In what follows in this section, the sum on the subscript k will be taken from k=1 to n. 33 We

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¯ n−1 = {cˆ = (ck ), 0 ≤ ck ≤ 1, 



ck = 1}

if the vanishing of at least one constituent is allowed, or its interior if the system must always have n constituents: ˆ M◦ = R + × R × n−1 = {(ρ, e, c),

cˆ = (ck ), 0 < ck < 1,



ck = 1}.

n Then the set of variables (e, (ρk )) is the space M = R × (R+ \ {0}), with R+ = [0, +∞[. Using “polar coordinates” for the specific masses, we transform the space n R+ \ {0} into the product space R + × n−1 (for the set of ρ, (ck ) or of τ, (ck )). The set of variables (P , T , (μck )) (or ( PT , T1 , (μck ))) is the convex space N = R × R+ × Rn . Thus the set of thermodynamic states of a system with n constituents is a subspace of F = M × R × N .

Differential Forms of Matter θ mat , of Heat θ Q We assume that the heat is still defined by the differential form θ Q = de + P dτ. We associate to the variation of matter the differential forms  ρ  μ˜ k dρk , θ c = θ mat,c = μck dck , θ mat = θ mat,ρ =

(1.170)

respectively, for the specific masses dρk , and for the mass fractions dck , having ρ as coefficients chemical potentials denoted by μ˜ k or μck or μ˜ k . The variables ck  are not independent, since ck = 1, whence the dual variables, i.e., the chemical potentials are not defined in a unique way: we can add to each of the chemical potentials μck the same function (of all variables) without any change in θ mat,c . In order to have only independent variables, we could eliminate one of the mass fractions (cn , for example) as a function of the others. The drawback of this method is that it breaks the symmetry of the formulas between constituents and leads to replacing the chemical potentials μk with relative chemical potentials such as μrk = μk − μn , k = 1, . . . , n − 1. We change the differential form θ mat,ρ into θ mat,c using “polar coordinates” for the specific masses, that is, ρk = ck ρ, ρ

We can define μck from μ˜ k by

with ρ =



ρk =

1 . τ

(1.171)

1.6 Thermosystems with Different Constituents

73

ρ

μck = μ˜ k ρ, k = 1, . . . , n.

(1.172)

Thus we obtain θ mat,ρ =



ρ

μ˜ k dρk =



ρ

μ˜ k ρ dck +



ρ

μ˜ k ck dρ.

(1.173)

We define the pressure P by  ρ  ρ μ˜ k ρk = μck ρk , μ˜ k ck = ρ  ρ  hence τ P = μ˜ k ρk = μck ck .

P =ρ 2



(1.174)

Then we have the relation θρ = θc +

P dρ = θ c − P dτ, ρ2

(1.175)

using (1.171). Thus we see that the formulas of the differential forms θ mat,ρ and θ Q both contain the usual term of work θ W = −P dτ .

Various Differential Forms and Parametrizations Parametrizations based on the energy.   With variables τ, s, (ck ); e; P , T , (μck ) , we have a straight extension of the parametrization (τ, s; e; P , T ) with one constituent. The differential form θ e,c , similar to θ e for one constituent, is  θ e,c = θ e + θ c = −de − P dτ + T ds + μck dck . (1.176)  ρ  ˜ k )) , and with ρˆ = (ρk ), the differential form With variables (s, (ρk )); e; (T , (μ θ e,ρ then is  ρ θ e,ρ = −de + T ds + θ ρ = −de + T ds + μ˜ k dρk . (1.177) Parametrization based on energy by unit of volume e˜ = ρe. Let s˜ = ρs. Multiplying the differential form (1.177) by ρ, we obtain ˜ = ρθ e,ρ = −d e˜ + T d s˜ + θ˜ e,ρ



μ˜ k dρk ,

(1.178)

with ρ

ρ

μ˜ k = ρ μ˜ k + e − T s = ρ μ˜ k + f = μck + f.

(1.179)

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Parametrization based on the entropy s. The differential forms based on the entropy s with the mass fractions are  μc P 1 1 k de − dτ + dck = θ s + θ mat,c , T T T T  μ˜ ρ 1 k = ds − de + dρk . T T

θ s,c = ds − θ

s,ρ

(1.180)

ρ

ˆ (so that θ s,ρ = Note that the chemical potential μ˜ k is linked to the entropy s(e, ρ) 1 ρ ∂s 0) by T μ˜ k = − ∂ρk . Then the pressure P , defined by (1.174), is obtained from s by  1 ∂ τ P = −s with  = ρk , T ∂ρk

(1.181)

where  is the Euler map (here with constant e). Parametrization based on the free energy f . The differential forms based on the free energy f (with f = e − T s) expressed with (ck ) or with (ρk ) are  μck dck , θ f,c = −df − P dτ − sdT +  ρ θ f,ρ = −df − sdT + μ˜ k dρk .

(1.182)

Parametrization based on the Gibbs free energy g. The differential form θ g,c based on the Gibbs free energy g = f + P τ = e − T s + P τ expressed with (ck ) is expressed by θ g,c = dg + sdT − τ dP −



μ˜ k dck .

(1.183)

If we want to return to variables ρk instead of ck , using relation ck = τρk , we derive once again the differential form θ f,ρ of (1.182). ¶ Remark 17 The Duhem differential form. We first give a relation between g and the chemical potentials. We have g = f + τP = f +



ρ

ρk μ˜ k =



ρ

ck (f + ρ μ˜ k ) =



ck μ˜ k .

(1.184)

Let  be a maximal integral manifold of θ g,c . Differentiating (1.184), then subtracting it from (1.183), we obtain the local Duhem formula θ D,c | = sdT − τ dP + Comparing with (1.176), we note that



ck d μ˜ k = 0.

(1.185)

1.6 Thermosystems with Different Constituents

θ e,c + θ D,c = d(T s − e − P τ +



75

ck μ˜ k ) = d(T s − e − P τ − g) = 0.

Multiplying (1.185) by ρ, the Duhem formula is, with s˜ = ρs, θ D,ρ | = dP − s˜ dT −



ρk d μ˜ k = 0.

(1.186)

With (1.178) we have θ˜ e˜,ρ − θ D,ρ = 0. Thus the Duhem formula is equivalent to P (T , (μ˜ k )) is the conjugate function of the weighted energy e, ˜ with e˜ a (convex)  function of (˜s , (ρk )) and such that P + e˜ = s˜T + ρk μ˜ k , according to the Legendre involution. ¶

State Equations and the Map  As in the case of one constituent, there are state equations coupling the temperature and the chemical potentials to the energy and the specific masses. These equations are independent of the entropy, and thus we have to use the differential form θ s,ρ . In most cases (ideal gases, van der Waals fluids), the state equations are summed by a map such that ρ

μ˜ 1 (e, (ρk )) = ( , − k ) T T

μ˜ k 1 ˜ e, or ( ˜ (ρk )) = ( , − ). T T

These maps are deduced from the individual state equations and from some required properties, also in relation to global thermodynamics. The first required property is that they be integrable, i.e., that ∗ (θ s,ρ ) be exact: there is a function s0 ((e, (ρk )) such that ds0 = ∗ (θ s,ρ ). Thus the Schwarz conditions must be satisfied, which implies conditions on the state equations with the chemical potential and the temperature.

Convexity Properties with Respect to Mass Fractions Here we give some local convexity properties that are deduced from the property of the map , whose opposite of its Jacobian J ( ) is positive definite. This will also be deduced from convexity properties of global thermodynamics. The energy e(τ, s, c) ˆ is a convex function of cˆ ∈ n−1 (and still of (τ, s)), and s(e, τ, c) ˆ is a concave function of cˆ ∈ n−1 . From e(τ, s, c), ˆ we can define by Legendre transformation with fixed mass fractions the free energy f (τ, T , c), ˆ and then the Gibbs free energy g(P , T , c) ˆ by f (τ, T , c) ˆ = inf (e(τ, s, c) ˆ − T s), s∈R

g(P , T , c) ˆ = inf(e(τ, s, c) ˆ + P τ − T s) = inf(f (τ, T , c) ˆ + P τ ). τ,s

τ

(1.187)

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1 Thermostructure

With index notation, we have ec,τ ˆ (s) − fc,τ ˆ (T ) = T s,

fc,T ˆ (τ ) − gc,T ˆ (P ) = −P τ,

and also ecˆ (τ, s) = sup(−P τ + T s + gcˆ (P , T ), P ,T

ecˆ (τ, s) − gcˆ (P , T ) = −P τ + T s.

Thus f (τ, T , c) ˆ is a concave function of T , a convex function of τ , and a convex function of c; ˆ g(P , T , c)) ˆ is a concave function of (P , T ) and a convex function of c. ˆ ¶

Relations Between Global Functions and Individual Functions We make the following hypotheses: (i) To each constituent Ck , we associate Fk = Mk × R × Nk , a 5-dimensional space, with thermodynamic variables (ek , τk ; sk ; Pk , Tk ). Now we assume with Hypotheses 1 and 2 that there is a diffeomorphism k from the space Mk (of variables (ek , τk )) onto the space Nk (of (Pk , Tk ), or ( PTkk , T1k )), given by two state equations: 1k , 2k . This determines a 3dimensional subspace  k of Fk . With the basic form θk = θ sk , on  k is associated, as previously, an ˜ k = ( k , Q k ) (with leaves integrable bundle Q k , and thus a foliation  35 (Ls k )), of the space  k . 0 (ii) The variables of the system denoted by (e, τ, s, T , P ), along with the mass fractions and the chemical potentials, have the following properties: -

the temperature Tk = T is the same for all constituents;  the pressure P of the system is the sum of partial pressures P = Pk ; the specific mass is such that ρ = ρk ; ρk = ck ρ, whence τ = ck τk , ∀k; the energy e of the system is e = ck ek ;  the entropy s of the system is such that s = ck sk .

Thus   Tk = T , P = Pk , ρ = ρk ,   e= ck ek , s = ck sk .

35 Recall

τ = ck τk , ∀k, (1.188)

that this is a family of maximal integral manifolds of Q k made up of the jets J (sk ) of the “partial” entropy sk (ek , τk ), with variable s0k ∈ R as index.

1.6 Thermosystems with Different Constituents

77

Relation between individual differential forms and the differential form of the system. Under these hypotheses, we have θs =



ck θ s k +

 μc gk ( k − )dck . T T

(1.189)

Proof Indeed, we have  Pk  μc 1  d( ck ek ) − d(ck τk ) + ( k dck T T T c   μ gk 1 Pk dτk ) + = ck (dsk − dek − ( k − )dck , T T T T

θ s = d(



ck sk ) −

(1.190)

with gk = ek +Pk τk −T sk . This implies (1.189). With the condition μ˜ k = μck +f = gk , we have the following relation between the individual differential forms and that of the system: θs =



ck θ s k .

(1.191)

θ -Admissible Evolutions and θ -Reversible Evolutions According to the second law, a necessary condition for the evolution γ of the thermodynamic variables of a system in F with vector Xγ (t ) to be admissible is that it satisfy (for every time) < θ s,c , Xγ (t ) > ≥ 0,

(1.192)

that is < θ s,c , X > =

1 de P dτ  μ˜ k dck ds − − + ≥ 0. dt T dt T dt T dt

(1.193)

Hence with fixed concentrations, the previous condition gives the usual relation < θ s,c , X > =

1 de P dτ ds − − ≥ 0. dt T dt T dt

(1.194)

In the general case, the problem is to know whether the differential form  μc  μ˜ P 1 1 k k de + dτ − dck = de − dρk T T T T T ρ

θ0 =

is exact. We will study this in the case of a system with ideal gases.

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1 Thermostructure

1.6.2 Global Thermodynamic Modelling In the framework of global thermodynamics, the volume V of the bounded domain  of the system is a variable that is a new parameter with respect to the local thermodynamics. The usual thermodynamic variables are V , E, (Nk ), S, P , T , (μk ), Nk being the mole numbers, μk = mk μ˜ k the chemical potentials. We denote by M the space of variables (V , E, (Nk )), by N the space of (P , T , (μk )). We also will use the variables Mk and μ˜ k instead of Nk and μk . The global free energy F and the global Gibbs free energy G are expressed by F = E − T S,

G = E + PV − T S = F + PV.

(1.195)

Global Basic Differential Forms   μ˜ k dMk = μk dNk represents the global The differential form θ Mat = variation of matter. The differential form of heat is θ Q = dE + P dV . The global basic differential forms are expressed by ˆ

θ E,M = θ E + θ Mat = −dE − P dV + T dS + ˆ

θ S,M = θ S +



μ˜ k dMk ,

1 P 1 Mat 1 θ μ˜ k dMk . = dS − dE − dV + T T T T ˆ

(1.196)

ˆ

Likewise, we use the notation θ S,N , θ E,N , . . . for the corresponding differential forms expressed with Nk and chemical potentials μk . The differential forms with global free energy F and global Gibbs free energy G are ˆ



G,Mˆ



θ F,M = θ F + θ Mat = −dF − SdT − P dV + θ

=θ +θ G

Mat

= −dG − SdT + V dP +

μ˜ k dMk , (1.197) μ˜ k dMk .

Additivity Properties Additivity properties are induced by the following hypothesis, which relates global thermodynamic   variables of the system to the individual ones, with Tk = T , M = Mk , N = Nk : P =



Pk ,

E=



Ek ,

S=



Sk , ∀k.

Let θ Sk be the differential form relative to the Ck component such that

(1.198)

1.6 Thermosystems with Different Constituents

θ Sk = dSk −

79

μk 1 Pk dEk − dV + dNk . T T T

(1.199)

Taking Sk = sk Nk , Ek = ek Nk , V = τk Nk ,36 we obtain θ Sk = Nk θ sk = Nk (dsk −

μk 1 Pk Gk dek − dτk ) + (− + )dNk , T T Nk T T

(1.200)

k with Gk = Ek +Pk V −T Sk . Then if μk = G Nk = gk , we obtain the relation between the global basic differential form and the individual differential forms

ˆ

θ S,N =



θ Sk =



Nk θ sk .

(1.201)

With Hypotheses 1 and 2 of a diffeomorphism k (due to the individual state equations) from Mk onto Nk , we have that T1 dek + PTk dτk is the differential of a function sk,rev . Then θ sk = dsk − dsk,rev = dsk,0 and ˆ

θ S,N =

 ˆ

d Thus the (irreversible) evolution < θ S,N , dt >=

according to the second law, but S1 =



Sk1 =



Nk sk,0 ,

ˆ θ S,N

(1.202)

Nk dsk,0 . 

Nk

dsk,0 dt

is only positive

is not an exact differential form. Let

G1k = Ek + Pk V − T (Sk − Sk1 ),

(1.203)

μ0k = μk + T sk,0 = gk + T sk,0 = gk1 . Then we have ˆ

θ S,N =



Nk dsk,0 = dS 1 −



sk,0 dNk ,

and ˆ

θ S,N +



sk,0 dNk = dS −

 μ0 P 1 k dE − dV + dNk = dS 1 , T T T

and hence d(S − S 1 ) −

36 The

 μ0 P 1 k dE − dV + dNk = 0. T T T

τk variable defined here is not the inverse of a specific mass.

(1.204)

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1 Thermostructure

 μ0k 2 Thus T1 dE + PT dV − T dNk = dS is an exact differential form, and therefore we can write S = S 1 + S 2 , but a priori this sum is not a decomposition into irreversible and reversible parts. From the second law, the evolution must satisfy dS 1  dNk − sk,0 ≥ 0. dt dt Homogeneity Properties Intensive and extensive variables, Duhem differential form In the frame of global thermodynamics, the dilation action on the domain of the system and on some associated quantities is expressed for certain variables, called extensive—such as volume, energy, entropy, number of moles of constituents—by multiplication by a parameter λ ∈ R + . In contrast, the intensive variables such as temperature, pressure, and chemical potential are invariant. The action of the dilation group (with parameter λ > 0) is given in the space F (for a system with n + 1 degrees of freedom) by Hλ (x, u, p) = (λx, λu, p),

(x, u, p) ∈ F, λ ∈ R + ,

(1.205)

with x, u the extensive variables, and p the intensive variables. Then the basic differential forms θ are changed by (1.205) into λθ , Hλ∗ (θ ) = λθ,

(1.206)

which implies the conservation of the contact structure. The state equations involving these extensive variables must be invariant by (1.205), and thus the map must be homogeneous of degree 0, that is, (λx) = (x),

Hλ∗ = .

(1.207)

Then this group keeps invariant every leaf of the foliated manifold for θ (for systems with n + 1 degrees of freedom). But note that is no longer a diffeomorphism. These properties imply the transformation of the thermodynamic functions. Positively homogeneous functions. A real function u such that u(λx) = λu(x), ∀λ > 0, x ∈ R n , is said to be positively homogeneous. Its conjugate function u∗ (of Fenchel) is such that u∗ (p) = sup (p.x −u(x)) = sup (p.λx −u(λx)) = sup λ(p.x −u(x)) = λu∗ (p), x∈R n

x∈R n

x∈R n

which implies u∗ (p) = 0 or +∞. The domain of u∗ is defined by Ku = Dom u∗ = {p ∈ R n , p.x ≤ u(x)} = {p ∈ R n , u∗ (p) is finite }.

1.6 Thermosystems with Different Constituents

81

∗ =σ Then u∗ = ψKu , the indicator function of Ku , and u∗∗ = ψK Ku is the support u function of Ku (see [Aub, ch. 1, 1.3, 3.1, pp. 10, 34]), and then we have u + u∗ = u|Ku , u(x)+u∗ (p) = p.x, p ∈ Ku . Thus if u is a positively homogeneous function, it satisfies the following properties:   ∂u = xj pj = x.p. (i) It satisfies the Euler formula:u(x) = xj ∂x j (ii) Its subdifferential ∂u(x) is such that ∂u(λx) = ∂u(x), ∀x, λ > 0, i.e. (subject to smoothness), the partial derivatives of u are homogeneous of order 0. (iii) The determinant of the Hessian matrix of u is null. Indeed, on taking the derivative of the Euler formula, we have

 j ∂u  ∂u ∂ 2u = δk + xj , ∂xk ∂xj ∂xj ∂xk  2u 2u thus xj ∂x∂j ∂x = 0, and therefore the Hessian matrix A = H (u) = ∂x∂j ∂x k k is such that Ax = 0, and hence x ∈ ker A, and A admits a null eigenvalue. The Legendre involution L mapping (x, u, p) into (p, U, x), with U = p.x − u transforms the differential form θ = p.dx − du into L∗ (θ ) = x.dp − dU with U positively homogeneous. Application to thermodynamic functions. Here we specify the properties of being positively homogeneous for the main functions in global thermodynamics, corresponding to the action of a parameter λ > 0, here with the mass variables37 ˆ denoted by M: ˆ S(λV , λE, λMˆ ) = λS(V , E, Mˆ ), E(λV , λS, λMˆ ) = λE(V , S, M), ˆ = λF (V , T , M), ˆ G(P , T , λM) ˆ = λG(P , T , M). ˆ F (λV , T , λM)

(1.208)

To the extensive (respectively intensive) variables correspond homogeneous functions of order 1 (respectively of order 0). In fact, S and E are homogeneous functions ˆ respectively of (V , S, M), ˆ whereas FT and GP ,T are of order 1 of (V , E, M), ˆ ˆ homogeneous functions of order 1 of (V , M), respectively of M. ˆ and on We specify the action of Hλ on the entropy S, a function of (V , E, M), ∗ its graph G(S). We have Hλ (Si ) = λSi , i = 1, 2, whence Hλ∗ (S) = S ◦ Hλ = λS,

ˆ S) = (λV , λE, λM, ˆ λS), and Hλ (V , E, M,

ˆ = S(λV , λE, λMˆ ), the graph G(S) of the entropy is and since λS(V , E, M) (globally) invariant under Hλ . We recall that these properties of being positively homogeneous are consequences of the invariance under the group (Hλ ), λ > 0,

37 Or

ˆ with the mole, denoted by N.

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1 Thermostructure

(i) of basic differential forms: Hλ∗ (θ S,M ) = λθ S,M ; (ii) of state equations: Hλ∗ = . Hence every leaf of the foliation due to ∗ (θ S ) is invariant under Hλ , or the group (Hλ ) still operates in a reversible way in  . Consequences: the Euler thermodynamic formula and the Duhem differential form. The action on entropy of the infinitesimal generator  of the homothetic group is given by the Euler operator ˆ = ∂ S(λV , λE, λMˆ )|λ=1 ; S(V , E, M) ∂λ

(1.209)

∂S  ∂S ∂S +E + Mk . ∂V ∂E ∂Mk

(1.210)

thus S = S = V

In a leaf  = LS0 of the foliation due to ∗ (θ S ), these derivatives are respectively μ˜ k P 1 T , T , − T , and thus we have in , ˆ =V S(V , E, M)

1  μ˜ k P +E − Mk , T T T

whence E = −P V + T S +



μ˜ k Mk ,

G=



μ˜ k Mk .

(1.211)

This homogeneity property is expressed by the global Duhem differential form (on every leaf  = LS0 ) ˆ

θ G,M | = −SdT + V dP −



Mk d μ˜ k = 0.

(1.212)

More generally, taking θ˜ S = θ S,M −



Mk dsk,0 ,

θ D = −SdT + V dP −



Mk d μ˜ k − T



Mk dsk,0 ,

we have T θ˜ S − θ D = 0; hence the relation θ˜ S = 0 implies θ D = 0, and thus −SdT + V dP −



Mk d μ˜ k = T



Mk dsk,0 .

These formulas are a direct consequence of the property of the energy u = E, or of the entropy u = S, of being positively homogeneous. Then for u = E a function of V , S, (Mk ), its conjugate function u∗ = U is such that U = 0 on its domain; thus

1.6 Thermosystems with Different Constituents

U = −P V + T S +



μ˜ k Mk − E =

83



μ˜ k Mk − G = 0,

and L∗ (θ ) is then reduced to the Duhem differential form on  = J (E). ¶ Example 1 The case of a system with one constituent is interesting to investigate. The present functions are functions of three variables, (such as E(V , S, M)) and not of two variables (here the mass M is a variable for an open system). The relation ˆ θ E,M = 0 (see (1.196)) is written with the number N of moles of the constituent (with M = Nm): dE + P dV − T dS − μdN = 0, with μ = mμt = m

G G = , M N

G equivalent to dG − μdN − V dP + SdT = 0, with μ = N , which can be written in the shape of the Duhem differential form or of the “Gibbs–Duhem equation”

SdT − V dP + Ndμ = 0. ¶

Convexity Properties of Global Thermodynamic Functions The convexity properties being strongly bound to stability, we are led to admit the following convexity properties:38 S(V , E, Nˆ ) is a concave function of (V , E, Nˆ ; E(V , S, Nˆ ) is a convex function of (V , S, Nˆ ); G(P , T , Nˆ ) is a convex function of Nˆ = (Nk ), and a concave function of (P , T ); F (V , T , Nˆ ) is a convex function of (V , Nˆ ). We will verify these properties in the case of systems of ideal gases. These properties are obviously linked among themselves. These convexity properties imply that the map of the state equations has its opposite of the Jacobian −J ( ) positive definite. Relations between global thermodynamic functions through the partial Legendre transformation. The Gibbs free energy G is defined as the opposite of the convex function that is conjugate to FT ,Mˆ , thus by ˆ = −F ∗ (−P ), G(P , T , M) T ,Mˆ whence 38 A

priori on every maximal integral manifold for the differential form θ; see below.

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1 Thermostructure

ˆ = − sup (P V − F (V , T , M)) ˆ = inf (F (V , T , M) ˆ − P V ). G(P , T , M) V >0

V >0

The relations between the functions energy, free energy, and Gibbs free energy are ˆ = E(V , S, M) ˆ − T S, F (V , T , M)

(1.213)

ˆ = E(V , S, M) ˆ + P V − T S = F (V , T , M) ˆ + PV. G(P , T , M) These relations are associated through the Legendre transformation.

¶

Remark 18 Positively homogeneous, subadditive, and convex functions. Let u be a positively homogeneous and convex function (as in the case of energy). This is equivalent to saying that u is positively homogeneous and subadditive, i.e., is such that u(x1 + x2 ) ≤ u(x1 ) + u(x1 ),

u(λx) = λu(x),

∀λ > 0.

(1.214)

Indeed, if u satisfies these relations, then u(λx1 + (1 − λ)x2 ) ≤ u(λx1 ) + u((1 − λ)x2 ), for all 0 ≤ λ ≤ 1, and thus u(λx1 + (1 − λ)x2 ) ≤ λu(x1 ) + (1 − λ)u(x2 ), which means that u is a convex function. The converse is evident. Let u = u(, (Nk )) be a function of the domain  of the system and of the numbers Nk of moles of the Ck constituent in , in fact depending only on the volume V of  and on the numbers Nk . By the subadditivity for the two domains 1 , 2 , with respectively Nk1 , Nk2 the numbers of moles of the Ck constituent, u(1 ∪ 2 , (Nk1 + Nk2 )) ≤ u(1 , (Nk1 )) + u(2 , (Nk2 )).

(1.215)

In the case of entropy, u = S is a positively homogeneous and concave function, and then S˜ = −S is a positively homogeneous and convex function, and thus subadditive. ¶

1.6.3 Going from Global to Local Thermodynamics The properties of homogeneity in global thermodynamics allow to obtain many properties in local thermodynamics by passing to the quotient.

1.6 Thermosystems with Different Constituents

85

Relations Between Global to Local Thermodynamic Quantities ˆ S) ∈ M × R be extensive variables of a system in global Let j = (E, V , M; thermodynamics. The action of the group Hλ on j gives (λE, λV , λMˆ ; λS). Passing to the quotient by the action of the group, we define π(j ) ∈ (M × R)/R + . We can choose a representative of the equivalence class of this element for the action of the group by taking λ = 1/M or λ = 1/V . So we obtain the element ˆ ˆ (E/M, V /M, M/M; S/M), or (E/V , 1, M/V ; S/V ). Passing to the quotient on the global state equations, we obtain the local state equations, and thus we pass G L in local from the manifold  in global thermodynamics to the manifold  thermodynamics, by eliminating the variable of volume. Then the relations39 between the global variables (E, V , Mk , S) and local variables (e, τ, ρk , s), or between (E, V , Nk , S) and (e, ˆ τˆ , xk , sˆ ), or also between (E, Mk , S) and (e, ˜ ρk , s˜ ), are E = Me,

V = Mτ,

Mk = Mck ,

E = N e, ˆ

V = N τˆ ,

Nk = Nxk ,

S = Ms, S = N sˆ ,

(1.216)

E = V e, ˜ Mk = Vρk , Nk = V nk , S = V s˜, with e˜ = ρe, s˜ = ρs, but we again denote e˜ and s˜ (and also e, ˆ e) ˆ by e and s if there is no misunderstanding. Likewise, we have F = Mf = V f˜, G = Mg = V g. ˜

Relations Between the Local and Global Maps  of State Equations Corresponding to (1.216), we have the relations μ˜ k P 1 , , − ), T T T μk P 1 (V , E, (Nk )) = (N τˆ , N e, ˆ (Nxk )) = (τˆ , e, ˆ (xk )) = ( , , − ), T T T

(V , E, (Mk )) = (Mτ, Me, (Mck )) = (τ, e, (ck )) = (

ρ

μ˜ 1 (V , E, (Mk )) = (V , V e, ˜ (Vρk )) = (1, e, ˜ (ρk )) = ( , − k ), T T ρ

μ˜ mk 1 (V , E, (Nk )) = (V , V e˜, (V nk )) = (1, e, ˜ (nk )) = ( , − k ). T T

relations come by homogeneity from the usual formulas E = we assume that the specific mass ρ is constant in the domain.

39 These

 

eρdx, S =

(1.217)

 

sρdx if

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1 Thermostructure

Relations Between Basic Local and Global Differential Forms ˆ

The relations between basic global differential forms θ S,N , θ E,M , and local forms θ s,n , θ e,ρ are due to Hλ∗ (θ S,N ) = λθ S,N , ∀λ > 0, and the choice of λ = V1 gives θ S,N = V θ s,n ,

ˆ

θ E,M = Mθ e,ρ .

(1.218)

Proof Similar to (1.200), using relations (1.216), we have  μk 1 P d(V e)) − dV + d(V nk ) T T T  μk  μk nk P 1 1 dnk ] + [s − e − + ]dV . = V [ds − de + T T T T T (1.219)

θ S,N = d(V s) −

But s−

 μk Nk 1 P  μk nk 1 1 PV e− + = [S − E − + ] T T T V T T T G  μk Nk 1 ] = 0, = [− + V T T ˆ

which implies the relation θ S,N = V θ s,n . We have a similar relation with θ E,M and ˆ ˆ we again have θ F,M = Mθ f,ρ , θ G,M = Mθ g,c . ¶

Relations Between Local and Global Thermodynamic Functions ˆ E(V , S, M) ˆ correspond to the functions s(e, ρ), The functions S(V , E, M), ˆ e(s, ρ) ˆ by ˆ = Vρe(s, ρ), E(V , S, M) ˆ

ˆ = Vρs(e, ρ). S(V , E, M) ˆ

(1.220)

ˆ G(P , T , M) ˆ and local f (T , ρ), Similarly, between global F (V , T , M), ˆ g(T , ρ) ˆ thermodynamic potentials, we have the relations ˆ = Vρf (T , ρ), F (V , T , M) ˆ

ˆ = Vρg(T , ρ). G(P , T , M) ˆ

(1.221)

These relations are obtained using the properties of homogeneity of extensive variables. Taking τ=

V , M

ck =

ρk Mk = , M ρ

ρk =

Mk , V

1.6 Thermosystems with Different Constituents

87

we obtain the relations between local and global thermodynamic functions (with cˆ = (ck )) E(τ, s, c) ˆ = e(τ, s, c), ˆ

F (τ, T , c) ˆ = f (τ, T , c), ˆ

G(P , T , c) ˆ = g(P , T , c), ˆ

(1.222)

S(τ, e, c) ˆ = s(τ, e, c), ˆ

which indicate that these functions are identical when M = 1.

Relations Between Local and Global Convexities ˆ with respect to Mˆ is The hypothesis of convexity of the energy E(V , S, M) E(V , S, λMˆ 1 + (1 − λ)Mˆ 2 ) ≤ λE(V , S, Mˆ 1 ) + (1 − λ)E(V , S, Mˆ 2 ). Dividing this inequality by V and using (1.220), we obtain that e(˜ ˜ s , ρ) ˆ is convex with respect to ρ. ˆ The action of the group Hλ on the epigraph of the global energy E, by passing to the quotient, gives the epigraph of the local energy e, ˜ which keeps the convexity property. We have the same property for the opposite of entropy. From ˆ with respect to M, ˆ we deduce that s˜ (e, the concavity of S(V , E, M) ˜ ρ) ˆ is concave with respect to ρ. ˆ ˆ respectively F (V , T , M), ˆ and G(P , T , M) ˆ From the convexities of E(V , S, M), ˆ ˆ ˆ with respect to (S, M), respectively M, and M, with e˜ = ρe, s˜ = ρs, . . . , we deduce the following properties of convexity: e(˜ ˜ s , ρ), ˆ respectively f˜(T , ρ), ˆ and g(T ˜ , ρ) ˆ is convex with respect to (˜s , ρ), ˆ respectively ρ, ˆ and ρ. ˆ Furthermore, dividing the relation G(P , T , λMˆ ) = λG(P , T , Mˆ ) by V , we obtain g(T ˜ , λρ) ˆ = λg(T ˜ , ρ), ˆ and thus g(T ˜ , ρ) ˆ is positively homogeneous with respect to ρ. ˆ By (1.222), we obtain the convexity of the local thermodynamic functions with respect to mass fractions by restriction to n−1 of the global functions. ˆ with respect to V is equivalent to that of f (T , ρ) The convexity of F (V , T , M) ˆ ˆ = Vρf (T , ( Mˆ )), and with respect to ρˆ with fixed T . Indeed, with F (V , T , M) V taking the derivative twice with respect to V , we obtain ρ  ∂ 2F = 3 Mk Mj ∂k ∂j f. 2 ∂V V ∂2F ∂V 2

≥ 0 if and only if f is convex with respect to ρˆ with fixed T . ˆ We again note the following property implied by the subadditivity of GP ,T (M):

Thus we have

ˆ ≤ GP ,T (M)



GP ,T (Mk ),

hence



ck μ˜ k ≤

ˆ = MgT (ρ) ˆ and GP ,T Mk ) = Mk gT (ρk ). since GP ,T (M)

¶



ck gk ,

88

1 Thermostructure

1.6.4 From n Independent Constituents to a Mixture Spaces in Semiglobal Thermodynamics Let (Ck ) be an open system of a family of n constituents in disjoint domains k of R 3 , whose variables (Ek , Vk , Nk ; Sk ; Tk , Pk , μk ) express the global thermodynamic modelling of each constituent. We agree that for each constituent there are three state equations, giving an integrable map k , k (Vk , Ek , Nk ) = (

Pk 1 μk , , − ). Tk Tk Tk

For example, if each constituent is an ideal gas, we have Pk Nk = R0 , Tk Vk

Nk 1 = Cvk , Tk Ek

−

μk Vk Ek = R0 log + Cvk log . Tk Nk Nk

The space, denoted by  k , a representative of each state satisfying these equations, is 4-dimensional. Then the differential form θ Sk is such that θ Sk = dSk − dSk,res = dSk,0 . The space  k is a foliated manifold by a family (FSk,0 ), Sk,0 ∈ R, of maximal integral manifolds for the differential form θ Sk , corresponding to the jet of the entropy Sk , with independent variables Ek , Vk , Nk , each leaf being 3-dimensional. Here we agree that there is no exchange between the different parts of the system, and the variables of a constituent are independent of those of another constituent. The representative of the total space of thermodynamic states with the entropy of each constituent (respectively with their state equations) is the 7n-dimensional  (respectively 4n-dimensional) space F = Fk (respectively  =  k ). With one variable of entropy for the system instead of n, we have a (3n + 1)-dimensional space. (Note that these spaces can be reduced if there are exchanges between the different subsystems, thanks to additive properties.) Then the basic differential form  Sk is θ S = θ , and the total map of thesystem (given by  the state equations) is given by = ( k )1≤k≤n from M = k Mk into N k Nk , which are 3ndimensional. The main question about a system with adjacent domains k is to specify these exchanges between them through their faces, which are models of walls, or of membranes, with different properties, either thermodynamic (for example, forbidding heat exchange, or giving constraints on temperature, pressure, or exchanges of numbers of moles), or dynamic (we can attach a mass, a velocity, . . . , to the material points x i ). Thus it may be difficult to specify the new independent thermodynamic variables of the system. We will study below the simple case of two adjacent domains, each with one fluid, in order to specify some properties of exchange, or constraints due to a membrane (or a wall).

1.6 Thermosystems with Different Constituents

89

Spaces for Mixture in Global Thermodynamics Now we assume that the domain occupied by each constituent is the same, with volume V , and that the temperature is the same, T .  For the system, have a total energy E = Ek , a total  by additive properties, we  entropy S = Sk , and a total pressure P = Pk . Without  the additive properties, the system would be modeled  by the fibre product V ,T Fk , with 5n + 2 degrees of freedom, or by  = V ,T  k for the set of states satisfying the state equations, which is (2n + 2)-dimensional. The basic Sk , and the maximal integral differential form of the system would be θ S = θ manifolds would be expressed by the fibre product V ,T Fsk,0 of the jets of the entropy (Sk ), and then the maximal integral manifolds of θ S would be (n + 2)dimensional, with n transverse independent variables sk,0 whose variation along with time must be positive. With the additive properties, the system is modeled by the space F with variables (E, V , (Nk ); S; T , P , (μk )), of odd dimension, that is, 2n + 5, with a contact structure due to the basic differential form θ S = dS −

 μk P 1 dE − dV + dNk . T T T

Hence the set  of the states of the system satisfying these equations is again of dimension 2n + 2, and can be identified with the previous space, also denoted by  . The maximal integral manifolds of θ S are of dimension n + 2, and they S are also maximal integral manifolds   S of θand are identified with the fibre products S k θ = Nk dsk,0 . V Fsk,0 . Then we have θ = Spaces for Mixture in Local Thermodynamics In a similar way, in local thermodynamics we can consider a system with n constituents. The thermodynamic state of each constituent is given by five variables, (ek , τk ; sk ; Tk , Pk ), satisfying two state equations. Let Fk (respectively  k ) be the representative space of states of each constituent (respectively satisfying the state equations). The space  k , of dimension three, is again a foliated manifold by a family of 2-dimensional leaves (Fsk,0 ), sk,0 ∈ R, which are maximal integral manifolds relative to the basic differential form θ sk . We assume that the temperature T is the same for all constituents; then the system is modeled by the fibre product, denoted by T Fk , of dimension 4n + 1, or by   = T  k for the set of states satisfying these state equations, of sdimension 2n + 1. Then the basic differential form of the system is  θs = θ k , and the maximal integral manifolds are given by the fibre product T Fsk,0 of the jets of entropy sk .   For this system, ck ek , an entropy s = ck sk , we can define an energy e = a pressure P = Pk , and τ = ck τk , ∀k the inverse of the specific mass. Then

90

1 Thermostructure

the system is modeled by the space F with variables (e, τ, (ck ); s; T , P , (μck )), of odd dimension 2n + 3, with a contact structure due to the basic differential form  μck θ s = ds − T1 de − PT dτ + we have two state equations (in the T dck . Then   k case of ideal gases P τ = rk ck T , e = cv ck T ). Hence the set  of states of the system satisfying these equations is again of dimension 2n + 1. The maximal s integral manifolds  for θ are of dimension n + 1, and they are identified with the fibre products T Fsk,0 , of transverse variables (sk,0 ) whose variation along with sk s time be  must  positive. Then we have θ = dsk − dsk,rev = dsk,0 , and θ = s k ck θ = ck dsk,0 . We can obtain these results of local thermodynamics by passing to the quotient the results of global thermodynamics thanks to the homogeneity properties of extensive variables, thus eliminating the volume V .

1.6.5 Semiglobal Thermodynamics The splitting of a system into interacting subsystems with a global modelling for each subsystem leads to various problems, notably of mixture. For instance, let us consider two disjoint connected bounded domains with a common part of boundary, denoted by , each domain j being occupied by Nj moles of a constituent Cj , j = 1, 2, in an equilibrium state with variables (Vj , Ej , Nj ), with entropy Sj (Vj , Ej , Nj ).

Problem of Mixing If the separation between the two domains is removed, what will be the new equilibrium state of the global system (if it exists)? The study of the transition to this state is relevant only to local thermodynamics. With respect to global thermodynamics, the transition seems to be irreversible, and then the entropy of the system would be higher than the sum of the entropies of each subsystem. Using the subadditivity of the opposite of the entropy (Sˆ = −S is a positively homogeneous convex function), we have S(V1 , E1 , N1 ) + S(V2 , E2 , N2 ) ≤ S(V1 + V2 , E1 + E2 , N1 + N2 ).

(1.223)

Then the total volume, energy, and matter (total number of moles) are conserved. The conjecture is that S(V1 + V2 , E1 + E2 , N1 + N2 ) is the entropy of the new equilibrium state of the mixture. We recall that in (1.223), S is in fact the thermodynamic function Srev . Now with S = Srev + S0 , we therefore have S0 = S − Srev . Then with the additive properties of the state variables E = E1 + E2 , S = S1 + S2 , we have with (1.223) for the global system

1.6 Thermosystems with Different Constituents

91

S0t ot = S1 + S2 − Srev (V1 + V2 , E1 + E2 , N1 + N2 ) ≤ (S1 − S(V1 , E1 , N1 )) + (S2 − S(V2 , E2 , N2 )), and thus the variable of irreversible entropy satisfies S0t ot ≤ S01 + S02 . If the constituents are ideal gases, or if they satisfy the van der Waals state equation, we can directly verify the property (1.223) on the entropy formulas ˆ notably obtained using the function ψ(u, v) = u log uv with u, v > 0 and its properties of subadditivity. Thus in the case of ideal gases, and if the two subsystems are at the same temperature T but at pressures P1 and P2 , the total system will be at 1 the temperature T and at the pressure P = ξ P1 + (1 − ξ )P2 with ξ = V1V+V . 2 Of course such a mixture state is not necessarily obtained, notably if there is a superficial tension at the separation surface of the two (closed) subsystems. In this case, we can find, for example, in [Pri-Kon] the differential form θ = dS −

P1 1 P2 γ dE − dV1 − dV2 + dA, T T T T

where A is the area of the surface . But we remark that the variables V1 , V2 , A are not necessarily independent. Example 2 It is interesting to specify the term of superficial tension, in a simple case, with two liquids (or a liquid and a gas) satisfying the van der Waals equations, in the domains 1 = Br , 2 = BR /Br with R > r. Thus 1 is the ball of radius r, 4π 3 3 3 and 2 is a spherical crown, each of volume V1 = 4π 3 r , V2 = 3 (R − r ). The 2 area of the surface of their joint boundary is A = Ar = 4πr , so that V1 and Ar are not independent variables. We assume that the mole numbers Ni are constant, that the temperatures T1 , T2 are equal. Now using the van der Waals equations, the basic differential form is Pˆ1 1 Pˆ2 γ dV1 − dV2 + dAr , θ = dS − (Cv1 N1 + Cv2 N2 ) dT − T T T T with Pˆi = Pi + a0i

Ni2 Vi2

=

R0 Ni , Vi − Ni bi

i = 1, 2.

Now if the temperature is constant, and if the total volume V = V1 + V2 is constant, then dV2 = −4πr 2 dr) = −Ar dr, dV1 = 4πr 2 dr = Ar dr, and dAr = 8πrdr. Thus we have

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1 Thermostructure

θ0 =

γ Pˆ1 − Pˆ2 Pˆ1 Pˆ2 γ dV1 + dV2 − dAr = [Ar − 8πr ]dr. T T T T T

The stability property at equilibrium implies that the superficial tension γ is given Vi as a function of T , τi = N , i = 1, 2, by i r Pˆ1 − Pˆ2 1 r 1 γ = = R0 [ − ]. T 2 T 2 τ1 − b 1 τ2 − b 2 We can compare to the usual Laplace formula (see [Pri-Kon, p. 108]).

(1.224) ¶

System of Two Constituents Separated by a Rigid Surface Here we can think of a cylinder  = ×]0, L[ of volume V = A × L, occupied by two fluids, the first in 1 = ×]0, x[, the second in 2 = ×]x, L[. But generalizsations to other geometries are obvious. In order to separate individual variables from those global variables, let V = V1 + V2 , E = E1 + E2 , N = N1 + N2 , δV = V1 − V2 , δE = E1 − E2 , δN = N1 − N2 , and then   P 1 P1 P2 1 1 1 1 μ 1 μ1 μ2 = ( + ), = ( + ), = ( + ), T m 2 T1 T2 T m 2 T1 T2 T m 2 T1 T2     μ P P1 P2 1 1 1 μ1 μ2 = − , = − , = − . T T1 T2 T T1 T2 T T1 T2 The basic differential form of the total system is given by  θ S = θ Sm + θ S = dS − −

1 2

P T 

 

μ 1 dE − dN T m T m m     μ 1 P dδV + dδE − dδN . T T T (1.225)

dV +

We assume that the state equations of each constituent give the integrable maps 1 and 2 , so that we can define the two maps t ot , δ , on the variables ((V , E, N), (δV , δE, δN)):

1.6 Thermosystems with Different Constituents

93

  P 1 −μ 1 =( , , ) = [ 1 (V1 , E1 , N1 ) + 2 (V2 , E2 , N2 )], T m T m T m 2       1 P 1 −μ 1 δ = ( , , ) = [ 1 (V1 , E1 , N1 ) − 2 (V2 , E2 , N2 )]. 2 T T T 2 

t ot

Of course, if the total variables of the system V , E, N are constant, the basic differential form of the system reduces to θ S = θ S = dS −

1 2



P T

 dδV +

   μ 1 dδE − dδN , T T

with dδV = 2Adx in the cylinder case, A being the area of the section. Now we consider some particular cases. (i) Isotherm. If T1 = T2 , the formulas are simple, first for θ S , θ S = dS −

 1 11 [Pm dV + dE − μm dN] − [P ] dδV − [μ] dδN . T 2T

(ii) Adiabatic. If there is no exchange of heat with the exterior, then θ Q1 +θ Q2 = 0, so that if the total volume is constant, we have with 12 dδV = Adx, dE1 + dE2 + (P1 − P2 )Adx = dE + [P ] Adx = 0. Then also if N is constant, the differential form θ S is reduced to       1 P 1 1 1 μ S θ = dS − dE − Adx − dδE + dδN . T m T 2 T 2 T

1.6.6 System of Ideal Gases, Global Thermodynamics We consider in a global model, a system of n ideal gases in a domain  of volume V . We denote by Nk the number of moles of the gas with subscript k, by Mk its mass, by mk its molar mass (thus Mk = Nk mk ), by μk = μ˜ k mk its chemical potential (dual variable of Nk ) for k = 1, . . . , n. Let us again have N = Nk , M = Mk , and xk = NNk the mole fractions for k = 1, . . . , n. We denote by Nˆ = (Nk ) and xˆ = (xk ), k = 1, . . . , n, the family of the Nk , xk . Then the two usual state equations of the system of ideal gases are (i) P V − NR0 T = 0, (ii) E − NCvm (x)T ˆ = 0,

(1.226)

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1 Thermostructure

using the variable xk rather than ck , ck =

Mk nk mk n N ρk x k mk = =  =  = x k mk = x k mk , nk mk x k mk ρ M ρ M

and thus Cvk = cvk mk , and with ˆ = Cvm (x)

 Nk N

mk cvk =



xk mk cvk =

M m c (c), ˆ N v

which is a homogeneous function of Nk of degree 0. Complete state equations from integrability. We have to find an expression for the chemical potentials with respect to the variables (V , E, Nˆ ) with Nˆ = (Nk ). The condition of integrability of the map such that (V , E, Nˆ ) = (

μk P 1 , , −( )) T T T

are ∂ P ∂ μk ( ), ( )=− ∂Nk T ∂V T

∂ 1 m ˆ ∂ μk ( ). ( Cv (N )) = − ∂Nk E ∂E T

Using (1.226), we have −

∂ μk R0 = , ∂V T V

−

∂ μk Ck = v. ∂E T E

Thus we obtain, up to an arbitrary function ζk of Nˆ , −

μk = R0 log V + Cvk log E + ζk (Nˆ ). T

This expression must be homogeneous of degree zero with respect to (V , E, Nˆ ). ˆ Then we must have, up to a new arbitrary40 function ζ˜k of N ˆ N = x, −

μk V E = R0 log + Cvk log − Cvm (x) ˆ − R0 + ζ˜k (x). ˆ T N N

Then the differential form θ S is

˜

∂ ζk this family of functions must be smooth and integrable, that is, ∂N = k  ˆ ˆ ˜ ˆ there is a function ϕ(N) such that dϕ = ζk dNk and ϕ(N) = Nϕ(x).

40 But

∂ ζ˜k  ∂Nk

, ∀k, k  , so that

1.6 Thermosystems with Different Constituents

95

N N dV − Cvm (x) dE ˆ V E  V E − [R0 log + Cvk log − Cvm (x) ˆ − R0 + ζ˜k (x)]dN ˆ k. N N

θ S =dS − R0

(1.227)

k

When θ S = 0, we have by integration S(V , E, Nˆ ) = NR0 log Thus with s =

S N,e

=

E N,τ

V E + NCvm (x) ˆ + S0 . ˆ log + Nϕ(x) N N

=

V N , s0

=

S0 N,

(1.228)

we obtain

ˆ log e + ϕ(x) ˆ + s0 . s(τ, e, x) ˆ = R0 log τ + Cvm (x)

(1.229)

We see that the map due to the state equations depends on an integrable map (ζ˜k ),  ˜ dependent only on (x), ˆ so that we can write = ϕ , with  dϕ = (ζk )dxk . These functions may be specified if we adopt the relation s = xk sk (see (1.249) below). Complete state equations from additivity, and entropy. We return to the differential forms (1.201), (1.200) with (1.199). Let Sk = Nk sk , Ek = Nk ek , V = Nk τk . The individual state equations are given by Pk Nk 1 = R0 = R0 , T V τk

Nk 1 1 = Cvk = Cvk . T Ek ek

(1.230)

θ sk = d[sk − Cvk log ek − R0 log τk ].

(1.231)

Then

Thus θ sk is an exact differential, and then θ sk = dsk,0 . The differential form θ Sk is θ Sk = dSk − = Nk θ sk Hence if μk = S1 = with

Gk Nk ,



G1k

μk 1 Pk dEk − dV + dNk , T T T Gk μk − +( )dNk . T T Nk

(1.232)

we have θ Sk = Nk dsk,0 . Let us again have Sk1 =



Nk sk,0 ,

μ0k = μk + T sk,0 =

= Gk + Nk T sk0 = Ek + Pk V

Then the differential form θ S becomes

G1k , Nk

− T (Sk − Sk1 ).

(1.233)

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1 Thermostructure

θS =



θ Sk =



[dSk −

 μk 1 Pk dEk − dV + dNk ] = dS 1 − sk0 dNk , T T T (1.234)

which finally gives the relation, in accordance with (1.204), d(S − S 1 ) −

 μ0 P 1 k dE − dV + dNk = 0. T T T

(1.235)

Thus following (1.204), the entropy S is S = S 2 + S 1 , i.e.,   S= Nk sk = [Cvk Nk log ek + R0 Nk log τk + Nk sk,0 ].

(1.236)

Using the state equations, we have ek =

E Ek = Cvk T = Cvk , m ˆ Nk Cv (N)

τk =

N T T V = R0 , = R0 Nk Nk P xk P

which gives the entropy formulas Sk = Nk [Cvk log

Ek V + R0 log + sk,0 ] Nk Nk

(1.237)

and S=



Sk =



Nk [Cvk log

Cvk E V + sk,0 ] + R0 log m Nk Cv (Nˆ )

(1.238)

V E ˆ ˆ log + ϕ(x)) = N(R0 log + Cvm (x) N N with ϕ(x) ˆ =



Cvk xk log

Cvk − R0 xk log xk + xk sk,0 ], Cvm (x) ˆ

(1.239)

or as a function of (P , T , Nˆ ), S(P , T , Nˆ ) =



Nk [Cvk log T − R0 log

P 1 − R0 log xk + sk,0 ], T

(1.240)

with 1 sk,0 = Cvk log Cvk + R0 log R0 + sk,0 .

(1.241)

1.6 Thermosystems with Different Constituents

We can also write

S N

97

= s as a function of P , T , x: ˆ

s = Cvm (x) ˆ log T − R0 log

  P 1 xk sk,0 . xk log xk + − R0 T

(1.242)

Equation of characteristics. Let θ S be the basic differential form and 0 the state equation of ideal gases 0 (x, u, p) = p2 x2 −



Nk R0 =

P .V − NR0 T

with variables (x; u; p) = (E, V , (Nk ); S; T1 , PT , (− μTk )). Then the characteristic vector ξ is expressed by ξ = (0, V , (0);

P P .V ; 0, − , R0 ), T T

and the corresponding evolution equations are dE dV dNk dS P dp1 dp2 μk d = 0, =V, = 0; = V; = 0, = − p2 , (− )) = R0 . dt dt dt dt T dt dt dt T Hence denoting the initial conditions using the exponent 0, the solution is given by E = E 0 , V = V 0 et , Nk = Nk0 , S = S 0 + N 0 R0 t, T = T 0 , P = P 0 e−t , −

μ0 μk = − k + R0 t, T T

and thus μk = μ0k − R0 T 0 t. Moreover, we deduce G = E + P V − T S = E 0 + P 0 V 0 − T 0 S 0 − T 0 N 0 R0 t = G0 − T 0 N 0 R0 t. Similarly as for the characteristic equation associated with the differential form θ Sk and with the state equation k = PTk .V − Nk R0 , we obtain Gk = G0k − Nk0 R0 t. Thus the evolution due to ξ is isothermal. If we take as initial manifold E0 =



Nk0 Cvk T 0 ,

then for all times, we have  E= Nk Cvk T ,

G0k = μ0k Nk0 ,

Gk = μk Nk ,

thus G0 =

hence G =





μ0k Nk0 ,

μk Nk .

Remark 19 Smoothness Properties of G. We note that with fixed (P , T ) (P and T not null), the function G(P , T , (xk )) (with xk = Nk /N) is continuous on the

98

1 Thermostructure

¯ n−1 , and moreover, C ∞ on the open set n−1 , but it is not of class C 1 simplex  ¯ n−1 , which is implied by the fact that μk → −∞ when xk → 0. This comes on  from S 1 (Nˆ ). The boundary of the simplex occurs in the chemical reactions with the disappearance or appearance of a constituent. ¶

Concavity and Convexity Properties (1) The function (V , E, Nˆ ) → S(V , E, Nˆ ) is concave, i.e., Sˆ = −S is convex. Thus E(V , S, Nˆ ) is a convex function of (V , S, Nˆ ).  Proof Since S = Sk , we have only to prove the concavity of Sk , and this will result in the concavity of the function ψ such that ψ(u, v) = u log uv with u, v > 0. Now the Hessian matrix H (ψ) of ψ is expressed by H (ψ) =

− u1 1 v

1 v − vu2



1 =− u



1 − uv 2 − uv uv 2

.

Thus for all ξ = (ξ1 , ξ2 ), we have (H (ψ)ξ, ξ ) = − u1 (ξ1 − uv ξ2 )2 ≤ 0. This implies the concavity of Sk , and thus of S. (2) The function (P , T , Nˆ ) → G(P , T , Nˆ ) has the two following properties: (i) G(P , T , Nˆ ) is a concave function of (P , T ). (ii) G(P , T , Nˆ ) is a convex function of (Nk ), with fixed T and P . Proof (subject to smoothness). (i) Indeed, we have NR0 T ∂ 2G ∂V =− = , ∂P 2 ∂P P2

∂ 2G N(R0 + Cvm ) , = − ∂T 2 T

∂ 2G NR0 = , ∂P ∂T P (1.243)

and the determinant (G) of the Hessian H (G) of G (for fixed (Nk )) is such that (G) =

∂ 2G ∂ 2G − ∂P 2 ∂T 2



∂ 2G ∂P ∂T

2 =

NR0 T N(R0 + Cvm ) − T P2



NR0 P

2 ≥ 0.

(ii) To say that G(P , T , Nˆ ) is a convex function of (Nk ) is equivalent to saying that S˜ 0 (Nˆ ) is a convex function (Nk ). We have ∂μk 1 ∂ 2G = = −R0 T , for j = k, ∂Nk ∂Nj ∂Nj N Hence for all (zj ) ∈ R n , we have

∂ 2G ∂μk 1 1 = = R0 T ( − ). 2 ∂Nk Nk N ∂Nk

1.6 Thermosystems with Different Constituents

def

H (G) =



∂ 2G R0 T zj zk = ∂Nk ∂Nj N

99



zk2

 N 2 −( zj ) . Nk

√ Thus with xk = Nk /N, yk = zk / xk , this expression is such that by the Cauchy–Schwarz inequality, H (G) =

  √ R0 T  2 yk xk )2 ≥ 0. yk − ( N

¶

1.6.7 System of Ideal Gases, Local Thermodynamics Here we take again a model of local thermodynamics for a system of ideal gases. Let xˆ = (xk ), xk =

 Nk ,N = Nk , N

cˆ = (ck ), ck =

ρk Mk = , M ρ

ρ=



ρk .

Let ek (respectively e˜k ) be the energy by mass unit (respectively by moles) of the constituent Ck . We have Ek = Mk ek = Nk mk ek = Nk e˜k , and thus e˜k = mk ek with mk the molar mass. The usual state equations of this constituent are Pk = nk R0 T ,

ek = cvk T ,

respectively e˜k = Cvk T , with Cvk = mk cvk . (1.244)    Recall the additivity properties P = Pk , e = ck ek , e˜ = xk e˜k . Then the rk partial pressure Pk due to Ck is such that Pk = nnk .P = xk P = cr(k c) .P . ˆ  ρk V ,n = , the state equations are With τ = ρ1 , respectively τ˜ = n1 = N mk  (i)P τ = r(c)T ˆ , with r(c) ˆ = R0 m−1 k ck ,  (ii) e = cvm (c)T ˆ , with cvm (c) ˆ = cvk ck ;

(1.245)

respectively (i)P τ˜ = R0 T , ˆ , with Cvm (x) ˆ = (ii) e˜ = Cvm (x)T



Cvk xk ;

(1.246)

r, cvm and Cvm are homogeneous functions of degree 0 of (ρk ). In the case of a pure gas, n = 2, we have r = R0 /m1 and cvm = cv . Below we will use (1.246) in an essential way, and in order to simplify notation, we suppress the tilde in τ˜ , e, ˜ and e˜k . The expressions for the local thermodynamic

100

1 Thermostructure

functions are here obtained by passing to the quotient on the extensive functions in global In particular, through division by V , the entropy formula thermodynamics.   S = Sk = Nk sk gives s = ck sk . In local thermodynamics, the entropy is thus given as in the expression (1.231) for θ sk , when θ sk = 0, by s=



xk sk =



xk [Cvk log ek + R0 log τk + sk,0 ].

(1.247)

Using the state equations ek = Cvk T = Cvk C me(x) , τ = τk xk , τk = R0 PTk , and ˆ Pk P

v

= xk , we obtain (with (1.241))

ˆ log e + ϕ(x), ˆ s(e, τ, x) ˆ = R0 log τ + Cvm (x)

(1.248)

with ϕ(x) ˆ = −R0



xk log xk +



Cvk xk log

 Cvk )+ xk sk,0 , m Cv (xˆ

(1.249)

or as a function of (P , T , x), ˆ s(P , T , x) ˆ =

 

xk [Cvk log(Cvk T ) + R0 log R0

T + sk,0 ] Pk

Pk 1 + sk,0 ] T   P 1 = Cvm (x) ˆ log T − R0 log − R0 xk sk,0 , xk log xk + T (1.250) =

xk [Cvk log T − R0 log

which could be directly obtained by the formula (1.242) for S. We immediately obtain the chemical potential μk = gk = ek + Pk τk − T sk , and hence μk Pk 1 = sk − Cvk − R0 = Cvk log(Cvk T ) − R0 log + sk,0 − Cvk − R0 T T P 1 = Cvk log T − R0 log − R0 log xk + sk,0 − Cvk − R0 , T (1.251) and thus we have −

−

μ0 μk = − k + sk,0 . T T

The usual thermodynamic functions, other than e and s, are obtained by taking the quotient of their corresponding global extensive variables, which gives h=



ck hk ,

f =



ck fk ,

g=



ck gk .

1.6 Thermosystems with Different Constituents

101

Remark 20 First integrals and compatibility of the state equations. Here we return to variables linked to the specific mass ρ of (1.245). Let ρ

μ˜ 1 y = (y0 , y ) = (y0 , (yk )) = ( , − k ), T T 

x = (x0 , x  ) = (x0 , (xk )) = (e, (ρk )),

(1.252)

z = s. The state equations of ideal gases with the energy–temperature  relation are defined here through the contact Hamiltonians K 1 and K 0 , with |x  | = xk = ρ:  1 1 (i)K (x, y, z) = x .y +  r(x  ) = xk yk +  r(x1 , . . . , xn ), |x | xk def

1





n

k=1

1 1 def (ii)K 0(x, y, z) = x0 y0 −  cvm (x  ) = x0 y0 −  cvm (x1 , . . . , xn ), |x | xk (1.253) P respectively corresponding to − ρT + r(c) ˆ and Te − cvm (c). ˆ The “commutator” of K 1 and K 0 (see [Arn1][App. 4]) is given by (the subscripts correspond to derivatives)

  [K 1 , K 0 ] = K 1 , K 0 + Kz1 EK 0 − Kz0 EK 1 ,   with K 1 , K 0 = Kx1 .Ky0 − Ky1 .Kx0 and EK = K − x.Kx .

(1.254)

Now we have Kx10 = 0, Kx1k = yk + Kx00

=

y0 , Kx0k

1 (rk − r(c)), ˆ Ky1 = (0, x  ), Kz1 = 0, ρ

1 = (cvm (c) ˆ − cvk ), Ky0 = (x0 , 0), Kz0 = 0, ρ

(1.255)

and thus    1 [K 1 , K 0 ] = K 1 , K 0 = − ˆ − cvk ) = 0. ¶ xk . (cvm (c) ρ

(1.256)

To recognize a usual Hamiltonian structure, we can pass to a space of even dimension (the “symplectization” of the contact space with 2n + 1 independent variables) using a new parameter λ, following [Arn1][App. 4]. Equations of the characteristics for a system of ideal gases. Here we use the parametrization (1.252), and we determine the characteristics for the law of ideal gases (1.253). The characteristic vector ξ for K 1 is given by

102

1 Thermostructure

ξ = (Ky1 ; y.Ky1 ; −(Kx1 + yKz1 )) = (0, x  ; y  .x  ; 0, −(yk +

1 (rk − r(c))), ˆ ρ

whence ξ = (0, (ρk ); −

 μ˜ ρ k

T

ρ

.ρk ; 0, −(

μ ˜k 1 + (rk − r(c))). ˆ T ρ

Then the equation of the characteristics is dx  dyk dy0 1 dz dx0 = 0, = x , = 0, = −(yk + (rk − r(c))), = x  .y  , ˆ dt dt dt dt ρ dt (1.257) that is, ρ

ρ

ρ

 μ˜ μ˜ de d 1 ds dρk d μ˜ k 1 k .ρ . ˆ = 0, = ρk , ( ) = 0, ( ) = − k + (rk − r(c)), =− k dt dt dt T dt T T ρ dt T

The solution is given as a function of initial conditions by e(t) = e0 , ρk (t) = ρk0 exp t, s = s 0 − r(c0 )t, ρ

ρ

T (t) = T 0 , μ˜ k (t) = ((μ˜ k )0 + λk t) exp −t,

∀k,

(1.258)

0

with λk = Tρ 0 (rk − r(cˆ0 )). The relation ρk (t) = ρk0 exp t implies ck (t) = ck0 , and hence the entropy evolution. Then the pressure evolution is expressed by P (t) = ρr(c)T ˆ = ρ 0 r(cˆ0 )T 0 exp t = P 0 exp t. The evolution of the system following the characteristic is thus made with constant temperature, energy and mass fractions. Then the differential form (1.180) is reduced to (1.17).

1.6.8 Real Solutions Lewis Formula Generally, the chemical potentials of a system with n constituents must be expressed by n relations, which are n state equations. Notably in the case of liquid solutions, various formulas of chemical potentials (more or less phenomenological) are proposed. An example is given by the Lewis formula (see [Pri-Kon, ch. II, 5.3, 8.1]) in global thermodynamics:

1.6 Thermosystems with Different Constituents

103

μk (P , T , Nˆ ) = R0 T log ak (Nˆ ) + μ0k (P , T ),

(1.259)

with ak > 0 a differentiable function of xj = 0, called the activity of constituent Ck , the chemical potential μk being associated with the constituent Ck (and with the system). When ak = xk , the solution is said to be ideal. This is the case of an ideal gas; see (1.251). Otherwise, it is said to be nonideal. State Equations for the Lewis Formula. Recall the Gibbs free energy, here expressed with the mole numbers with the differential form (see (1.197))  ˆ θ G,N = −dG − SdT + V dP + μk dNk ,   G= μk Nk = (Nk R0 T log ak (Nˆ ) + Nk μ0k (P , T )).

(1.260)

Let  be the map (T , P , (Nk )) = (−S, V , (μk )). In order that  be integrable ˆ (with respect to θ G,N = 0), we must have ∂μj ∂μk = , ∀k = j, ∂Nj ∂Nk

∂V ∂μk . = ∂Nk ∂V

(1.261)

Using the second formula of (1.260), we have V =

∂G  ∂μ0k = Nk . ∂P ∂P

(1.262)

We can also obtain the energy E as a function of (P , T ), thanks to the Euler operator ∂ ∂ P ,T = P ∂P + T ∂T : E = G − P V + T S = (I − P ,T )G =



Nk (I − P ,T )μ0k ,

(1.263)

 since Nk (I − P ,T )R0 T log ak = 0. The relations (1.262), (1.263) are identifik able with state equations, just like the relations μk = G Nk . Moreover, G must be a convex function of the (Nk ), and with the hypothesis of smoothness C 2 , the μk must be such that 

zk zj

∂μk ≥ 0, ∂Nj

∀zj ∈ R, j = 1, . . . , n.

When Nk → 0, the contribution of Ck to G must tend to 0; thus μk Nk → 0, whereas we must have μk → −∞ (as for ideal gases), notably for chemical reactions. For j = k, the chemical potential μj must be such that μj (N1 , . . . , Nk , . . . , Nn ) → μj (N1 , . . . , Nˆ k , . . . , Nn ) when Nk → 0, where Nˆ k means that the term Nk is missing. Since G is a homogeneous function of (Nk ), the activity coefficients ak are homogeneous functions of degree 0 of (Nk ), and thus depend only on (xk ).

104

1 Thermostructure

If μ0k (P , T ) is a concave function of (P , T ), for all k, then G also is a concave function of (P , T ), and we can define E(V , S) by the Legendre transformation E(V , S) = (−G)∗ (V , S), with fixed Nk . k Remark 21 With hypothesis μk = G Nk , the Gibbs function Gk is a function of the variables Pk , T ,  and Nk , and thus Gk (Pk , T , Nk ). Now Nk = xk N, and we let γk = PPk with P = Pk . A priori, γk is determined by the state equations for the pressures Pk (with γk = xk for an ideal gas). Therefore, μk = μk (γk P , T , xk N). We see that the Lewis formula assumes a certain separation of variables, which is not necessarily satisfied, especially with the van der Waals equation.

Real van der Waals Solutions Consider a mixture with n liquid constituents. We assume the additivity Properties (1.198) and that the modelling of each constituent is made through the van der Waals state equations (1.91), (1.98). We have to determine the chemical potentials, which we will find first using the global thermodynamics. We first assume that we have liquid constituents, with the following conditions: T < Tck ,

P > PSk (T ),

bk < τk < τl,k (T ), ∀k.

The temperature is less than the critical temperature for each constituent, and the pressure is higher than the saturation pressure. Then the van der Waals equations for the total energy E and the total pressure P are given, with different constants Cvk , bk , a0k , by  k ˆ Cv Nk Cvm (N) 1 = , =  1 k 2 1 m T E+V a0 Nk E + V a0 (Nˆ 2 )   Nk P 1 1 = R0 −( a0k Nk2 ) 2 , T V − bk Nk V T = R0



(1.264)

1 Nk Cvm Nˆ − a0m (Nˆ 2 ) 2 , V − bk Nk V E + 1 a0m (Nˆ 2 ) V

with Cvm (Nˆ ) = form (1.196) is



Cvk Nk , a0m (Nˆ 2 ) = ˆ

θ S,N = dS −



a0k Nk2 . Now the basic differential

 μk P 1 dE − dV + dNk . T T T

Thus we must have the Schwarz conditions of integrability

1.6 Thermosystems with Different Constituents

∂ μk ∂ 1 ( ), ( )=− ∂Nk T ∂E T

∂ P ∂ μk ( ). ( )=− ∂Nk T ∂V T

105

(1.265)

By integration, we obtain, up to arbitrary functions fk (Nˆ ), −

μk bk Nk V − bk Nk = R0 log − R0 + R0 log Nk T Nk V − bk Nk (a k Nk ) 1 1 ˆ − Cvk log − Cvk log Cvm (Nˆ ) + fk (N). +2 0 V T T

(1.266)

Thus we obtain μk as a function of (V , T , (Nk )), or using (1.264), of (V , E, (Nk )). k m ˆ We can verify that the family φk =  R0 log Nk − Cv log Cv (N ) is integrable, that is, φk dNk is an exact differential φk dNk = dφ, with φ(Nˆ ) = R0



(Nk log Nk − Nk ) − [Cvm (Nˆ ) log Cvm (Nˆ ) − Cvm (Nˆ )],

(1.267)

that is, with the usual function ϕ(u) = u log u − u, φ(Nˆ ) = R0



ˆ ϕ(Nk ) − ϕ(Cvm (N)).

(1.268)

Thus we can eliminate the term φk from − μTk . If we keep it, we have to add the term (−R0 + Cvk ) log N) to it in order to have a homogeneous chemical potential of zero order. Now using the additivity of entropy of each constituent, we have S=



Sk =



R0 log

V − bk Nk  k + Cv Nk log(Cvk T ). Nk

(1.269)

Then we obtain the chemical potential by −

∂S V − bk Nk bk Nk μk = = R0 [log − ] T ∂Nk Nk V − bk Nk + Cvk

(a k Nk ) + Cvk log Cvk − Cvk − R0 . log T + 2 0 VT

(1.270)

If we compare (1.266) or (1.272) with (1.270), we obtain that the difference is given by the constant terms (Cvk log Cvk − Cvk − R0 ) if we have suppressed the term φk , and if not, by the function fk (x), ˆ fk (x) ˆ = (Cvk log Cvk − Cvk − R0 ) − R0 log xk + Cvk log Cvm (x), ˆ

(1.271)

106

1 Thermostructure

with xk =

Nk τ = , N τk

 1 1 N . = = τ V τk

Now, using the homogeneity, we can express μk as a function of the local variables, since μk (V , E, (Nk )) = μk (V , V e, ˜ (V τk )) = μk (1, e˜, (τk )) ˆ N(xk )) = μk (τ, e, ˆ (xk )). = μk (Nτ, N e,  V Then with respect to τk = NVk , τ = N , N = Nk , and xk = function of (T , (τk )) or (τ, T , (xk )), using (1.270) −

(a k ) 1 μk bk ]+2 0 = R0 [log(τk − bk ) − T τk − b k τk T + Cvk

log T +

Cvk

log Cvk

− Cvk

τ τk ,

we obtain μk as a

(1.272)

− R0 .

Of course with the first relation (1.264), we can express μk as a function of e, ˜ (τk ). In order to compare (1.270) to the Lewis formula, we would have to express V with respect to the total pressure P . But V would be obtained by solving the second equation (1.264) with (P , (Nk ), T ) given, which is an equation of degree n + 2. Note that as with a mixture of ideal gases, we have obtained for a mixture of van der Waals liquids, from (1.264), the chemical potentials (up to a family of functions φk of Nˆ ), and thus an integrable map φ (up to a function φ such that  dφ = φk dNk ), with φ (V , E, Nˆ ) = ( PT , T1 , ( μTk )), giving the entropy S up to a “constant” S0 . With the hypothesis of additivity of entropy, we see that the φk are constant. Starting from other state equations of the constituents and using additivity properties of the pressure and equality of temperature, we could generalize in order to find the chemical potentials (up to an integrable function of (Nk )).

1.7 Chemical Reactions Consider a system with p constituents C1 , . . . , Cp . It is possible that in certain domains of temperature and pressure, some chemical reactions appear, changing the system into a new system with other constituents. For the study of these reactions, it is necessary to take into account the complete family of the n occurring constituents in these reactions.

1.7 Chemical Reactions

107

1.7.1 The Case of One Chemical Reaction Stoichiometric Coefficients Let a system consisting of ν− constituents Ci and ν+ constituents Cj in a chemical reaction R given with stoichiometric integer coefficients αi > 0 and αj > 0 be given by α1 C1 + · · · + αν− Cν−  α1 C1 + · · · + αν + Cν + .

(1.273)

Let R− , R+ be the sets that form a partition of the set R defined by R− = {Ci , i = 1, . . . , ν− }, R+ = {Cj , j = 1, . . . , ν+ }, R = R− ∪ R+ . Let41 ν = ν− + ν+ . Let Cj +ν− = Cj with j = 1, . . . , ν+ . The stoichiometric coefficients are obtained through the balance law of atoms that are the components of molecules (Ci ) and (Cj ). Let (Ak ), k = 1, . . . , p, be the atoms in the set of molecules of the reaction. Let Ci = (A1 )ni,1 · · · (Ap )ni,p , be the molecule Ci made up of ni,1 atoms A1 ,. . . , ni,p atoms Ap , with ni,k a positive integer or zero. Similarly for Cj , with nj,k . Balancing a reaction consists in determining the coefficients αi and αj in order to keep each species of atoms, which correspond to the equations ν− 

αi nik =

i=1

ν+ 

αj nj k ,

k = 1, . . . , p.

(1.274)

j =1

Let nj +ν− ,k = nj k . Thus we define the occupation matrix of molecules by the atoms: N = (nl,k ), l = 1, . . . , ν = ν− + ν+ , k = 1, . . . , p. Let Nk be the vector with components nl,k , l = 1, . . . , ν. Let αˆ = (−α1 , . . . , −αν− ; α1 , . . . , αν + ).

(1.275)

Then equation (1.274) corresponds to < α, ˆ Nk > = 0,

k = 1, . . . , p,

or α.N ˆ = 0.

(1.276)

generally we have ν− = 2, since the chemical reactions come from the meeting of two different molecules, and a chemical reaction with ν− ≥ 2 constituents is obtained by a succession of “elementary” chemical reactions.

41 Very

108

1 Thermostructure

Let E be the vector space generated by the Nk . Then the problem is to determine αˆ ∈ E ⊥ with integer components in Z (and moreover, negative for the ν− first components, positive for the others). These are linear Diophantine equations. We immediately remark (with the required positivity conditions) the following: (i) If αˆ is a solution of (1.276), then mαˆ is also a solution of (1.276) with m ∈ N ∗ ; for m = −1, this is equivalent to exchanging the direct reaction with the inverse reaction. (ii) If αˆ 1 and αˆ 2 are solutions of (1.276), then αˆ 1 + αˆ 2 is a solution of (1.276). Let pE be the dimension of the space E. Then if pE ≥ ν, there is no solution of (1.276). If pE < ν, then dim E ⊥ = ν − pE . The problem is unspecified. If we give integer values to the extra αi , we obtain a linear system that can be solved using Cramer’s formulas, so that we have a solution in the space Q of fractions. But multiplying by a positive integer gives a solution in Z ν . Then it comes down to a matter of positivity of αi and αj . Except for that, we have ν − pE independent solutions αˆ (with integer components). In fact, we search for “minimal” solutions in the sense that if αˆ is a solution, then m1 αˆ is not a solution for every m ∈ N, m > 1. This means that the αi and αj are mutually relatively prime (when they are distinct). Other consequences. Let mA k be the mass of atom Ak . Then on multiplying relations (1.274) by mA , we obtain the conservation of the mass of the system for k the atoms   αi nik mA αj nj k mA (1.277) k = k, i,k

j,k

which gives the conservation of mass of the system for the molecules, with mi the mass of the molecule of constituent Ci : M=

 i

αi mi =



αj mj .

(1.278)

j

Example 3 Consider the chemical reaction (between gases at high temperature; see [Pri-Kon][ch. II, 9.2]) Cl + H2  H Cl + H. Here we have p = 2, ν = 4, V1 = VCl = (1, 0, 1, 0), V2 = VH = (0, 2, 1, 1) with the above notation. The stoichiometric coefficients of the above chemical reaction correspond to αˆ = (−1, −1; 1, 1). But there is another independent family, αˆ  = (−1, −2; 1, 3), which is the reaction Cl + 2H2  H Cl + 3H.

1.7 Chemical Reactions

109

Hence there is no uniqueness of the stoichiometric coefficients to balance the reaction (which gives a priori another reaction). Of course we can suggest other choices of coefficients such that Cl + nH2  H Cl + (2n − 1)H. (But it is only a linear combination of the two previous reactions: the coefficients of the first reaction being multiplied by n1 = n − 1, and those of the second reaction by n2 = 2 − n.) ¶

Degree of Reaction We first choose to study chemical reactions in global thermodynamics as it is usually done. A priori, the system is described by the variables E, V , (Nl ), S, T , P , (μl ) with l = 1, . . . , n, where Nl with l = 1, . . . , ν− is the mole number of Cl , and Nl = Nj with l = j + ν− , j = 1, . . . , ν+ , is the mole number of Cj during the chemical reaction. The variations in the mole numbers Ni , Nj , and the differentials dNi , dNj are then proportional to the stoichiometric coefficients, which allows us to define the degree of reaction ξ or its differential dξ by dξ =

dNν dN  dNν− dN1 = ··· = = 1 = ··· =  + . −α1 −αν− α1 αν+

(1.279)

Note that if the stoichiometric coefficients are multiplied by an integer m, and the mole numbers also by m, the degree of reaction is not changed. Taking N− =



Ni , N+ =



Nj , N = N− + N+ ,

α− =



αi , α+ =



αj ,

we also have dξ =

dN− dN+ = . −α− α+

(1.280)

If α− = α+ , we again have dξ = α+dN −α− , which connects the degree of reaction ξ to the total mole number. The relations (1.279) imply Ni = −αi ξ + N0,i , i = 1, . . . , ν− ,

 Nj = αj ξ + N0,j , j = 1, . . . , ν+ , (1.281)

 represent the mole numbers of C , C  at the start of the reaction. where N0,i , N0,j i j

Let ξ0,i =

N0,i αi ;

then we have Ni = −αi (ξ − ξ0,i ),

 Nj = αj (ξ + ξ0,j ).

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1 Thermostructure

Naturally ξ must be such that 0 ≥ ξ ≥ ξ0,m = min ξ0,i (the chemical reaction stops when one of the left constituents vanishes). If we replace the numbers Ni by the molar density ni = NVi , and if we set ξ˜ = Vξ , then the relations (1.280) become d(Ni + αi ξ ) = d(V (ni + αi ξ˜ )) = 0,

(1.282)

and thus V (ni + αi ξ˜ ) is a constant denoted by Ni0 . Hence with “constants” n0i and ξ˜i0 dependent only on V , we have ni + αi ξ˜ =

Ni0 = n0i . V

(1.283)

New Basic Differential Form. Affinity of the Reaction Let α, ˆ with components αˆ l , 1 ≤ l ≤ ν− + ν+ , be defined by αˆ l = −αl , 1 ≤ l ≤ ν− , αˆ l = −αl−ν− ,

ν− + 1 ≤ l ≤ ν− + ν+ .

(1.284)

Let μˆ = (μ1 , . . . , μn− ; μ1 , . . . , μn+ ),

αˆ = (−α1 , . . . , −αn− ; α1 , . . . , αn + ).

We define the affinity of the reaction by A=−



μl αˆ l = −μ. ˆ α. ˆ

(1.285)

The differential form of matter θ mat is expressed by θ mat =



μl dNl =



μl αˆ l dξ = −Adξ = θ ξ .

(1.286)

ˆ

Then the basic differential form θ S,N , see (1.196), here with the mole numbers, becomes θ S = dS −

 1 P P 1 1 1 dE − dV + μk dNk = dS − dE − dV − Adξ. T T T T T T (1.287)

Now if we have a map such that (V , E, Nˆ ) = ( PT , T1 , Tμˆ ) from the state ˆ such that equations, then using (1.281), we obtain a new map ˆ , E, ξ ) = ( (V

P 1 A , , ). T T T

(1.288)

1.7 Chemical Reactions

111

ˆ is integrable with respect to the differential form θ S when We have to verify that ˆ ˆ is integrable with respect to θ S,N . The function S(V , E, (Nk ) that satisfies θ S,N = 0 0 becomes with (1.281) and (1.284) S(V , E, (Nk ) = S(V , E, (Nk + αˆ k ξ ), so that we have  ∂S  1 1 ∂S = αˆ k = − μk αˆ k = A. ∂ξ ∂Nk T T Now an evolution of ξ does not a priori determine an evolution of the other variables. But this is possible if there are some constraints, for instance if V and E, or P and ˆ we have T , are constant. In this last case, from the state equations, that is, from , P 1 ˆ ˆ 1 (V , E, ξ ) = T and 2 (V , E, ξ ) = T , giving V and E as functions of ξ . Then from (1.287) we can obtain the entropy if θ S = 0. We can also determine the heat of reaction. Example When all constituents are ideal gases, with the state equation (1.226), the hypothesis that temperature and pressure (or temperature and volume) are invariant during the chemical reaction allows us to give the heat of the chemical reaction. (i) Volume V and T invariant. The received heat quantity during the evolution of ξ from 0 to a degree of reaction ξ0 is expressed by 



ξ0

Q=

θ

Q

ξ0

=

0

 (dE + P dV ) =

0

0

ξ0

dE = E(ξ0 ) − E(0) = T Cvm (α)ξ ˆ 0.

(ii) Pressure P and T invariant. The heat quantity is expressed thanks to the enthalpy by 

ξ0

Q=



ξ0

d(E+P V ) =

0

0

dH = H (ξ0 )−H (0) = T [Cvm (α)+R ˆ 0 (α+ −α− )ξ0 ].

Chemical Kinetics The reaction rate in the direct sense is expressed as a function of concentrations ni = Ni /V of the constituents Ci , i = 1, . . . , ν− , with a coefficient kdir > 0 that depends42 on T , by (see [Pri-Kon][ch. II.9.5.5]) Nν 1 dξ N1 = kdir ( )α1 · · · ( − )αν− . V dt V V

(1.289)

Here there is no invariance by changing α and N into mα and mN, and we see the importance of having αˆ “minimal.” In the relations (1.279), extensive (or global)

42 It

is given by the Arrhenius law; see [Pri-Kon].

112

1 Thermostructure

quantities Nk are involved, and thus ξ , whereas on the right-hand side of (1.289), the intensive (or local) quantities nk are involved (and therefore the term with 1/V on the left-hand side of (1.289)). Let |α| =



αi ,

αν

0 Kdir = kdir α α (−1)|α| .

α α = α1α1 · · · αν−− ,

Then the kinetic equation (1.289) becomes dξ αν = R− = kdir V 1−|α| N1α1 · · · Nν−− dt

(1.290)

0 = Kdir V 1−|α| (ξ − ξi0 )α1 · · · (ξ − ξν0− )αν− .

Equation (1.290) is not sufficient to determine the evolution of the reaction and that of the system. We have to specify some constraint on the system, as indicated below. Solving equation (1.290) with constant volume. We assume that V = 1. Let P− (ξ ) = Kdir (ξ − ξ10 )α1 · · · (ξ − ξν0− )αν− ,

0 with Kdir = Kdir V 1−|α| .

With (1.290), we obtain dξ = P− (ξ ). dt

(1.291)

We assume that the ξi0 are all distinct. If we split into rational fractions the product (ξ − ξ10 )−α1 · · · (ξ − ξν0− )−αν− =





i

νi =1,...,αi

λi,νi (ξ − ξi0 )−νi ,

 with λi,αi = j =i (ξi0 − ξj0 )−αj , then the solution of (1.291) is expressed in a neighborhood of 0 by ϕ(ξ ) − ϕ(0) = t

(1.292)

with −1 [ ϕ(ξ ) = Kdir

ν− αi   i=1 νi =2

(ξ − ξi0 )−νi +1 ×

λi,νi + λi,1 log |ξ − ξi0 |]. −νi + 1

Let ψ(ξ ) = ϕ(ξ ) − ϕ(0). For 0 ≤ ξ < ξ0,m , we have dϕ(ξ ) 1 dψ(ξ ) = = = 0, dξ dξ P− (ξ )

(1.293)

1.7 Chemical Reactions

113

and we can apply the theorem of implicit functions (see, for example, [Dieud2][ch. 10.2]), which provides the solution of (1.291) for ξ ∈ [0, ξm0 [ by ξ = ψ −1 (t)

 hence Ni = −αi ψ −1 (t) + N0i , Nj = αj ψ −1 (t) + N0j .

(1.294)

Moreover, we see that for ξ → ξ0,m , we have t → +∞, which means that the time when one of constituents disappears is infinite. We specify the behavior toward infinity. For αm > 1, we have λm,αm (ξ − ξm0 )−αm ≈ Kdir t; thus (ξ − ξm0 ) ≈ (

Kdir t −1/αm ) . λm,αm

(1.295)

For αm = 1, we have λm,αm log |ξ − ξm0 | ≈ Kdir t; thus |ξ − ξm0 | ≈ exp(

Kdir t ). λm,αm

¶

(1.296)

Taking the inverse chemical reaction into account gives the global reaction rate in the form (with a coefficient kinv > 0) 1 dξ  αν α = kdir nα1 1 · · · nν−− − kinv (n1 )α1 · · · (nν+ ) ν+ . V dt

(1.297)

Solving equation (1.297) with constant volume. Let 





Kinv = kinv (−1)α (α  )α V 1−|α | ,



 α1  P+ (ξ ) = Kinv (ξ + ξ0,1 ) · · · (ξ + ξ0,ν ) +

αν +

.

If we transfer this into (1.297), we obtain dξ = P (ξ ) = P− (ξ ) − P+ (ξ ); dt

(1.298)

P (ξ ) is a polynomial of degree d ◦ P = max(d ◦ P− , d ◦ P+ ) = max(|α|, |α  |). The decomposition into irreducible rational fractions of the inverse of P (ξ ) gives a solution of (1.298) in a form similar to (1.292) with (1.293): ψ(ξ ) = t. With the hypothesis P (0) > 0, let ξm = minP (ξ )=0,ξ >0 be the first positive zero of P (ξ ). Then we have P− (ξm ) = P+ (ξm ), which corresponds to equilibrium. In ) 1 the interval [0, ξm [, we have dψ(ξ dξ = P (ξ ) = 0, and the application of the implicit function theorem gives the solution of (1.298) when ξ ∈ [0, ξm [ by ξ = ψ −1 (t),

(1.299)

114

1 Thermostructure

which also gives Ni and Nj , as in (1.294). When ξ → ξm , we again have t → ∞. Thus the system reaches equilibrium in infinite time. We can also give the behavior at infinite time: if ξm is a simple zero of P (ξ ), we will have exponential behavior, similar to (1.296). The condition for ξm to be a double zero of P (ξ ) is given by P  (ξm ) = P− (ξm ) − P+ (ξm ) = 0, or even by (log P− ) (ξm ) − (log P+ ) (ξm ) = 0, and thus αν + α1 αν− α1 + ···+ = + · · · + ,   ξm − ξ0,1 ξm − ξ0,ν− ξm + ξ0,1 ξm + ξ0,ν + which gives asymptotic behavior of type (1.295). We remark that (i) if P− (0) > P+ (0) (for example if one of the constituents Cj is missing at the start, which gives P+ (0) = 0), then we have dξ dt (0) > 0, and thus ξ is increasing, hence positive, and the reaction goes essentially in the direct sense. Furthermore, we note that ξm < ξ0,m (by comparing the present chemical reaction to the direct chemical reaction). Indeed, we have dξ dξ− = P− (ξ ) − P+ (ξ ) < = P− (ξ ). dt dt Integrating over time, we obtain 

∞

ξm = 0

dξ dt < dt



∞ 0

dξ− dt = ξ0,m . dt

(ii) if P− (0) < P+ (0), then dξ dt (0) < 0, whence ξ is decreasing, therefore negative, and the reaction goes into reverse. ¶ Remark 22 The chemical reaction rate is not always expressed in the form (1.297): instead of stoichiometric exponents, we can have fractional exponents (see [Pri-Kon]). Moreover, instead of molar fractions, there may be activities ai , aj , with stoichiometric exponents, so that  1 dξ  αν = kdir a1α1 · · · aν−− − kinv (a1 )α1 · · · (aν + )αν+ . V dt

¶

(1.300)

Note that the opposite of the affinity of the reaction is expressed thanks to the formula (1.259) of the chemical potential for nonideal solutions by

1.7 Chemical Reactions

115

  −A = μ. ˆ αˆ = R0 T α˜ l log al + α˜ l μ0l (P , T )  α˜  = R0 T log al l + α˜ l μ0l (P , T )   α  ai i ] + α. ˆ μ˜ 0 (P , T ). = R0 T [log (aj )αj − log

(1.301) ¶

Evolution of the entropy with constant T and P . Thanks to the state equations, the variables (E, V ) depend on ξ , and there is a map ξ ∈ R → (E, V , (Nk )ξ ), and thus a path in M that can be lifted in a leaf Ls0 ⊂ M × R if θ S = 0, and so < θS,

 μk dNk dS d d >= − < θ Q, > + = 0, dξ dξ dξ T dξ

and hence < θS,

 μk αˆ k dS 1 dE P dV d >= − − + = 0. dξ dξ T dξ T dξ T

(1.302)

d P dV Let Q = < θ Q , dξ > = dE dξ + T dξ , which gives the rate of heat of the reaction. Then with A the affinity of the chemical reaction, we have

A Q dS = + . dξ T T

(1.303)

We note that if E and V are affine functions of ξ , then Q is independent of ξ , and S, which is a priori a concave function of (E, V , (Nk )), is again a concave function of ξ . The evolution equation of the entropy S is expressed by dS dS dξ Q A dξ = =[ + ] . dt dξ dt T T dt

¶

(1.304)

Chemical reaction for a system of ideal gases. Let Cˆ v = (cvk ), with k = 1, . . . , n, and   α= αi , α  = αj , δα = −α + α  ; and thus with the duality notation, < 1, αˆ > = δα, < Cˆ v , αˆ > =



−cvi αi +



cvj αj , −A = < μ, ˆ αˆ > = μ. ˆ α. ˆ

Recall that the chemical potential μk of an ideal gas is expressed by (1.251), which is written here (with PTk = NVk R0 = nk R0 ) μk = −Cvk log T + R0 log (nk R0 ) − S0 . T

(1.305)

116

1 Thermostructure

The affinity is given by    A =( Cvk αˆ k ) log T + R0 ( αˆ k log(nk R0 ) − αˆ k S0k . T

(1.306)

With (T , P ) constant, with initial conditions V 0 = N 0 R0 PT , E 0 = Cvk Nk0 T , the state equations of ideal gases are V = NR0

T T = V 0 + ξ δαR0 , P P

E=



Cvk Nk T = E 0 + ξ Cvk αˆ k T ,

and thus Q = < Cˆ v , αˆ > + R0 δα. T   N  α Let A− = kdir ( NVi )αi , A+ = kinv ( Vj ) j . Then the evolution equation of entropy S is expressed by (see (1.304)) A Q dξ A1 + Q dS =[ + ] = R0 V [log A− −log A+ ].[A− −A+ ] +V .[A− −A+ ]. dt T T dt T The first term, i.e., AT 0 dξ dt , is positive, since (log A− − log A+ )(A− − A+ ) ≥ 0, the dS logarithm being an increasing function. If dξ dt = 0, whence A− = A+ , then dξ is a constant, which can be null for a particular value of S0 if δα = 0.

Chemical Reactions in Local Thermodynamics First, it is advisable to write the formulas concerning the degree of reaction in local thermodynamics according to the formulas (1.279) in global thermodynamics. With the molar masses mi and the concentrations ci = ρρi , by conservation of the total mass M, we again have dξ =

dNi mi dMi dci dNi = = =M , −αi −αi mi −αi mi −αi mi

and similarly for the cj . Thus setting ξ˜ = d ξ˜ =

Thus we have

dcj dci =  , −αi mi αj mj 1 dξ V dt

=

M d ξ˜ V dt

ξ M,

we have

i = 1, . . . , ν− , j = 1, . . . , ν+ . ˜

= ρ ddtξ . Let

(1.307)

(1.308)

1.7 Chemical Reactions

117 −αν−

1 m ˆ −α− = m−α · · · mν − 1







, (m ˆ  )−α− = (m1 )−α1 · · · (mν+ )−αν+ .

Then the local chemical kinetic equation is d ξ˜ 1  αi  1  α = kdir m ˆ −α− ˆ  )−α+ ρi − kinv (m (ρj ) j dt ρ ρ  α    α = kdir m ˆ −α− ρ α−1 ˆ  )−α+ ρ α −1 (cj ) j . ci i − kinv (m

(1.309) ¶

Here also we don’t have any information on dρ dt , and the evolution of the system can be specified if we know the constraints on the system.

1.7.2 System of r Independent Chemical Reactions Consider a system with m constituents (family of molecules) Cl , l = 1, . . . , m, likely to chemically react through r “independent” reactions and stable with respect to chemical reactions (i.e., all constituents that occur in the system thanks to these reactions are taken into account). The independence of the reactions implies r ≤ m − 1. Let E be the set of Cl , l = 1, . . . , m. For each reaction Rk , we note that R− k (respectively R+ k ) is the set of constituents of the left-hand (respectively right-hand) side of the reaction Rk . Thus we have for each k a partition of the set of constituents + 0 0 of the system: R− k ∪Rk ∪Rk , with Rk corresponding to the constituents that do not occur in the kth reaction. Here νk,− (respectively νk,+ ) is the number of constituents + − − + + of R− k (respectively Rk ); Ck,i ∈ Rk , i = l, . . . , νk,− (respectively Ck,j ∈ Rk , j = 1, . . . , νk,+ ), are the constituents on the left-hand (respectively right-hand) side of Rk . Thus the chemical reaction Rk is denoted with stoichiometric coefficients αk,i = −  = α + by αk,i and αk,j k,j − − − + + + αk,1 Ck,1 + · · · + αk,ν C−  αk,1 Ck,1 + · · · + αk,ν C+ . k,− k,νk,− k,+ k,νk,+

Here we use a global thermodynamic modelling of the system. The balance of variation of the mole number of each constituent through the reaction Rk is, with the degree of reaction ξk , dξk =

 dNν dNk,1 dNk,νk,− dNk,1 = ··· = =  = · · · =  k,+ , −αk,1 −ανk,− αk,1 ανk,+

(1.310)

or with the new notation, with  = ±, i− = i ∈ [1, νk,− ], i+ ∈ [1, νk,+ ] (also denoted by j ):

118

1 Thermostructure

dξk =

 dNk,i  αk,i 

,

∀, i ∈ [1, νk, ].

(1.311)

Then we define a straight line k in the space of mole numbers, of direction vector    αˆ k = (−αk,1 , . . . , −αk,νk,− ; αk,1 , . . . , αk,ν ) = (αk,i ). k,+ 

Passing from the (k, i ) numbering to that with l, 1 ≤ l ≤ m, can be made through the maps (with  = ±):  ϕk, of Rk in E, ϕk, (k, i ) = l such that Ck,i = Cl , for i ∈ [1, νk, ]  − − ϕk,− of Rk in E, ϕk,− (k, i) = l such that Ck,i = Cl , +  ϕk,+ of R+ k in E, ϕk,+ (k, j ) = l such that Ck,j = Cl  . Furthermore, if each constituent appears only on at most one side of each reaction, we can define a map on [1, r] × [1, m] such that (k, l) = −1 if Cl ∈ Rk− ,

(k, l) = +1 if Cl ∈ Rk+ ,

(k, l) = 0 if Cl ∈ Rk0 .

Let dNl be the differential of the mole number Nl of Cl . If Cl takes part in the reaction Rk , let dk Nl be the restriction of the differential to k . Thus we have dk Nl = dNϕ −1 (l) = −αϕ −1 (l) dξk , if Cl ∈ R− k, k,−

dk Nl =

dN  −1 ϕk,+ (l)

k,−

=

α  −1 dξk , ϕk,+ (l)

(1.312)

si Cl ∈ R+ k,

and dk Nl = 0 if Cl does not take part in the reaction Rk . We set  = α  −1 α˜ k,l

ϕk, (l)

Recall that α  −1

ϕk, (l)

(1.313)

is positive, whence

− = −αϕ −1 (l) if Cl ∈ R− α˜ k,l k, k,−

Then we deduce the relations    dk Nl = α˜ k,l dξk , dNl = k, Cl ∈Rk

if Cl ∈ Rk .

+ α˜ k,l = α  −1

ϕk,+ (l)

if Cl ∈ R+ k.

with k such that Cl ∈ Rk .

(1.314)

=± k

Therefore, with constants Nl0 for 1 ≤ l ≤ m,, we have Nl = Nl0 +





=±, k, Cl ∈Rk

 α˜ k,l ξk .

(1.315)

1.7 Chemical Reactions

119

From the map such that (V , E, (Nk )) = ( PT , T1 , (− μTk )), due to the state A ˆ such that ((V ˆ equations, we deduce a map , E, (ξj )) = ( PT , T1 , ( Tj )). Then ˆ since from an integrable map , we obtain an integrable map , S(V , E, (Nk )) = S(V , E, (Nk0 ) +



 α˜ k,l ξl ,

and thus  ∂S ∂Nk  μk ∂S Al  α˜ k,l . = =− ξl = ∂ξl ∂Nk ∂ξl T T Now the chemical kinetic equations (see (1.309) and (1.297)) with coefficients kdir,k and kinv,k , here denoted by Kk, (with  = − for dir,  = + for inv), are  α 1 dξk =− Kk, nl k,l V dt 

with l such that Cl ∈ Rk .

(1.316)

l

Since r < m − 1, the evolution equations of r degrees of reaction with l such that Cl ∈ Rk are  dξk  Kk, V 1−αk =− dt =± Nl = Nl0 +





 αk,l

Nl

with αk =

,

l, Cl ∈Rk



 αk,l

l

(1.317)



α˜ k ,l ξk  . 

  ,k  , Cl ∈Rk

We also have the evolution relations of mole numbers as functions of the r degrees of reaction, for 1 ≤ l ≤ m, dNl = dt



 α˜ k,l

=±,k, Cl ∈Rk

dξk , dt

(1.318)

and evolution equations of the m mole numbers Nl , 1 dNl = V dt

 , k, Cl ∈Rk

 α˜ k,l





(

  ,Kk ,  l  , C  ∈R  l

Nl  α    ) k,l . V

(1.319)

k

Remark 23 Chemical reactions with local thermodynamics. We are induced to  define the coefficients νk,l , which we call mass stoichiometric coefficients such that (with ml the molar mass of constituent Cl )   = α˜ k,l ml . νk,l

120

1 Thermostructure

The conservation of mass for the kth reaction is given by 

 νk,l = 0,

∀k = 1, . . . , r.

(1.320)

,l

The variations in concentration are related to the degrees of reaction by  dk cl = νk,l d ξ˜k ,

dcl =



d k cl =

k



 νk,l d ξ˜k .

(1.321)

,k

We write d cˆ = νd ξˆ , with the matrix43 ν = (νk,lk,l ). Then we have 

cl =



 ˜ νk,l ξk + cl0

,k

with (cl0 ) positive constants such that



cl0 = 1 (from (1.320)).

¶

Entropy of the system in global thermodynamics. As in the case of a system with only one chemical reaction, we have to know how the energy and the volume depend on the degrees of reaction ξk in order to have (i) a complete system of equations: recall that the volume V occurs in the evolution equation (1.317), (ii) the heat due to the chemical reactions and the entropy variation. This can be obtained from the state equations with constant T and P , as in the ˆ ) = (Nj (ξ )) ∈ R m has case of ideal gases. Then the map ξ = (ξk ) ∈ R r → N(ξ m+2 of variables (E, V , Nˆ ). When the map ψ is affine (as a lifting ψ in the space R in the case of ideal gases), if S is a concave function of (E, V , Nˆ ), then the entropy S ◦ ψ is again a concave function of ξ = (ξk ) ∈ R r . Then with constants (T , P ), the differential form θ S , with the differential form θ Q associated with the heat, becomes ψ ∗ (θ S ) = dS − ψ ∗ (θ Q ) − ψ ∗ (θ Q ) =

 Ak T

 ∂E ∂V ( +P ) dξk , ∂ξk ∂ξk

dξk , with Ak = −



 α˜ k,l μl ,

(1.322)

, l, Cl ∈Rk

where Ak is the affinity of the kth chemical reaction.44 Then we deduce the evolution equation of entropy

43 If we assume that each constituent appears in only at most one side of each reaction; if not, we must consider  as an additional parameter. 44 From relation (1.320), we see that the choice of the (μ ) up to an additive function does not i matter; we can take μci for μi , in agreement to the Duhem equation.

1.7 Chemical Reactions

121

 ∂S dξk dS = dt ∂ξk dt k

with the evolution equations of ξk given by (1.317).  l  Example 4 Using the state equations V = R0 PT Cv Nl of ideal Nl and E = T gases, then with (T , P ) constant, we have dV = R0

T   α˜ k,l dξk , P

dE = T

k,,l

Let λVk =

 ,l

 α˜ k,l , λE k =

dV =

 ,l



 Cvl α˜ k,l dξk .

k,,l

 Cvl α˜ k,l . Thus we write

T  V λk dξk , P k

dE = T



λE k dξk .

k

Then the differential form of heat becomes  V (λE ψ ∗ (θ Q ) = T k + λk )dξk . k

Thus for a system of ideal gases, the differential of entropy is expressed, with constants (T , P ), by dS =



V (λE k + λk + Ak )dξk ,

(1.323)

k

with the affinity Ak and the chemical potential μk given by (1.322) and (1.251). Remark 24 On the possibility of cycles in chemical reactions. If the evolution of a system during chemical reactions is irreversible, it is impossible to have cycles in the whole space. But in projection, cycles may exist, notably for the concentration of certain constituents. If the evolution of the system is reversible, then it is possible to have cycles in leaves (FS0 ). For examples we refer to [Pri-Kon][ch. V, 19.4, 19.6]), notably for the Brusselator, the three-molecule model, the Belusov–Zhabotinsky reaction, and the simplified model called FKN (Field, Koros, Noyes). In the threemolecule model, there are, in fact, six molecules, denoted by A, B, D, E and X, Y . The concentrations of constituents A, B, D, E, are kept constant, the mole numbers NA , NB , ND , NE are constant, and we assume that the volume V is constant. The evolutions of NX , NY , of the constituents X, Y , are obtained through the chemical kinetic equations

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1 Thermostructure

dNX = k1 NA − k2 NB NX + k3 NX2 NY − k4 NX , dt dNY = k2 NB NX − k3 NX2 NY . dt

(1.324)

The stationary solution is (NX )S =

k1 NA , k4

(NY )S =

k4 k2 NB . k1 k3 NA

The stability of this state depends on the eigenvalues of the Jacobian matrix at the stationary solution, the matrix being given by 

k2 NB − k4 k3 (NX )2S −k2 NB −k3 (NX )2S

.

Let a = k2 NB , b = k3 (NX )2S . The eigenvalues are the roots of the equation (a − k4 − λ)(−b − λ) + ab = 0, therefore of λ2 + (−a + b + k4 )λ + bk4 = 0. The product and the sum of the roots being given by λ1 λ2 = bk4 > 0,

λ1 + λ2 = a − b − k4 ,

these two roots have a positive real part if a − b − k4 > 0, which is expressed by NB > kk42 + kk32 (NX )2S , thus by NB >

k4 k3 k12 2 + N . k2 k2 k42 A

If this inequality is satisfied, this leads to an oscillating state, for which we refer to [Pri-Kon][ch. V, 19]. ¶

1.8 Thermodynamic Fields 1.8.1 Thermodynamics Depending on Space Consider a system occupying a bounded connected domain M ⊂ R 3 for which we want to give a model of the thermodynamic state at every point x ∈ M and at every time t ∈ R of the system in evolution. Here the system is static, that is, the evolution depends only on diffusion.

1.8 Thermodynamic Fields

123

At first, we are led to use the product space M × F with variables (x, j ), with j = (τ, e, s, P , T ) for a system with one constituent, with two degrees of freedom in local thermodynamics. Let π be the canonical projection of the space M × F onto M; thus (M × F, M, π) has the trivial structure of a fibre bundle of base M. A smooth section ϕt of the fibre bundle indicates the thermodynamic state ϕt (x) ∈ F at time t and at each point x ∈ M of the (configuration space of the) system. Taking the state equations into account, we can replace the space F by the space  (of dimension three), and thus consider the space M ×  . Let S 0 be the foliated manifold of  with leaves s 0 = (Ls0 ), s0 ∈ R,, and let M × S 0 be the foliation of M ×  , with leaves M × Ls0 , with the trivial structure of a fibre bundle on M. If the evolution of the thermodynamic state expressed by ϕ(t, x) = ϕt (x) ∈ F, t ∈ J, x ∈ M, is compatible with the state equations, then ϕt (x) ∈  , so that we can keep only the independent variables (τ, e, s) ∈ F0 = M × R, and thus the foliation of this space. Then the evolution ϕt splits as ϕt = (ζt , ◦ ζt ),

ζt (x) = (τ, e, s)t,x , ◦ ζt (x) = ((τ, e)t,x ) = (

P 1 , )t,x . T T

The evolution ϕt (or ζt ) is θ -reversible if (ϕt (x))t ∈J is contained in a leaf Ls0 . With ϕx (t) = ϕt (x), the θ -reversible condition is equivalent to < θ s , (ϕx )∗ (

d ) > = 0, ∀t ∈ J. dt

More generally, an evolution (x(t), ϕ(x(t)), t ∈ J, is θ -admissible if < θ s , (ϕx )∗ (

d ) > ≥ 0, dt

∀t ∈ J,

(1.325)

that is, with the variable s0 transverse to the foliation, < ds0 , (ϕx )∗ (

d d ) > = (s0 (ϕx (t)) ≥ 0, dt dt

(1.326)

i.e., the irreversible entropy is increasing with evolution. A θ -reversible evolution at x is such that s0 (x) is time independent. Generalization. Consider a system occupying a domain M ⊂ E3 , whose thermodynamic state is described by (2n + 1) state variables denoted by ((uk ), s, (αk )), with k = 1, ·, n,45 and s the entropy, functions of the position x ∈ M, which is given as previously by the map ϕt ; the thermodynamic space F of the system is equipped with the differential form  θ = ds − αk duk . (1.327) 45 We

must change the previous notation of the thermodynamic variables in order to keep the variable x for the position of a point in M.

124

1 Thermostructure

We assume that the state equations give the map  , (u1 , . . . , un ) = (α1 , . . . , αn ), so that the restriction of the differential θ 0 = αk duk to the manifold  is exact θ 0 | = dsrev , or that ∗ (θ 0 ) is exact, that is, is integrable. Thus dsrev =



k (u1 , . . . , un )duk ,

therefore ∗ (θ ) = ds − dsrev = ds0 . (1.328)

Let ϕ be the map of R × M into  (or into F0 = M × R) such that ϕ(t, x) represents the thermodynamic state at (t, x). Then the pullback ϕ ∗ ( ∗ (θ )) of ∗ (θ ) by ϕ is expressed by ϕ ∗ ( ∗ (θ )) =



    ∂s  ∂uk ∂uk ∂s  − k dt + dx j . − k ∂t ∂t ∂x j ∂x j

We will simply denote ϕ ∗ ( ∗ (θ )) by θ . The condition of the physical evolution of the system (the second law) is expressed by < θ,

∂s  ∂s0 ∂ ∂uk >= − k = ≥ 0. ∂t ∂t ∂t ∂t

(1.329)

1.8.2 Diffusion Process and Flux Fick Law Let us consider a system with n constituents Ck , of molar mass mk , whose state variables are (e, (ρk ), s, T , (μk )), with ρk the specific mass, (μk ) the chemical ρ potential of Ck , the basic differential form being (see (1.180) with μk = μ˜ k ) θ = ds −

 μk 1 de + dρk . T T

Then we set u0 = e, uk = ρk , α0 = T1 , αk = − μTk , k = 1, . . . , n. We assume that there is no chemical reaction, and that each constituent evolves thanks to a diffusion process: ∂ρk + div Jk = 0, ∂t

(1.330)

with Jk (x) ∈ Tx M (identified with R 3 ); Jk is said to be the flux of ρk ; Jk is given by the Fick law, with a diffusion constant Dk > 0: Jk = −Dk ∇ρk .

(1.331)

1.8 Thermodynamic Fields

125

Thus equation (1.330) with (1.331) of ρk is parabolic. The Cauchy problem to solve (1.330) with (1.331), with given initial condition ρk (0) = ρk0 in its natural functional frame L1 (M), has a unique solution ρk (t) ∈ L1 (M) for positive t, and it is obtained by a semigroup G(t), t ≥ 0, so that the evolution of ρk in itself is irreversible. We can find the solution of the Cauchy problem thanks to the R 3 -valued associated diffusion process Xk (see, for instance, [Par][ch. II.6.2, Thm. 18, 19, 20]), the solution of the stochastic differential equation, with 0 ≤ s ≤ t, i = 1, 2, 3, and k = 1, . . . , n: 1

k i 2 d(Xik )t,x s = bi ds + (2Dk ) dWs

(1.332)

3 (Xk )t,x t =x ∈R

k with bik = ∂D and Ws the standard Wiener process with values in R 3 . Then in a ∂x i first step, the backward Kolmogorov equation is associated with this process,

∂uk + div (Dk ∇uk ) = 0, ∂t 3

0 ≤ t ≤ t1

(1.333)

3

uk (t1 ) = u¯ k ∈ Cb (R ) ∩ L (R ), 1 ≤ p ≤ ∞, p

which has a unique solution, given with the expectation value E, by the formula   uk (t, x) = E u¯ k ((Xk )t,x t1 ) . With a source term σk , the balance of the state variable uk is expressed by ∂uk + div Jk = σk , ∂t

k = 0, 1, . . . , n,

(1.334)

which is also, for every subdomain ω ⊂ M of boundary ∂ω, d dt







uk (x, t)dx + ω

Jk (x, t).ν(x)d∂ω = ∂ω

σk (x, t)dx, ω

with Jk (x) = −Dk ∇uk or a linear combination at x of the gradients of the thermodynamic functions uk  , but Dk and σk can be nonlinear functions of uk . The study of these conservation laws with nonsmooth (even discontinuous) conditions and functions ϕ is outside the scope of this book. We refer for this to [God-Rav] and [Serre]. In a second step, using duality, if ρk (0) is the density of the probability law of (Xk )0 with respect to the Lebesgue measure, then ρk (t) is the density of the probability law of the process (Xk )t with respect to the Lebesgue measure, and it is the solution in C([0, ∞[; L1(R 3 )) of the Fokker–Planck equation

126

1 Thermostructure

∂ρk − div (Dk ∇ρk ) = 0, ∂t

t ≥ 0,

(1.335)

ρk (0) = ρk0 .   Moreover, we have the relation R 3 ρk (t, x)dx = R 3 ρk0 (x)dx. Of course the Fokker–Planck equation can be solved with a source term. We refer, for instance, to [Par][ch. II.5.6, Prop. 7,7*]. More generally, if the system is in a domain M ⊂ R 3 , with boundary conditions, we also have uniqueness of the solution, with similar properties (see also [Par][ch. II.6.2, Thm. 19, 20]). Remark 25 In the case of a moving fluid (with one constituent), with velocity v, the diffusion phenomenon is taken into account by ∂ρ + div (Jρ + ρv) = 0 ∂t

with Jρ = −D1 ∇ρ,

with D1 the diffusion coefficient. In this case, we have to change in the diffusion process (X1 )t (see (1.332)) the term bi into b˜i = bi + vi , that is, the drift of the process. In this case the Fokker–Planck equation becomes ∂ρ − div (D1 ∇ρ + vρ) = 0, ∂t ρ(0) = ρ 0 .

(1.336)

¶

Fourier Law The heat balance is given with a source term σQ by < ρθ Q ,

∂ > = − div JQ + σQ , ∂t

(1.337)

with the heat flux JQ also denoted by q, given by the Fourier law: JQ = −κ∇T .

(1.338)

First consequence. From the state equation (1.246) for ideal gases, or for van der ˆ . Waals fluids, we have, with constant ρˆ = (ρk ), d e˜ = Cvm (ρ)dT (i) If ρˆ = (ρk ) is constant, we have ρθ Q = d e˜ = ρde, and the weighted internal energy e˜ evolves as a diffusion process ∂ e˜ + div Je˜ = σe˜ , ∂t

Je˜ = −

κ Cvm (ρ) ˆ

∇ e, ˜

σe˜ = σQ ;

1.8 Thermodynamic Fields

127

then we obtain the “heat equation,” that is, the equation of temperature ∂T + div JT = σT , ∂t

σT =

σQ . m Cv (ρ) ˆ

JT = −

κ Cρm (ρ) ˆ

∇T .

(1.339)

(ii) If ρˆ = (ρk ) is not constant, we obtain the conservation of the weighted internal energy e˜ = ρe from the heat (1.337) and from (1.26). Then the conservation of e˜ is given by ∂ e˜ = − div Je˜ + σe˜ , ∂t ˜ ρ, Je˜ = JQ − μJ

with

(1.340)

σe˜ = σQ − ∇ μ.J ˜ ρ + μσ ˜ ρ.

Second consequence. Then the second law (see (1.325)) implies with (1.337) that < θs,

∂s 1 ∂ 1 >=ρ + div JQ − σQ ≥ 0, ∂t ∂t T T

whence the Clausius–Duhem relation: ρ

JQ 1 ∂s 1 + div − ∇( ).JQ − σQ ≥ 0. ¶ ∂t T T T

(1.341)

Maximum Parabolic Principle for the Entropy s or s˜ = ρs Taking back (1.330), (1.340), we obtain (with (1.21)) < θ, Applying

∂ ∂x j

 ∂s  ∂ >= + k div Jk − k σk . ∂t ∂t

(1.342)

to θ , with j = 1, 2, 3, gives < θ,

 ∂uk ∂s ∂s0 ∂ >= j − k j = , j ∂x ∂x ∂x ∂x j

(1.343)

or again ∇s =



k ∇uk + ∇s0 .

(1.344)

We apply the div operator to (1.344). With  = div ∇, we obtain s =



div ( k ∇uk ) + s0 ,

(1.345)

128

1 Thermostructure

and with (1.342), (

 ∂s0 ∂s ∂uk − s) − ( k − div ( k ∇uk )) = − s0 . ∂t ∂t ∂t

(1.346)

With (1.330), let Pk = k

∂uk − div ( k ∇uk ) = −[ k div Jk + div ( k ∇uk )] + k σk ∂t

= −div ( k (Jk + ∇uk )) + ∇ k .Jk + k σk . (1.347) Then (1.346) becomes, with srev = s − s0 ,  ∂srev − srev = Pk . ∂t

(1.348)

If η is a rate of production of irreversible entropy, let ∂s0 − s0 = η. ∂t

(1.349)

Applying the maximum parabolic principle (see [D-L2][ch. V.5.2]) to (1.349) multiplied by −1 on the domain Q =]0, t0 [×M allows us to obtain the following information: If η ≥ 0 (thus −η ≤ 0), then for all time intervals (0, t0 ), s0 (t, x) ≥

inf

(t,x)∈∂p Q

s0 (t, x), with ∂p Q = {0} × M ∪ [0, t0 ] × ∂M,

or again inf s0 (t, x) =

(t,x)∈Q

inf

(t,x)∈∂p Q

s0 (t, x).

Similarly, we obtain a positivity property for uk by (1.330). But in order to verify that the variation of irreversible entropy is only positive, we have to apply the maximum parabolic principle to the expression for the derivative of (1.349) with respect to time. Denoting by s0 and η the derivatives of s0 and η, we obtain ∂s0 − s0 = η . ∂t

(1.350)

Then if η ≥ 0 (hence −η ≤ 0), the maximum parabolic principle allows us to assert that s0 ≥ 0 if it is such on the boundary ∂p Q. Moreover, thanks to the strong maximum parabolic principle, we have for all (t0 , x) ∈ Q,

1.8 Thermodynamic Fields

129

s0 (t0 , x) >

inf

(t,x)∈∂p Q

s0 (t, x).

By the positivity hypothesis on η , we obtain that η = the connverse is not true.

t 0

η (t  )dt  is positive, but

Thermodynamic Forces and Fluxes As with the state variables uk , it is usual (see [Pri-Kon][ch. IV.15.4]) to write a balance of entropy s˜ (which in this section, we simply denote by s, so that is to ˜ with (x) ˜ be changed into = ρ in the following) by ∂s + div Js = σs , ∂t

(1.351)

with Js and σs expressed by Js =



k J k ,

σs = σˆs + σs0 ,

σˆs = σ + σs1 ,

(1.352)

with σ = (∇ k ).Jk ,

σs1 =



k σk ,

σs0 =

∂s0 ≥ 0. ∂t

Then (1.342) implies the relation (similar to the Duhem relation) ∂s + div Js − σˆs ≥ 0. ∂t

(1.353)

Let Fk = ∇ k ∈ Tx M, k = 1, . . . , n, be called thermodynamic forces. The following hypothesis is usually made (see, for example, [Pri-Kon][ch. IV, 15.4]): the fluxes depend linearly on the thermodynamic forces through a positive definite matrix Lj k (with real constants Lj k = Lkj ) by Jj =



Lj k Fk .

(1.354)

Then σ is a positive quantity:46 σ =



Fk .Jk =



Lkj Fk .Fj =



Lkj ∇ k .∇ j ≥ 0,

(1.355)

46 In order for σ to be positive, it is not necessary that the relations (1.354) be satisfied, as we see with the Fourier law for the heat or the Fick law for diffusion.

130

1 Thermostructure

and Js =





k J k =

Lkj k .∇ j =

1  ∇( Lkj k j ). 2

(1.356)

Besides, we also have with the hypothesis (1.354), div ( whence div ( becomes





k J k ) =

k J k ) −





Lkj ∇ k ∇ j +

∇ k Jk =





k Lkj  j ,

Lkj k  j . Then the relation (1.353)

 ∂s  + Lkj k  j − k σk ≥ 0. ∂t

¶

(1.357)

Relative to the previous remark, a conjecture is to admit the relation with (1.350) and (1.355): η = σ.

(1.358)

Balance of the Dual Variables αk = k (u1 , . . . , un ) With the notation of (1.327) and the usual smoothness assumption on k , we obtain by derivation  ∂ k ∂uj  ∂ k ∂αk = = (−div Jj + σj ). ∂t ∂uj ∂t ∂uj j

j

Thus corresponding to the balance of the variables (uk ), see (1.330), we have ∂αk (α) + div J(α) k = σk , ∂t

(1.359)

with the Jacobian matrix J ( ) and (α)

Jk =

 ∂ k j

σk(α) =

∂uj

Jj =

 j,m=1,2,3



Jk,j ( )Jj ,

∂ ∂ k m  ∂ k ( )J + σj . ∂x m ∂uj j ∂uj

(1.360)

j

Now we can consider that (Fˆ k = −J(α) k ) are in fact thermodynamic forces, because k of their relations (1.360) with fluxes, and with the matrix −( ∂ ∂uj ), which is positive

1.8 Thermodynamic Fields

131

definite, therefore giving the concavity of the entropy. Then taking (Lj k ) the inverse matrix of the opposite of J ( ), we have the relation (see (1.354)) Jj =



Lj k Fˆ k .

(1.361)

We also have, with the scalar product in R 3 , 

Fˆ k .Jk = −



Jk,j ( )Jj .Jk = −



Jk,j ( )aj,k ≥ 0,

(1.362)

k

since aj,k = Jj .Jk defines a positive definite matrix A = (aj,k ), 

aj,k xj xk = (



xj Jj ).(



xk Jk ) > 0,

∀xj = 0, xj ∈ R,

 and then k Fˆ k .Jk is the trace of the matrix J (− )A, which is also the trace of 1 1 A 2 J (− )A 2 , which is positive.

1.8.3 Local Balance and Flux for a Moving Fluid Consider a moving fluid with one constituent and without chemical reaction. We assume for notational simplicity that the domain M of the fluid is time-independent. Here we want to follow the thermodynamic state of the particle fluid in its motion. Knowing the state of the moving medium is expressed by a (smooth) section ϕ˜ of the bundle T (R × M) × F of the base space R × M, such that ϕ(t, ˜ x) = ((1, v(t, x)), ϕ(t, x)); ϕ(t, x) gives the thermodynamic state of the fluid at (t, x), and v(t, x) is the fluid velocity at (t, x). The motion of the particle fluid is expressed ) by a (smooth) function ζ : (t, x) ∈ R × M → (t, x(t)) such that dx(t dt = v(t, x(t)), with x(0) = x. Therefore, the thermodynamic state of the moving fluid particle is expressed by ϕ(t, x(t))) = ϕ(ζ (t, x)). We again assume state equations that define a manifold  ⊂ F, and we also assume as previously that the differential form θ 0 | is exact, so that we have an entropic foliation (Ls0 ), s0 ∈ R of  . The evolution of the fluid particle is θ -reversible if Im ζ ⊂ ϕ −1 (Ls0 ), or again for every time < ϕ ∗ θ s , ζ∗ (

ds0 ∂ d ∂ ) > = < ϕ∗θ s , > = < ϕ∗θ s , + v.∇ > = = 0, ∂t dt ∂t dt

∂ where we denote as usual by dψ dt (t, x(t)) = (( ∂t + v∇)ψ)(t, x(t))) the “material derivative” of every real function ψ on R × M. Then the thermodynamic evolution of the fluid particle is θ -admissible if

∂ ds0 = ( + v.∇)s0 ≥ 0. dt ∂t

(1.363)

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1 Thermostructure

We can assume similar properties with the weighted entropy s˜ = ρs and s˜0 . Thus with (1.349), v and s˜0 have to satisfy ˜s0 + v.∇ s˜0 ≥ −η.

(1.364)

In this simple situation, the determination of the thermodynamic state of the moving fluid is based on the balance of specific mass and of heat ρθQ = θ˜Q . The conservation of the specific mass of the moving fluid is given (apart from diffusion phenomena) by ∂ρ + div (ρv) = 0. ∂t

(1.365)

dρ + ρ div v = 0. dt

(1.366)

Thus

Local Weighted Balance and Clausius–Duhem Inequality The balance of work is again expressed by ρ < θW ,

dτ d > = −ρP = −P div v. dt dt

More generally, for a viscous fluid, it is given by ρ < θW ,

d > = −P div v + rW , dt

(1.367)

with rW , which depends on the strain tensor v , and thus ρ < θW ,

d > = − div JW + σW , with JW = P v, σW = v.∇P + rW . dt

(1.368)

Similarly, we express the balance of heat by ρ < θ Q,

d > = − div JQ + σQ , dt

(1.369)

with JQ given by the Fourier law: JQ = −κ∇T (see (1.338)). From (1.369) and (1.368), we deduce the conservation of internal energy by ρ

de + div Je = σe , dt

(1.370)

1.8 Thermodynamic Fields

133

with Je = JW + JQ = P v − κ∇T , σe = σW + σQ .

(1.371)

Balance of kinetic energy and of total energy. Let ecin = 12 v.v. Then e˜cin = ρecin represents the kinetic energy of the fluid at velocity v. The balance of kinetic energy (according to the weak formulation of the Navier–Stokes equation) is expressed by ρ

decin + div Jecin = σecin dt

, with Jecin = −τˆv .v, σecin = v.f − σe ,

(1.372)

where f is the field of external forces and τˆv is the tensor field of type (1, 1), with components τji = λ (div v)δji + 2μji ,

with ji =

1 (∂i v j + ∂j v i ), 2

(ji ) being the components of the strain tensor v . Adding the balances (1.370) and (1.372) of internal energy and kinetic energy, we obtain the (local) balance of total energy et ot = ecin + e (see (1.371)): ρ

det ot + div Jetot = v.f, dt

with Jetot = Je + Jecin = JW + JQ − τˆv .v

(1.373)

Local balance of weighted energy e. ˜ With the weighted internal energy e˜ = ρe and d e˜ ∂ e˜ = + e ˜ div v = + div ( ev), ˜ the “weighted” balance of internal energy is ρ de dt dt ∂t ∂ e˜ ˜ = σe , + div (Je + ev) ∂t or

d e˜ dt

+ div Je˜ = σe˜ , with Je˜ = Je + ev, ˜ σe˜ = σe + (∇ e˜).v. ¶ A consequence of (1.369). The second law (see (1.325)) implies with (1.337) that < θs,

ds 1 d 1 >=ρ + div JQ − σQ ≥ 0, dt dt T T

whence the usual Clausius–Duhem relation: ρ

JQ 1 1 ∂s + div − ∇( ).JQ − σQ ≥ 0. ∂t T T T

(1.374)

1.8.4 Connecting Local and Global Differential Forms Let Mt ⊂ R 3 be the domain of the fluid at time t, Vt its volume, and let the global thermodynamic variables at time t be as follows: energy Et , entropy St , when these

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1 Thermostructure

local thermodynamic variables are et , st , the specific mass being ρt . We denote by dx the Lebesgue measure in R 3 , identified with the 3-form dx 1 ∧ dx 2 ∧ dx 3 . The motion of the fluid particle is given by a function ζt : x ∈ M → x(t) ∈ Mt ⊂ R 3 generated by the vector field vt . We have 



Vt =

dx,



Et =

Mt

ρt et dx,

St =

Mt

ρt st dx. Mt

Let M = M0 , V = V0 , E = E0 , S = S0 , and e = e0 , s = s0 . The local and global differential forms are connected by formally writing 

 ρdτ ⊗ dx,

dV =



dE =

M

ρde ⊗ dx, M

dS =

ρds ⊗ dx

(1.375)

M

(with the  tensor product), where the meaning of these global forms is given with d v˜ = ∂t∂ + v j ∂x∂ j = dt and < v, ˜ dV > = dV dt , . . ., by 

dτ dx, < v, ρ ˜ dE > = < v, ˜ dV > = M dt



de dx, < v, ρ ˜ dS > = M dt

 ρ M

ds dx. dt

Proof

  (i) Since Vt = ζt (M) dx = M ζt∗ (dx), then taking the derivative with respect to time, we obtain, with the Lie derivative Lv ,    1 dVt |t =0 = (ρ div v) dx Lv (dx) = div v dx = dt ρ M M M   1 dρ dτ dx = dx. = − ρ ρ dt M M dt

(ii) In a similar way, we have  ∂ d(ρe) (ρe) dx + Lv (ρe dx)] = + (ρe)div v] dx [ dt M ∂t M   dρ d(ρe) de − e ] dx dx. = [ = ρ dt dt M M dt

dEt |t =0 = dt



[

(iii) The last relation of (1.375) is similar, on changing e for s. W = Then with the notation θG relation (a formal first principle)

dE =

W θG

Q −P dV , θG

Q + θG

= dE + P dV , we have the formal

 =

ρ(θ W + θ Q ) ⊗ dx. M

¶

1.8 Thermodynamic Fields

135

 W Of course if P does not depend on x, then θG = M ρθ W ⊗ dx. If P and T are S and θ s are linked by x-independent, then the basic differential forms θ S = θG  S s θ = M ρθ ⊗ dx.

1.8.5 Functional Convex Analysis The previous functional thermodynamic modelling requires a certain smoothness of functions. But we have often to consider more general situations and to specify the functions with respect to the natural functional frame L1 (M), or L∞ (M) by duality, in order to define a global internal energy and a global entropy of the system by integration. Here we will use the parametrization with (ρ, s˜ , e, ˜ μ, ˜ T ) or (ρ, e, ˜ s˜ , − Tμ˜ , T1 ), with the chemical potential μ. ˜ With the state equation e = cv T , thus e˜ = cv ρT , we must have, up to an arbitrary function f (ρ), − Tμ˜ = cv log e˜ + f (ρ). In the ideal gas case, we have f (ρ) = −(r + cv )[1 + log ρ]. Let ξ be a map of M into R × R such that ξ(x) = (ρ(x), s˜ (x)) is of class C 1 . We assume that the internal energy e(x, ˜ ξ ) with ξ = (˜s , ρ) ∈ R × R, e˜ = ρe, s˜ = ρs, is a Carathéodory function, i.e., (i) x ∈ M → e(x, ˜ ξ ) ∈ R, ∀ξ ∈ R × R is a measurable function, (ii) ξ ∈ R × R + → e˜(x, ξ ) ∈ R, a.e. x ∈ M, is a continuous function. We again assume that the function x → e˜(x, ξ(x)) (with ξ(x) = (˜s (x), ρ(x))) is ν in L1 (M) if the map ξ is such that s˜ ∈ Lν (M) and  ρ ∈ L (M) with ν ∈ [1, +∞[. (A priori, the total mass is expressed by m0 = M ρ(x) dx, which gives ν = 1.) This allows us to define the total internal energy of the system by 

 e(x, ˜ ξ(x)) dx =

E(ξ ) = M

e(x, (τ (x), s(x)))ρ(x) dx. M

Then (see [Eke-Tem][ch. IV.1.2, Prop. 1.1]) the map ˜ ξ(x))} ∈ L1 (M) ξ = (˜s , ρ) ∈ V = Lν (M) × Lν (M) → {x ∈ M → e(x, is continuous on V into L1 (M), and the map E : ξ ∈ V → E(ξ ) is continuous on V into R. Moreover, the conjugate functional E ∗ (η) with η ∈ V∗ is given by (see [Eke-Tem][ch. IV, Prop. 1.2]) E ∗ (η) =



e˜ ∗ (x, η(x)) dx,

M

e˜∗ (x, η) = sup (η.ξ − e(x, ˜ ξ )). ξ ∈R 2

with (1.376)

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1 Thermostructure

Similarly, with ζ = (ρ, e) ˜ ∈ Lν (M) × Lν (M), we define the total entropy of the system S(ζ ) from the map x → s˜ (x, ζ (x)) ∈ L1 (M) by 

 s˜ (x, ζ (x)) dx =

S(ζ ) = M

s(x, (τ (x), e(x)))ρ(x) dx.

(1.377)

M

From the local convexity hypothesis of e(˜ ˜ s , ρ) (see Lemma 1), the total internal energy E(˜s , ρ) is a convex functional of the pair of functions (˜s , ρ), and we have the equivalence relations (T , μ) ˜ ∈ ∂E(˜s , ρ) ⇐⇒ (˜s , ρ) ∈ ∂E ∗ (T , μ) ˜  [T (x)˜s (x) + μ(x)ρ(x)] ˜ dx = E(˜s , ρ) + E ∗ (T , μ). ˜ ⇐⇒

(1.378)

M

Recall that (T , μ) ˜ ∈ ∂E(˜s , ρ) means that E(˜s , ρ) is finite and that ∀(˜s  , ρ  ) ∈ V  E(˜s , ρ) + [(˜s  − s˜ )(x)T (x) + (ρ  − ρ)(x)μ(x)] ˜ dx ≤ E(˜s  , ρ  ). (1.379) M

We also write with δE = E(˜s  , ρ  ) − E(˜s , ρ), δ s˜ = s˜ − s˜, δρ = ρ  − ρ (in a formal manner),  δE − (T δ s˜ + μ˜ δρ)dx ≥ 0. (1.380) M

Similarly, S(ζ ) is a concave function of ζ = (e, ˜ ρ). We also have the equivalence (with almost everywhere x ∈ M) (T , μ) ˜ ∈ ∂E(˜s , ρ) ⇐⇒ (T (x), μ(x)) ˜ ∈ ∂ e˜(˜s , ρ),

a.e. x ∈ M,

(1.381)

which is again written η ∈ ∂E(ξ ) ⇐⇒ η(x) ∈ ∂ e˜x (ξ(x)), a.e.x ∈ M, with previous notation. This property is due to the relation (by the definition of the conjugate function) for all x ∈ M, e˜x∗ (η(x)) − η(x).ξ(x) + e˜x (ξ(x)) ≥ 0.    Now η ∈ ∂E(ξ ) implies M η(x).ξ(x)dx = M e˜x (ξ(x))dx + M e˜x∗ (η(x))dx, whence, by comparison, the equivalence. ¶

1.9 The Use of the Free Helmholtz Energy The usual strategy in many domains (such as, for instance, elasticity coupled with electromagnetism) is to define a priori a free Helmholtz energy f = e − T s, or

1.9 The Use of the Free Helmholtz Energy

137

f˜ = ρf = ρ(e − T s) (according to many axioms), as a function of T and variables uk , k = 1, . . . n, thus f (T , u1 , . . . , un ). The basic differential form θ f being θ f = df + sdT −



pk duk ,

(1.382)

∂f ∂f then θ f = 0 gives the state equations pk = ∂u k , s = − ∂T . The differential of f gives the state equations for pk , uk by comparison with the basic differential form θ f . If the state equations are given by an integrable  map , so that (e, u1 , . . . , un ) = ( T1 , pT1 , . . . , pTn ) with ds = 0 de + k=1,...,n k duk , then we have

k = −

1 ∂f , T ∂uk

k = 1, . . . , n.

(1.383)

Moreover, with fixed (uk ), the function e(s) is the (Legendre) conjugate function of (−f )(T ). Let β = T1 , fˆ(β) = T1 f (T ). Thus fˆ(β) = βf ( β1 ), whence 1 1 fˆ (β) = 3 f  ( ) = T 3 f  (T ), β β so that fˆ is a concave function of β if f is a concave function of T . Then we have fˆ + s = βe,

with β =

∂s , ∂e

the last relation being due to e = f + T s = f − T differential of (1.384) gives

e= ∂f ∂T

∂ fˆ , ∂β

= f + β ∂f ∂β =

(1.384) ∂(βf ) ∂β .

The

d fˆ − edβ = −(ds − βde), so that β =

∂s ∂e

is equivalent to e =

∂ fˆ ∂β .

Thus s(e) is the “conjugate” function of fˆ(β), that is, s = fˆ∗ , since s and fˆ are concave functions respectively of e and β, so that this gives the state equation β = T1 = 0 (e, (uk )). This is equivalent to inverting the relation e(uk ) (T ) = e(T , (uk )) = f − T

∂f . ∂T

Remark 26 (i) In fact, the free Helmholtz energy must also be a function of the parameter s0 in order to take irreversible evolutions into account, so that we have fs0 (T , u1 , . . . , un ), for instance fs0 = f − s0 T , thus fs0 = e − (s + s0 )T : the free Helmholtz energy must be defined up to an affine function of T .

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1 Thermostructure

(ii) In many cases (especially with interaction of the medium with electromagnetism), the basic differential form with free energy as main variable is changed into the weighted differential of ρdf , θ˜ f = ρdf + ρsdT −



pk duk .

Of course we can divide  by ρ (if it cannot be zero), so that we have the basic form θ f = df + sdT − τpk duk . ¶ The free energy f must be a priori a convex function of (u1 , . . . , un ), but concave with respect to T . For a linear theory, f is a quadratic form of (T , (uk )); with constants ak,k  , ak , a, e0 , s0 , then f must be given by  1  f (T , (uk )) = e0 − s0 T − aT 2 + T ak uk + ak,k  uk uk , 2

(1.385)

with a > 0 and the matrix (ak,k  ) positive definite. Then this gives the entropy by s=−

 ∂f = s0 + aT − a k uk ; ∂T

(1.386)

thus we obtain the internal energy e = f + T s by  1  e = e0 + aT 2 + ak,k  uk uk . 2 Then T 2 = a2 [e − e0 − so that





(1.387)

ak,k  uk uk ], and we must have e − e0 −





ak,k  uk uk > 0,

 2 1  1 T = [ ] 2 [e − e0 − ak,k  uk uk ] 2 , a

(1.388)

1

and thus with a  = (2a) 2 , s(e, (uk )) = s0 + a  [e − e0 −



 1

ak,k  uk uk ] 2 −



a k uk .

(1.389)

It is equivalent to use the free Helmholtz equation f given by (1.385) or the entropy given by (1.389). Of course with the constitutive relation (1.388), we also have the constitutive relations from (1.382), (1.385), and with (1.383), pk =

 ∂f = 2ak,k uk + ak,k  uk + ak T , k ∂u  k =k

Using (1.386) and (1.387), the internal energy is given by

1 k = − pk . T

1.10 Thermoelectromagnetism

e(s, (uk )) = e0 +

139

  1  (s − s0 + a k u k )2 + ak,k  uk uk . 2a

(1.390)

From this relation (1.390), we can verify that e(s, (uk )) is a convex function, and thus that s(e, uk )) is a concave function.

1.10 Thermoelectromagnetism The classical model of electromagnetism in a medium at rest uses four vectors in R 3 , D, E, B, H, respectively called electric induction, electric field, magnetic induction, and magnetic field, which are a priori functions of space and time, thus sections of a vector bundle, with standard fibre Fem diffeomorphic to (R 3 )4 . For the time being, we do not consider their dependence with respect to the position and time variables, but generally D and B depend on E and H by state equations. Then the vectors P and M, respectively called electric polarization and magnetization, are defined by D = 0 E + P,

B = μ0 H + M,

with constants 0 , μ0 called permittivity and permeability of free space. Using the notation dD and dB as differential forms on Fem , we define θ em = E.dD + H.dB =



Ej dDj + Hj dBj ,

θ˜ em = D.dE + B.dH,

j =1,2,3

and again the differential form 0 θ em = dWem + θˆ em ,

0 with Wem =

1 (0 E2 + μ0 H2 ), θˆ em = E.dP + H.dM. 2

In the free space, the electromagnetic energy is defined by Wem = 12 (0 E.E + μ0 H.H). In a medium, the definition of the electromagnetic energy Wem from D, E, B, H is more difficult, since the separation with the internal energy of the medium is not evident. A usual hypothesis is that Wem is a convex function ∗ the of (E, H). With this hypothesis alone, we can only write, denoting by Wem conjugate function of Wem , ∗ Wem (D, B) = sup((D.E + B.H) − Wem (E, H)), E,H

∗ Wem (E, H) = sup((D.E + B.H) − Wem (D, B)), D,B

and then ∗ Wem + Wem = (D.E + B.H),

with (D, B) ∈ ∂Wem (E, H).

(1.391)

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1 Thermostructure

We define the differential forms ∗ ∗ θem = dWem − (E.dD + H.dB),

θem = dWem − (D.dE + B.dH),

∗ +θ thus θem em = 0.

1.10.1 Constitutive Electromagnetic Relations The constitutive relations between D, E, B, H or P, E, M, H depend on the interaction with the medium and can be very varied, even if the medium is not moving: If the relations are linear, so that D = E, B = μH, with , μ positive constants, then the electromagnetic energy is, with E 2 = E.E = Ej2 , 1 1 (E 2 + μH 2 ) = (D.E + B.H), 2 2 1 1 1 ∗ Wem (D, B) = ( D 2 + B 2 ) = Wem (E, H); 2  μ Wem (E, H) =

(1.392)

In the nonlinear case, , μ depend on E and on H. There are also “nonlocal” relations depending on x, t, with delay times or hysteresis phenomena. Thermoelectric and thermomagnetic phenomena lead one to admit constitutive relations coupling electromagnetism and thermodynamics of matter and to join the differential form θ em and that of matter θ e˜ . But in a local thermodynamic model, the units of θ˜ e = ρθ e = d e˜ − T d s˜ − μdρ ˜ (see (1.21)) are [θ˜ e ] = [θ em ] = ML−1 T −2 . Of course, we have to consider more sophisticated differential forms θ˜ e for solids in elasticity, for moving fluids. We refer to [Eri-Mau] for many examples in these domains. We remark that the choice of a local thermodynamic model is necessary with fluctuating electromagnetic fields. We have different species of models of the internal energy of a system depending on whether e represents both the energy of the constituent and the energy of electromagnetism.

1.10.2 Differential Forms in Thermoelectromagnetism System Consisting of Gas or Liquid with Electromagnetism The basic differential form θ = θ e,em relative to such a system may be written in many different ways. At first, we have with weighted energy and entropy

1.10 Thermoelectromagnetism

141

∗ ∗ = d(e˜ + Wem ) − T d s˜ − μdρ ˜ − (E.dD + H.dB) θ˜ e + θem

˜ + (D.dE + B.dH). θ˜ e − θ˜em = d(e˜ − Wem ) − T d s˜ − μdρ

(1.393)

The term (E.dD + H.dB) corresponds to the work (see [Lan-Lif3][ch. II, 10]). ∗ . Then e˜ − e˜ = W ∗ Let e˜1 = e˜ − Wem , e˜2 = e˜ + Wem 2 1 em + Wem = D.E + B.H. Keeping the second differential form of (1.393), with eˆ = e˜ − Wem , we write θ = d eˆ − T d s˜ − μdρ ˜ + D.dE + B.dH, or with the entropy as main variable, θ s˜,em = d s˜ −

μ˜ 1 1 d eˆ + dρ − (D.dE + B.dH). T T T

(1.394)

Now from the state equations we must have a map such that 1 μ˜ 1 1 (e, ˆ ρ, E, H) = ( , − , D, B), T T T T and with M = R × R + × R 3 × R 3 , N = R + × R × R 3 × R 3 , (e, ˆ ρ, E, H) ∈ M,

μ˜ 1 1 1 ( , − , D, B) ∈ N . T T T T

Furthermore, this map must be integrable, so that we must have the Schwarz relations between the derivatives of the components ( j ), 1 ≤ j ≤ 8, of . But usually, this is avoided by the hypothesis of the free Helmholtz energy f (T , ρ, E, H), as seen before. Example 5 We assume the linear constitutive relations: D = E, B = μH, with permittivity and permeability , μ. Let eE =

1 0 E 2 , 2

eH =

1 μ0 H 2 , 2

r =

 μ , μr = . 0 μ0

Then the differential form (1.394) becomes θ s˜,em = d s˜ −

μ˜ r 1 μr d eˆ + dρ − deE − deH . T T T T

¶

(1.395)

Another basic differential form may be (without specifying what the internal energy is) θ s,em = ds −

P τ 1 de + dτ − (D.dE + B.dH), T T T

(1.396)

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1 Thermostructure

or also (in the spirit of [Eri-Mau][ch. V.8, p. 157])47 θ s,em = ds −

P τ 1 de + dτ − (P.dE − M.dB), T T T

(1.397)

where P.dE − M.dB) does not contribute to the electromagnetic energy, since 1 P.dE − M.dB = (D.dE + H.dB) − d(E.E + B.B). 2 Elastic Solid Coupled with Electromagnetism We refer to [Eri-Mau]k which gives many examples, choosing an a priori knowledge of the free Helmholtz energy. By derivation with respect to T , we obtain the entropy as a function of T and the other basic variables, the components of the deformation tensor , and of E, B. But the entropy is obtained as a function of e, since s = fˆ∗ is the conjugate function of fˆ with fˆ(β) = βf (β), and β = T1 , as seen previously.

1.10.3 Functional Thermoelectromagnetism Now we assume that we have a medium in a domain M ⊂ R 3 with one constituent, in an electromagnetic field. The space is the fibre product of the (trivial) thermodynamic bundle M ×F with the electromagnetic bundle M ×Fem (in fact, the bundle ∧(T ∗ M) of differential forms on M). The electromagnetic field (E, H, D, B) satisfies the Maxwell equations with a current J. Then we have < θ em ,

∂D ∂B ∂ > = E. + H. = −div (E ∧ H) − J.E, ∂t ∂t ∂t

with E ∧ H = P the vectorial product of E and H, called the Poynting vector. If ∗ , ∂ > = 0, then < θem ∂t −

∗ ∂Wem = div (E ∧ H) + J.E. ∂t

(1.398)

∗ is decreasing in M. Now if the medium is isotropic and This is the rate at which Wem homogeneous, we have the linear constitutive relations D = E, B = μH, so that ∗ = dW . Then (1.398) is also the rate at which the energy E.dD + H.dB = dWem em Wem is decreasing in M. According to the Ohm law, E and J are connected by

47 In fact, it is the generalized free Helmholtz energy that is given in [Eri-Mau][ch. V.8, p. 157] in a more general setting.

1.11 Thermal Radiation

143

J = σ E, with the conductivity σ > 0. Then J.E “is the rate at which heat is produced per unit volume by conduction” [Jones][ch. I.27, p. 51]. There are constitutive relations, more general than the Ohm law for J , and the Fourier law for JQ , such as [Eri-Mau][ch. V.8, p. 170] with constants σT and κE : J = σ E + σT ∇T ,

JQ = κ∇T + κE E.

1.11 Thermal Radiation 1.11.1 Planck Law The interaction between matter (for example a gas at temperature T ) and electromagnetic field seen as a photon gas leads to states of thermal equilibrium. This was studied by Planck, leading to quantum-statistical mechanics. We call the field produced by this interaction thermal radiation. The radiation energy (by unit of volume), denoted by e(T ), a priori depends on the absolute temperature T , and is defined by its density e(ν, T ) relative to the frequency ν. This energy density is expressed by the Planck law (with the Boltzmann constant kB , the Planck constant h h, or h¯ = 2π ): e(ν, T )dν = μ0 (ν, T )dn(ν), μ0 (ν, T ) =

hν exp

hν kB T

−1

,

with dn(ν) = ζ (hν)2 d(hν), ζ =

8π . (hc)3

(1.399)

hν We usually set β = kB1T , and thus μ0 (ν, T ) = exp (βhν)−1 . This study by statistical mechanics is well known and leads to the first determination of the free energy F and of the entropy S. We now consider this study from a probabilistic point of view; the modelling of the electromagnetic field and its energy is done with the notion of photon, with a random variable, and the modelling of the thermodynamic variables is also done with real random variables, but there are two independent “deterministic” variables: temperature and frequency. Consider the Hilbert space H = L2 (R, C), and let Hω be the Hamiltonian operator of the harmonic oscillator with one dimension (with ω = 2πν, and m the mass of the particle)

Hω = −

h¯ 2 d 2 1 + mω2 x 2 . 2 2m dx 2

1/2 x, we obtain the operator With the change of variable y = ( mω h¯ )

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1 Thermostructure

Hω1 = h¯ ωH1 = hνH1 ,

H1 = −

1 d2 1 1 + y 2 = H0 + I, 2 2 dy 2 2

(1.400)

and the spectrum σ (H0 ) of the operator H0 is made up of the eigenvalues n ∈ N, i.e., σ (H0 ) = N, and the corresponding eigenfunctions, which form an orthonormal basis of H, are the Hermite functions (hn ). Let ραG be the density operator associated with the operator H˜ ω1 = h¯ ωH0 , called the Gibbs operator, defined by ραG =

1 1 exp (−β H˜ ω1 ) = exp (−αH0 ) Zα Zα

(1.401)

with α = β h¯ ω = βhν (without unit), and Zα the normalization factor, such that tr ραG = 1. We use λα = 1/Zα as normalization factor for the following probabilities. The probability of the state hn for a system at equilibrium at the temperature T = kB1β is expressed by Pα (n) = λα exp(−αn) =

1 exp(−αn), Zα

with λα = 1 − exp(−α).

(1.402)

 Then the Gibbs operator is expressed by ραG = Pα (n)n , with n the  spectral projector onto hn , to which is associated the probability measure Pα = Pα (n)δn on N. Remark 27 Instead of the Hilbert space L2 (R, C), we can also use the space 2 L2 (R, γ ) with the measure γ = (2π)−1/2 exp(− ξ2 )dξ , and use a Gaussian probability space, following [Mall2], which leads one to replace the Hermite functions (hn ) by the Hermite polynomials. This more probabilistic point of view avoids using the H0 operator.

1.11.2 Probability Measure and Expectation Values We assume that the probability of existence of a state with n photons at frequency ν is associated with that of state hn , and thus that there exists a real random function (r.r.f.)48 (Nˆ α ), α > 0, with integer values, such that P (Nˆ α = n) = Pα (n) = λα exp(−αn).

(1.403)

The family of probabilities (Pα ), α ∈ R + , allows us to define a probability measure + P on the product space NR such that πα (P ) = Pα . Moreover, we can also define the following r.r.f.:

48 See

[Nev][ch. III.4, III.3].

1.11 Thermal Radiation

145

Sˆα = − log Pα = α Nˆ α − log λα ,

α Eˆ α = hν Nˆ α = Nˆ α , β

Sˆα (n) = − log Pα (n) = αn − log λα ,

(1.404)

Eˆ α (n) = nhν.

The expectation value of Nˆ α is N α = E(Nˆ α ) =



nPα (n) = λα



n e−nα = −λα

d 1 ( ) dα λα

e−α 1 d log λα = , = α = dα 1 − e−α e −1

(1.405)

and then the expectation value of Eˆ α , which is the density of internal energy, is E α (hν) = hνN α =

α α e−α Nα = = μ0 (ν, T ) β β 1 − e−α

(1.406)

(at temperature T and frequency ν), which is the Planck formula. The expectation value of Sˆα is S α = E(Sˆα ) = −E(log Pα ) = −



Pα (n) log Pα (n)

= αN α − log λα = αN α + log N α + α = β E¯ α − log λα .

(1.407)

The differential of N α is dN α = −N α (N α + 1), dα

dN α dS α =α = −αN α (N α + 1), dα dα

(1.408)

and the differential of S α is dS α =

dS α dα = −αN α (N α + 1)(hνdβ + βd(hν)). dα

(1.409)

The second-order derivatives are given by d 2N α = N α (N α + 1)(2N α + 1), dα 2 d 2Sα d 2N α dN α + α = = N α (N α + 1)[−1 + α(2N α + 1)] dα 2 dα dα 2 y(α) , = N α (N α + 1) α e −1

(1.410)

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1 Thermostructure

with y(α) = eα (α − 1) + α + 1. Since y(0) = 0, y  (α) = eα (α + 1 ≥ 0, we have y(α) ≥ 0. Thus N α and S α are convex functions of α. Then we can prove that S α is a convex function of (hν, β). Now we can express S α as a function of (E α , hν), but it is not a concave function of this pair of variables. Compendium of results. Implicitly, at the outset we have considered the random + function Xˆ α,β on  = N (or on  = NR , or on H) into the space F ⊂ R 5 such that Xˆ α,β = (Sˆα , Eˆ α , hν, β, β Nˆ α ), which is a function of the two independent variables α, β (hence of ν and T ) and of the random variable Nˆ α corresponding to the “quantization.” Then the expectation value of that function Xˆ α,β is, with μ¯ α = βN α , E(Xα,β ) = X¯ α,β = (S α , E α , hν, β, μ¯ α ) = (S α , hνN α , hν, β, βN α ). We can join to it the differential form θ α,β = dS α − βdE α + μ¯ α d(hν) = dS α − βdE α + βN α d(hν) = dS α − αdN α , −α

with the state equations E α = βα N α , μ¯ α = βN α , N α = eλα , and μ¯ α can be seen as a “frequency potential.” From a formal point of view, we can consider θ α,β the expectation value of the random differential form θˆα,β = d Sˆα − βd Eˆ α + β Nˆ α d(hν). Integration on frequencies. The frequency density being given by (1.399), this leads us to define through integration with respect to α (thus with respect to frequencies) E˜ =



∞

E α α 2 dα,

S˜ =

0



∞

S α α 2 dα,

0

N˜ =



∞

N α α 2 dα.

(1.411)

π4 , 15

(1.412)

0

Using (1.406), we have49 1 E˜ = β



∞

eα

0

1 1 α 3 dα = C0 , −1 β

with C0 =

and S˜ − β E˜ = −



∞

(log λα )α 2 dα =

0

49 We

also have N˜ =

∞ 0

1 2 eα −1 α dα

1 3

 0

∞

1 1 ˜ α 3 dα = β E, eα − 1 3

(1.413)

= (3)ζ(3) = 2 × 1.202 with ζ the Riemann zeta function.

1.11 Thermal Radiation

147

and thus 4 4 S˜ = β E˜ = C0 . 3 3

(1.414)

1.11.3 Global Thermodynamics If the system is in a domain  of volume V , we define the global energy and global entropy at equilibrium by ˜ E = ζ V β −3 E,

˜ S = ζ V β −3 S,

(1.415)

with ζ given by (1.399). Note that ζ V β −3 is without unit, so that E and E˜ have the same unit (energy), and S like S˜ is without unit. Thus the global energy is such that E = ζ C0 β −4 = σ T 4 , V

σ = ζ C0 kB4 ,

(1.416)

which is the Stefan–Boltzmann law. We define the pressure P by P = 13 VE .50 Then we differentiate the expression S − βE = 13 βE, derived from (1.413) and (1.415), and since 1 1 1 E d(βE) = d(σ T 3 V ) = V σ T 2 dT + σ T 3 dV = −Edβ + β dV , 3 3 3 3V we obtain dS = βdE + β

E dV = βdE + βP dV , 3V

(1.417)

which is the usual differential form with the entropy, which finally gives (with kB = 1) dS =

P 1 dE + dV . T T

(1.418)

The entropy may be expressed as a function of (E, V ) by (1.414) and (1.416): S=

1 4 4 3 βE = E 4 (σ V ) 4 . 3 3

can also define the pressure by P = − ∂F , from the free energy F˜ = E˜ − β1 S˜ = − 13 β E˜  ∞ ∂V 2 −3 F α α dα, and F α = E α − 1 S α = 1 log λα , which also defined by F = ζ V β F˜ with F˜ =

50 We

0

is deterministic.

β

β

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1 Thermostructure

˜ then dN = NdV ˜ If the total number of photons is defined by N = V N, , and we can write the differential form with the variable N instead of V , with μ = P˜ , by N

dS = βdE + β

P dN = β(dE + μdN). N˜

(1.419)

Through integration over frequencies, using the volume V of the domain, we have obtained the usual expression of the global thermodynamics with the usual differential form, but with the state equations: P =

1E , 3V

E = σ T 4, V

(1.420)

which defines the map by (V , E) = (

P 1 1 1 E 3 E −1 σ 1 , ) = ( σ 4 ( )4 , ( ) 4 ) = ( T 3 , ). T T 3 V σV 3 T

(1.421)

The Jacobian matrix J ( ) is given by

− 4VE2 T 1 4V T

1 4V T 1 − 4ET

.

The determinant of the matrix is zero, and thus the entropy S is a concave function of (V , E). Remark 28 Recall that S, E, V are extensive variables, and E is a positively homogeneous function of V , S, and thus its conjugate function is an indicator function (i.e., equal to 0 or to +∞). Moreover, we can verify that E is a convex function of V , S; thus the Gibbs function is null on its domain, G(P , T ) = −E ∗ (−P , T ) = E + P V − T S = 0, which implies F + P V = 0, and with hypothesis P = ¶ S = 43 E T.

1E 3V,

we have F = − 13 E and

1.11.4 Local Thermodynamics Let s = J =

S V

, e=

E V.

Then P = 13 e. Then with (1.417), we have

S E P 1 1 (dS − βdE − βP dV ) = d − βd − β dV − d [S − βE). V V V V V

1.11 Thermal Radiation

149

Thus J = ds − βde +

dV V

(s − βe − βP , and we obtain ds − βde = 0.

(1.422)

But we also have S = 43 βE, and thus s = 43 βe. Then we differentiate, and we obtain βde + 4edβ = 0. By integration we have e = Cβ −4 , which is the law e = σ T 4 . Now let s˜ =

S S V = = sτ, N V N

e˜ =

E EV = = eτ. N V N

Then ds − βde = d

s˜ e˜ 1 1 − βd = (d s˜ − β e˜) + (˜s − β e)d ˜ . τ τ τ τ

Thus with s = 43 βe, we obtain, with P = 13 e, 1 d s˜ − βd e˜ − βedτ = d s˜ − β(d e˜ + P dτ ) = 0. 3

(1.423)

If the volume is constant, then dτ = − NV2 dN = −τ dN N , and thus we can write d s˜ − βd e˜ + μdN = 0,

with μ = β

Pτ . N

(1.424)

Chapter 2

Classical Mechanics

Let  be a physical system whose evolution is to be studied. Generally, many models are possible. Here, we consider modelling at a macroscopic level (or classical microscopic, without quantum physics), with a deterministic point of view. Studying a system  by a family of observers leads one to represent the system using frames and atlases, that is, a family of charts that leads to a modelling by a set M equipped with a manifold structure that is often Riemannian, with a notion of distance.

2.1 Time Setting In the present models, time is an essential “exterior” parameter to the system, a parameter of the evolution group (or pseudogroup) of the system. But in a physical system, time may be an internal variable1 that is taken into account by the usual statements of the principles of mechanics about the existence of absolute or Galilean frames. Also, time can occur in an enlarged phase space, with even dimension (the dual variable being the energy), or odd (in a contact structure). Such a point of view seems very natural if the evolution is led by a time-dependent vector field. The major importance of time then appears in the notion of oriented foliation: the spacetime is split into leaves with a time subscript. We also have to keep in mind that time occurs in several transformation groups, such as the Galilean group or the Lorentz group. But a principal importance of time is to fix the modelling order, i.e., the order of time derivatives of the functions (i.e., the use of jet functions) specifying the system model: in the evolution equations of “usual” mechanics (such as Newtonian), at most second-order time derivatives occur. Then we can hope to determine the

1 As in relativity, time occurs in a metric giving the distance between two events (pseudoRiemannian metric of Lorentz or Minkowski).

© Springer International Publishing AG, part of Springer Nature 2018 M. Cessenat, Mathematical Modelling of Physical Systems, https://doi.org/10.1007/978-3-319-94758-7_2

151

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system evolution from the knowledge of positions and velocities at the initial time, that is, a model of the system at second order, from a deterministic point of view. Finally, we must not forget the causality and time local interaction principles, which restrict the possible type of evolution equation in mechanics (notably in the modelling possibilities of delay phenomena). There are many questions linked with the evolution, such as smoothness (which we will generally assume), the possibility of returning to a point, and the comparison with a free evolution that leads to the notion of geodesics, which are submanifolds of the space M modelling the system. We will have to distinguish between paths and curves in M, a question that may seem technical, but it is important to understand the trajectorial nature of the system, which is linked to minimization questions. We will study two modelling types that can be viewed as different steps in modelling a physical system.

2.2 Two Main Modelling Types (i) Modelling based on a geometric representation of the system  in the Euclidean space E3 . The system  is a part of a larger system (such as the universe), but we want to know only the evolution of . Thus the “exterior” of  is modeled only by its interaction with . The  configuration is modeled by a subset M of a Euclidean space E3 , which is the natural geometric space occupied by the system. A usual case is M being a bounded open set of E3 (but M may be a finite number of points, a closed “regular” set, or a manifold with boundary). Furthermore, we notice that a priori, M is time-dependent, M = Mt , which corresponds to the global evolution of the system in space. Furthermore, the system  has also an internal structure due to internal variables and to supplementary degrees of freedom, notably: an internal geometric structure (for instance a frame associated with each fluid particle); a purely physical structure with quantities such as mass, specific mass, in the case of a “continuous” system, the electric charge, etc.; thermodynamic, electromagnetic structures, etc. We can (in part) represent these internal structures by a fibre bundle E on M, whose fibres Ex are all the possible values of the internal variables of  (which depend on each point x of M). The consequences of the action on  of the exterior of the system are as follows: The existence of so-called exterior forces, although they apply either to interior points of M (“volume” exterior forces) or to the boundary (“surface” exterior forces), these forces are seen as known functions of the position x in M (or of the position and the velocity), or of the boundary of M. This action may be, for example, gravitational or electromagnetic, and is sometimes derived from a potential

2.2 Two Main Modelling Types

153

The existence of a given heat flux, or more generally of a heat exchange, that is not considered in rational mechanics. In the system, there can be so-called interior forces, links, or cohesion forces of the system. These forces are a priori unknown and thus need to be eliminated to fix the evolution problem. The possibilities of motion of the system split into the following three types: global motion, hence of Mt in E3 , internal motion in Mt (and also internal variables of the system), deformation motion of the boundary of Mt . The determination of motion enters very generally in the frame of problems with free boundary (the boundary of Mt being an unknown of the problem), which often occurs in elasticity and in fluid mechanics. (ii) Associated modelling with a configuration space different from E3 . The variables specifying the system  are given by a set M˜ called configuration space (in a model at order 1 that includes the positions of  but not the velocities), ˜ The set M˜ has the natural and the  evolution is modeled by a trajectory in M. structure of a differentiable manifold (but may be of infinite dimension). In a second˜ both order model, the configuration space M˜ is changed into the tangent bundle T M, ∗ specifying positions and velocity of the system, or by its cotangent bundle T M˜ (or phase space). But the existence of links can reduce that space: if the links are holonomic, we can boil things down to work in a new space T N or T ∗ N, with N a submanifold of ˜ This is not possible in the case of nonholonomic links (velocity-dependent). M. The space M˜ is directly linked to the set of possibilities of motion of ; if  is a rigid body, we can identify M˜ with the displacement group in E3 , and thus M˜ = R 3 × SO(3) is a 6-dimensional manifold. The possible motion of  can be seen as global (motion set of all system); if  is an incompressible fluid in a fixed domain M of E3 , then M˜ is identified with the group of transformations (or diffeomorphisms) that preserve the Lebesgue measure. Here the possible motion of  is only internal. Then the set M˜ has the structure of a continuous group (which is a Lie group if it is of finite dimension, as in the case of rigid bodies). ˜ All the internal structure, and all degrees of freedom, have been transferred to M; ˜ we have thus obtained with M a model that has no more internal structure. Change the modelling of  from M to M˜ essentially leads to two problems: (1) find the equivalent of masses that intervene in the modelling of  by M: we must define an inertial tensor in M˜ for the rigid body; this leads to a Riemannian ˜ structure of M. ˜ We see the appearance of various (2) find the equivalent of forces that apply to M. species of force: exterior forces, linking force, inertial forces, or driving forces. In a Newtonian point of view, acceleration and force F are proportional; when the configuration space is a Riemannian manifold M, acceleration and force are

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not a priori modeled by similar mathematical objects, and thus they cannot be proportional. A fairly natural modelling of forces is the use of differential forms on the configuration space.

2.3 Reminder of Mechanics with an Example We first give an elementary example of a system consisting of n material points (particles) in E = E3 , in a field of exterior forces, and with mutual interactions. The starting point is a model of type (i), as defined in Section 1.2, with a finite set of n material points with given interaction forces, called interior forces of the system. At first, we use a classical presentation of the notion of force, as a field of vectors in Euclidean space. Let mi be the mass of the ith particle, xi ∈ E its position, vi its velocity, γi its acceleration. We define its momentum by pi = mi vi . Let Fij (xi , xj ) ∈ E (respectively Fie (xi ) ∈ E) be the force on the ith particle due to the j th particle (respectively by the exterior). This interaction force is generally defined for xi = xj , and is such that2 Fij (xi , xj ) = −

 ∂Ui,j  (xi − xj )  φ (r), ( x i − x j ) =  xi − xj  ij ∂xi

  ∂Ui,j (r), Uij (r) ∈ R r = xi − xj  . with φij (r) = − ∂r

(2.1)

Notice that Fij (xi , xj ) = −Fj i (xi , xj ) if Uij (r) = Uj i (r). The force field on the ith particle by the exterior system is assumed to be such that Fie (xi ) = −

∂Ui (xi ) , with Ui (xi ) ∈ R ; ∂xi

(2.2)

we say that the (interaction and exterior) forces are the derivative of a potential. Then the total force field on the ith particle also depends on the positions of the other material points of the system and is given by Fi (x1 , . . . , xn ) = Fie (xi ) +



Fi,j (xi , xj ).

j =i

The fundamental principle of mechanics (the Newton principle)—applied to the ith particle—consists in stating that acceleration is proportional to the force, with the proportionality coefficient being the inverse of the mass of the particle:

notation ∂x∂ i means the derivative in E with respect to the components of xi and will be used several times below.

2 The

2.3 Reminder of Mechanics with an Example

mi

155

dpi dvi = = Fi . dt dt

(2.3)

The configuration space of M˜ in a more precise modelling (of type (ii) from Section 1.2) is in fact a subdomain of E n , defined by   M˜ = (x1 , x2 , . . . , xn ), xi ∈ E, xi = xj ∀i, j, i = j ; M˜ is an open set of E n , and thus has a Riemannian structure.3 Let ˜ v = (v1 , . . . , vn ) ∈ Tx M˜ = E n , m = (m1 , . . . , mn ); x = (x1 , . . . , xn ) ∈ M, from the velocity v we successively define for the system its momentum p = (p1 , . . . , pn ) = (m1 v1 , . . . , mn vn ), then its total momentum P =  mi vi ; then its kinetic energy as Ekin (v) = K(v) =

1 mi vi2 . 2

(2.4)

We define the (“total”) force on the system at a point x ∈ M˜ by F (x) = (F1 (x1 , . . . , xn ), . . . , Fn (x1 , . . . , xn )), and also the so-called exterior and interior forces: F e (x) = (F1e (x1 ), Fne (xn )),   F1,j (x1 , xj ), . . . , Fn,j (xn , xj )). F˜ (x) = ( j =n

j =1

Thus the total force applied to the system at the point x ∈ M˜ is the sum of the exterior and interior forces, the two forces having now a similar use: F (x) = F e (x) + F˜ (x). If we define U (x) = U e (x) + U˜ (x), with    1 U e (x) = Ui (xi ), U˜ (x) = Ui,j (xi − xj ), 2

(2.5)

i=j

i

we see that the total force F (x) on x is such that F (x) = −

∂U ∂U (x), i.e. Fi (x) = − (x1 , . . . , xn ), i = 1, . . . , n. ∂x ∂xi

that its metric is given by g(v) = relative masses of each particle.

3 Notice



mi vi2 , with mi = mi /



mi , and thus depends on the

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2 Classical Mechanics

Thus as a consequence of hypotheses (2.1) and (2.2), the total force field F is a derivative of a potential U that is the potential of the system, with each exterior or interior force field the derivative of the respective potential U e or U˜ . The fundamental equation of mechanics for the system modeled by the point x (at time t) is (see (2.3)) dp (t) = F (x(t)). dt

(2.6)

We successively define the mechanical energy4 Em of the system modeled at the point (x, v), Em (x, v) = K(v) + U (x);

(2.7)

the work of the field of forces F in a Euclidean space E (in a coordinate system (x i )) along a path τ in E is given by the integral  F.dx =

W = τ

 

Fi .dx i .

(2.8)

τ

The power PvF of the force field F (along the velocity v) is defined by PvF =
= Fi .v i . j ∂x

Below we simply denote by Pve and Pvi the power of the exterior and interior forces. We have the following basic property: Theorem 1 During the motion of the system, the mechanical energy is preserved. The time derivative of the kinetic energy is equal to the power of all (exterior and interior) forces. Proof We have only to take the scalar product of (2.6) and v, and use the expressions dv d  vi2 dp .v = (m ).v = ( mi ), dt dt dt 2

F (x).v = −

∂U dx dU . =− (x(t)). ∂x dt dt

We thus obtain dK = F.v = PvF = Pve + Pvi . dt

¶

4 This term may be confusing, because here it is the sum of the system energy and of the potential energy of “exterior” forces.

2.4 Mechanics of Newton, Lagrange, and Hamilton

157

˜ which is the projection of By integrating (2.8) along a “path” τ in the space M, the motion onto T M˜ with origin x0 = τ (t0 ) and end x1 = τ (t1 ), we obtain that the increase in the kinetic energy of the system is equal to the work of the force field F along the path, and it depends only on the endpoints of the path 



K(t1 ) − K(t0 ) =

F.dx = τ

t1 t0

PvF dt = −(U (x1 ) − U (x0 )).

Energy of a system, and internal energy. The internal energy Eint of a system is a difficult notion to specify. It expresses the capacity of the system to store energy at the macroscopic level from its properties at the microscopic level, and therefore is an internal degree of freedom (a “state variable”). In the current example, the point of view is quite different; at the elementary level of this example, in a conservative frame, and without the possibility of exchange of heat, the internal (mechanical) energy is here defined as the potential of interior forces U˜ (see (2.5)). The differential of the internal mechanical energy is thus equal to the opposite of the work of interior forces. Therefore, we have dEint = −Pvi . dt

(2.9)

The time derivative of the internal energy (during the motion of the system) is equal to the opposite of the power of the interior forces. The (proper) energy E of the system  is equal to the sum of its kinetic energy K and its internal energy U˜ : E = Ekin + Eint = K + U˜ .

(2.10)

Therefore, the mechanical energy Em of  (see (2.7)) is equal to the sum of the energy E of  and of the potential “exterior” forces (which correspond to the interaction energy between the system  and the exterior system): Em = E + U e .

(2.11)

2.4 Mechanics of Newton, Lagrange, and Hamilton The modelling of an evolution system (for instance following the scheme (ii)), relative to evolution equations, may be accomplished in many ways, each with its proper interest (besides the historical interest). Along with the position x(t) of the system at time t, quantities such as velocity, acceleration, momentum, and force are used.

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(i) Newtonian mechanics based on a Euclidean space It is the former and in a sense the more general method (which allows one to define nonconservative systems, i.e., so that the total energy is not preserved). But as configuration space, it uses a Euclidean space structure M˜ = En , which is easier to set out, but diminishes the generality of applications and the intrinsic character of mechanical quantities, and (probably) a deeper understanding of physical quantities such as the notion of force. The Newton principle applied to a material point in a Euclidean space consists in writing the proportionality of acceleration with respect to force. In a Riemannian space M, the basic quantities are (besides the position of the system) the velocity and the acceleration, so that we have to use the spaces T M˜ and ˜ which are the tangent bundles of M˜ and T M. ˜ To follow the Newtonian T (T M), ˜ mechanics would lead to viewing the force field as a section of the bundle T (T M) ∗ ˜ or of T (T M), which is incompatible with the expression of virtual power. Here we give only some general guidelines to be developed below. ˜ called Lagrangian mechanics (ii) Mechanics based on the tangent bundle T M, Here the equation of motion comes from a “variational principle” whereby the tangent bundle T M˜ is of basic utility. We search for an extremum of an integral I (τ ) defined in terms of a real differentiable function L defined on the tangent bundle T M˜ (called a Lagrange function or Lagrangian function) among the set of paths τ of C 1 , the class of J = [t0 , t1 ] in M˜ with fixed endpoints, so that τ (t0 ) = x0 and τ (t1 ) = x1 . We write  I (τ ) =

t1

 L(τ˙ (t))dt =

t0

t1

L(x(t), v(t))dt,

(2.12)

t0

˜ with v(t) = dx dt (t), x(t) = τ (t), t ∈ J. We are led to lift the paths in M to paths in the tangent (or cotangent) bundle. In the case of a material point of mass m in a Euclidean space M˜ subject to a force field, the derivative of a potential U , the Lagrangian L is defined by L(x, v) =

1 m v 2 − U (x). 2

(2.13)

This can be generalized without difficulty if M˜ has a Riemannian structure, the kinetic energy being proportional to the Riemannian metric. ˜ When the potential U is null (we then say that the material point is free in M), the extrema of I (τ ) also correspond to the extrema of the lengths of paths, and are ˜ then we can change L into its square root in (2.12), I (τ ) being then geodesics of M; the length of the path τ . In the general case (U nonnull), the so-called Euler–Lagrange equations of the motion for the extremal path are given by d ∂L ∂L ( j ) − j = 0, dt ∂ x˙ ∂x

j = 1, . . . , n,

(2.14)

2.4 Mechanics of Newton, Lagrange, and Hamilton

159

n ˜ ˜ with x˙ = v = dx dt , (x, v) ∈ T M. Taking the Lagrangian (2.13), with M = R , we immediately obtain the Newton equation in the “conservative” case.

Remark 1 Various generalizations are possible: (a) The Lagrangian can explicitly depend on time; then the system is said to be nonautonomous Lagrangian. Then equation (2.14) is still valid. The system can also have time-dependent holonomic links. The configuration space is then a (time-dependent) submanifold of the configuration space of the free system (see [Arn1, ch. 4,§19E]). (b) We can give an intrinsic formulation of the evolution equation of the system, ˜ of accelerations (see called the Euler–Cartan equation, within the space T (T M) [Mall, ch. IV, 2.2]). ¶ ˜ called Hamilton mechanics (iii) Mechanics based on the phase space T ∗ M, ˜ whereas the force fields Here the momenta are of basic use as covectors on M, ∗ are modeled as sections of the bundle T (T M). As in section (ii), we can still refer to a variational principle with the search for ˜ on a fixed interval J = [t0 , t1 ]. There an extremum among the set of paths in T ∗ M, are many reasons behind this choice of space. ˜ An essential point is the existence of a canonical form on the phase space T ∗ M, ∗ j ˜ ˜ which is given in (q, p) ∈ T M in a coordinate system (x ) of M by θ = p.dx =  pj dx j , and then also the existence of a natural symplectic form ω = dθ . Then the variational principle can be written in T ∗ M˜ or in an enlarged space (with time) using the canonical form θ to obtain the least action principle, also called the Maupertuis–Euler–Lagrange–Jacobi principle, leading to the Hamilton equations and to the Hamilton–Jacobi equations. An “intrinsic” formulation is given by the Hamilton–Cartan equations using the notion of characteristic. Another main point is the writing of the virtual work principle (using a duality), which is essential to a weak formulation of a problem. It seems to be at the origin of the d’Alembert–Lagrange principle on links. ¶ These varied points of view of these mechanics are in fact very close to one ˜ and their tangent spaces another, and we are led to use both bundles T M˜ and T ∗ M, ˜ and T (T ∗ M). ˜ We can go from one space to another thanks to one of the T (T M) following transformations: Legendre transformation, Hamiltonian transformation (see [Mall]), and transformation by the Riemannian metric tensor. We insist on the fact that Lagrangian and Hamiltonian mechanics deal with conservative systems only: the field of forces is the derivative of a potential, which allows us to define a Lagrangian or a Hamiltonian. The mechanics of continuous media, which is modeled from the conservation laws and the principle of virtual work (or of virtual powers), does not use the same processing. In the virtual powers principle, forces appear as dual variables of velocities, which leads the notion of connection (in a Riemannian manifold). Let  be a system whose state is given by a point in a configuration space, here denoted by M. This space, which has the structure of a differentiable manifold, is also equipped with a Riemannian metric. This metric is given by the kinetic energy

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of the system and thus depends a priori on the masses of the system (or on the inertia tensor for a rigid body) and of course on the geometry of the system. A goal of this section is to study the notion of force field in order to find an application of Newtonian mechanics in the Riemannian frame. The notion of force field will be modeled by a differential form on the space M˜ (or more generally on ˜ T ∗ M). We deduce the modelling of accelerations, as a substitute for Newton’s principle, and then that of the motion. This is related to the theory of second-order differential ˜ Below we will denote equations on a manifold, using the tangent bundle T (T M). ˜ the space M simply by M.

2.4.1 Metric Map Changing the velocities into momenta leads to changing the space T M into the phase space T ∗ M using the Riemannian metric g on M and the mass of the system m. (i) For all v ∈ Tx M, we define the covector, denoted by Gv, by g(v, w) = < Gv, w >,

∀v, w ∈ Tx (M),

(2.15)

where < ., . > is the duality Tx∗ (M), Tx (M). We thus define a map (called a “metric” map) denoted by G from T M onto T ∗ M. In a coordinate system (x j ) where the metric is given by g =  gij dx i dx j , this map gives p˜ = Gv =



p˜ i dx i , with p˜ i =



gij v j , and v =



vj

∂ . ∂x j

(2.16)

(ii) The Riemannian metric must take the mass of each part of the system into account (the specific mass in the case of a rigid body). In order to preserve the notion of distance in the Riemannian metric with a length unit, we multiply by the inverse of the total mass of the system. Then the momentum p associated with the velocity v is the covector p = mGv = mp. ˜ In a coordinate system (x j ) of a Riemannian space, we write p=



pi dx i , with pi = mp˜i = m



gij v j .

In a Euclidean space E3 , for a particle (a material point) with mass m, the momentum p is identified with mv. For a system with n particles of masses (mi ), the total momentum is given by p = pi = mi v i = m mi v i , where m

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is the total mass and mi = mmi the reduced mass. This implies that the Euclidean space E3n is equipped with the Euclidean metric weighted by the relative masses  of particles, g(v) = mi vi2 and the distance between two configurations of the  1 system x = (xi ), y = (yi ) ∈ E3n , given by d(y, x) = ( mi |yi − xi |2 ) 2 .

2.4.2 First-Order Equation The motion of a system is given by a function τ : t ∈ J ⊂ R → τ (t) = x(t) ∈ M (assumed to be of class C 2 on a time interval J ), with x(t) giving the position. Then we can define its velocity by derivation with respect to time as τ˙ : t ∈ J → τ˙ (t) = (x(t), x  (t)) ∈ T M.

(2.17)

Let  be an open set of R × M, and let v˜ be a field of vectors on , thus timedependent, called a section of pr2 :  → M, where pr2 is the projection from R × M onto M, so that v˜ ◦ πM = pr2 (see [Bour.var, 9.1.7]), i.e., a map from  into T (M) such that v(t, ˜ x) ∈ Tx M, ∀(t, x) ∈ , of class C s . An integral curve of v˜ is a map τ (of class C k , k ≤ s − 1) from an open set J of R into M such that ∀t ∈ J , we have (t, x(t)) ∈  and ˜ x(t)). τ˙ (t) = v(τ ˜ (t)), we also write x(t) ˙ = x  (t) = v(t,

(2.18)

This definition can be viewed in the usual frame of integral curves, using the vector field η defined by η(t, x) = ((t, 1), v(t, ˜ x))

(2.19)

on  with values in T (R × M). In order that τ be an integral curve in M of v, ˜ it is necessary and sufficient that the map t → (t, τ (t)) be an integral curve of η in the usual sense. It is equivalent to having an integral curve of η or of v. ˜ Then we have the following classical properties with respect to the initial conditions, subject to the regularity conditions C r , r ≥ 1, of v˜ with the indicated reference. (1) Existence. For all x0 ∈ M, there exists, in an open interval J in R with 0 ∈ J , an integral curve τ of v˜ on J such that τ (0) = (0, x0 ). (2) Uniqueness. Let τ1 and τ2 be two integral curves of v˜ on open intervals J1 and J2 respectively, containing 0, such that τ1 (0) = τ2 (0). Then τ1 = τ2 on J1 ∩ J2 . We again note that if v(0, ˜ x0 ) = v0 ∈ Tx0 M, then the integral curve τ such that τ (0) = (0, x0 ) moreover satisfies x  (0) = v(0, ˜ x(0)) = v0 through (2.18). Often the field v˜ is obtained by solving the fundamental equation of mechanics, which will be presented below.

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2.4.3 Second-Order Equation Notions of second-order jets (in T (T M)). Let π = πM be the canonical projection of T M onto M, and let πT M be the canonical projection of T (T M) onto T M; thus in a chart U of T (T M), πT M (z) = (x, v) with z = ((x, v), (w, γ )) ∈ U . The derivative map of π, denoted by π∗ (or π  ), from T (T M) to T M is such that π∗ ((x, v), (w, γ )) = (x, w), ∀(w, γ ) ∈ Tx,v (T M), (x, v) ∈ T M. The set JM of elements of T (T M)) such that π∗ ((x, v), (w, γ )) = (x, v) (thus such that w = v) is called a jet manifold; JM is thus defined by JM = {z ∈ T (T M), π∗ (z) = πT M (z)} . It is a submanifold of T (T M)), which also has a vector bundle structure on T M, with the n-dimensional fibres (JM )x,v . There exists an involutive diffeomorphism j from T (T M) onto itself called the canonical involution of T (T M), such that5 π∗ ◦ j = πT M ,

πT M ◦ j = π∗ ,

j 2 = I dT (T M) ,

(2.20)

and the jet bundle may be still defined by JM = {z ∈ T (T M), j (z) = z} . Let be a vector field on T M that is a section of the jet bundle JM (it is called “rabatteur” (see [Dieud3][ch. XVIII]) or “special.” It satisfies π∗ (x, v) = (x, v),

∀(x, v) ∈ T M,

and thus J = j ◦ J . We recall that the acceleration is defined from the velocity by derivation: τ¨ : t ∈ J → τ¨ (t) = ((x(t), x  (t)), (x  (t), x  (t))) ∈ T (T M);

(2.21)

we define γ (t) = x  (t). We have τ¨ (t) ∈ JM for all t ∈ J . A second-order differential equation is needed to determine the motion of the system from a field of vectors on τ˙ (J ) (with initial conditions): τ¨ (t) = (τ˙ (t)).

(2.22)

The field must be such that (x, v) is a jet, hence such that (x, v) ∈ JM , ∀(x, v) in τ˙ (J ). Thus it is a section of the jet bundle on τ˙ (J ).

5 See

[Dieud3, ch. XVI.20, Pb 2].

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Remark 2 Isochronic fields. In order that the differential equation (2.22) does not depend on the unit of time, the acceleration field must be isochronic, i.e., it must satisfy the relation (see [Dieud3, ch. XVIII.4]) (hλ v) = λhλ (v),

∀v ∈ T M, λ ∈ R + , or = λ(hλ )∗ ,

(2.23)

where hλ is the transformation hλ (x, v) = (x, λv). Indeed, if (x, v) = (v, γ ), then must satisfy (x, λv) = (λv, λ2 γ ): ¶ hλ

((x, v), (w, γ )) ∈ Tv (T M) −→ ((x, λv), (w, λγ )) ∈ Thλ v (T M) ↑↓ π hλ∗ ( ) ↑↓ π˜ hλ

−→

(x, v) ∈ Tx M

(x, λv) ∈ Tx M

Transformation of JM and T (T M) by the Metric G∗ (i)Transformation from T (T M) onto T (T ∗ M). Let X ∈ Tx,v (T M) be a vector given in the coordinate system (x i , v i ) by X=



wi

∂ ∂ + γi i ; ∂x i ∂v

(2.24)

the vector G∗ X ∈ Tx,p (T ∗ M) is defined by its action on a differentiable function f from T ∗ M into R by (G∗ X)f = X(f ◦ G). Thus (G∗ X)f = X(x,v)(f ◦ G) =



(wi

∂ ∂ + γ i i )f (x, Gv). ∂x i ∂v

Since Gv is given by (2.16), we have (G∗ X)f =

 ∂pj ∂ ∂pj ∂ ∂ (wi i + γ i i + wi i )f (x, p), ∂x ∂v ∂pj ∂x ∂pj

and thus (G∗ X)x,p = with v k = (G∗



 ∂gj k ∂ ∂ (wi i + (γ i gij + wi v k ) ), ∂x ∂x i ∂pj

gkl pl , and therefore,

 ∂gj k ∂ ∂ ∂ )x,p = + vk ), i i ∂x ∂x ∂x i ∂pj

(G∗

 ∂ ∂ )x,p = gij . i ∂v ∂pj

(2.25)

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(ii) Image of the jet manifold. The image of the jet manifold JM by mG∗ is the submanifold T (T ∗ M), which we call an mG-jet manifold:6 J˜M = mG∗ (JM ) = {˜z ∈ T (T ∗ M), z˜ = (x, p; v, p), ˙ with p = mGv}.

(2.26)

If τ˜ : t ∈ J → (x(t), p(t)) ∈ T ∗ M is an embedding from J into T ∗ M, then τ˜J = τ˜ (J ) is a submanifold of T ∗ M. Let τ˙˜ be its time derivative: dp τ˙˜ : t ∈ J → P (t) = (x(t), p(t)), (v(t), (t))) ∈ T (T ∗ M). dt Then XP = τ˙˜ ◦ τ˜ −1 is a section of the bundle J˜M from τ˜J in T (T ∗ M).7 Remark 3 Transformation of the vector fields by the metric. From a vector field on T M, we can define the vector field G on T ∗ M, the image of , by the diffeomorphism G: G G = G∗ G−1 , i.e., x,p = G∗ x,v , with x,v =



wi

∂ ∂ + γ i (x, v) i , ∂x i ∂v

with v = G−1 p. The image X of a section of the jet manifold by mG∗ is a section of J˜M , since it satisfies π˜ ∗ ◦ X = IT ∗ M , with π˜ ∗ = Gπ∗ G∗ . To the path τ˙ in T M there corresponds the path τ˜ in T ∗ M such that τ˜ (t) = (x(t), p(t)) = (x(t), mGx(t )v(t)). Let = (x, v, v, γ ) be the velocity vector of τ˙ ; then the velocity vector XP (t) of τ˜ in T ∗ M at (x(t), p(t)) is XP (t ) = τ˙˜ (t) = (x(t), ˙ p(t)). ˙ Then mG∗ = XP .

(2.27)

Proof Let m = 1. By components, we have pl = is p˙ l =



glk v k ; then the time derivative

  ∂glk (g˙lk v k + glk v˙ k ) = ( j v j v k + glk γ k ). ∂x

But for all smooth functions f on T ∗ M, we have (G∗ )f = (f (G(x, v))) = whence (G∗ )f =





vk

∂f ∂f ∂gj k k ∂f + vm v + γj glj , k m ∂x ∂pj ∂x ∂pl

∂f ∂f v k ∂x ˙l ∂p = XP f. k +p l

¶

canonical projections πT (T ∗ M) (from T (T ∗ M) onto T ∗ M) and πT ∗ (T M) (from T ∗ (T M) onto T M), we have J˜M = {z ∈ T (T ∗ M), πT (T ∗ M) (z) = mG(πT ∗ (T M) (z)). 7 A section of the bundle J˜ may be called an mG-special field (or simply a G-special field). M 6 With the

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165

2.4.4 Hamiltonian Vector Fields Let M be the configuration space of a system, with the structure of a Riemannian manifold. Then let T ∗ M be the phase space, π˜ its canonical projection onto M. The main interest of the phase space T ∗ M is the existence of a canonical form θ defined in an intrinsic way thanks to the derivative of the canonical projection π˜ of T ∗ M, for all p ∈ T ∗ M by ∀X ∈ Tx,p (T ∗ M).

< θx,p , X > = < p, π˜ ∗ X >,

(2.28)

Thus θp = p ◦ π˜ ∗ . For X = ((x, p), (v, p)), ˙ we have < θx,p , X > = < p, v >. In a coordinate system (x i ), the canonical form is given by θx,p =



pi dx i .

The exterior differential ω = dθ of θ (called a symplectic form) is given by ωx,p =



dpi ∧ dx i .

Let X be a vector field on T ∗ M given in a coordinate system (x i , pi ) by X=



vi

∂ ∂ + f˜i ∈ Tx,p (T ∗ M). i ∂x ∂pi

(2.29)

The interior product i(X)ω of ω by X is a differential form on T ∗ M, given in the coordinate system by i(X)ω =



(−v i dpi + f˜i dx i ).

(2.30)

An interesting particular case for a conservative system is given by the following: Definition 1 A vector field X on T ∗ M is called Hamiltonian if there exists a differentiable function H on T ∗ M (called a Hamiltonian) such that i(X)ω = −dH.

(2.31)

This is equivalent to saying that the differential form i(X)ω is exact, or that the system is conservative. Then we set X = XH . We can have a weaker property, which is local: when i(X)ω is closed, i.e., is such that d i(X)ω = 0 or that LX ω = 0, with the Lie derivative LX , the vector field X is said to be a locally Hamiltonian field.

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Example 6 Consider a particle of mass m, subject to a potential U in the space M = R 3 . We define the Hamiltonian H by H (x, p) =

1 2 p + U (x). 2m

(2.32)

The associated vector field XH in the frame ( ∂x∂ i , ∂p∂ i ), i = 1, 2, 3, must also be the generator of the motion, whence (

dx dp , ) = XF , dt dt

i.e., XF =

 dx i ∂ dpi ∂ + . dt ∂x i dt ∂pi

(2.33)

Then by identification with (2.31), and (2.32), we obtain (

∂H ∂U ∂H pi dx i dpi , )=( , − i ) = ( , − i ), dt dt ∂pi ∂x m ∂x

which is the evolution equation of the particle in T ∗ M, and the Hamiltonian vector field is XF = (

∂U p ,− ) = (v, f ). m ∂x

The differential of the kinetic energy is dEkin =



v i dpi ,

with Ecin =

 1 p2 . 2mi i

Moreover, F =  fi dx i is a differential form on M, and thus if πT ∗ M is the canonical projection of T ∗ M onto M, then (πT ∗ M )∗ F is a differential form on T ∗ M, still denoted by F , that represents the force field on the particle. Then the force field F is an exact differential form F = −dU ; it is a potential derivative. For a Hamiltonian vector field XH , we thus have the basic formula i(XH )ω = −dH = −dEcin + F.

¶

(2.34)

Remark 4 Isochronic fields. Let XP be a vector field on the phase space corresponding to an isochronic field ; it must satisfy XP = λ(hλ )∗ XP ,

∀λ > 0,

(2.35)

and thus XP (x, λp) = (λv, λ2 p) ˙ for XP (x, p) = (v, p). ˙ The question is whether XP is a Hamiltonian vector field.

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If the answer is affirmative, let H be the Hamiltonian corresponding to (2.32); p then with (2.29), we have XP (x, λp) = (λ m , − ∂U ∂x ), and thus (2.35) is satisfied only if the potential U is null, which corresponds to the geodesics. ¶

2.4.5 Time-Dependent Force and Hamiltonian Time-Dependent Field of Forces Let  be an open set in T ∗ (M) × R and X a map from  into T (T ∗ M) that is a section of class C k of the projection pr1 from T ∗ (M) × R onto T (T ∗ M), such that X(x, p, t) ∈ Tx,p (T ∗ M), ∀(x, p, t) ∈ . We can represent this map by X(x, p, t) =



aj

∂ ∂ + bj , j ∂x ∂pj

 with coefficients a j , bj ∈ R that are (x, p, t)-dependent. Let F = j fj dx j be a  time-dependent field of forces. Let v = m1 G−1 (p); thus v j = m1 k g j k pk . Then we define a section XF of the projection pr1 by a j = vj ,

bj = f˜j = fj −

1  ∂g ik pi j pk . 2 ∂x i,k

We have i(XF )dθ = −

1 −1 G (p).dp + F˜ = −v.dp + F˜ , m

 with F˜ = f˜j dx j and θ = pdx. An integral curve of XF is a map u˜ (of class C r ) from an interval J ⊂ R into ∗ T M × R (or u from J into T ∗ M) such that for all t ∈ J , u(t) ˜ = (u(t), t) = (x(t), p(t), t) ∈ ,

and u (t) = XF (u(t), t) = XF (u(t)). ˜

j 1 −1 ˜ This corresponds to dx dt = v = m G (p), dt = fj , j = 1, . . . , n. 2 If F = −dU (x, p, t) (U of class C ), then we can associate with it a timedependent Hamiltonian Ht (x, p) = H (x, p, t) = 12 E + U and the vector field XF = XHt , and thus

dp

aj =

∂Ht , ∂pj

bj = −

∂Ht , ∂x j

∂ X˜ F = XHt + . ∂t

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Furthermore, if we denote by dHt the differential of Ht with respect to (x, p) only, we still have i(XHt )dθ = −dHt = −(dH − ∂H ∂t dt). Evolution in the Space T ∗ (M × R), Hamilton–Cartan Equation We can also consider the evolution in the phase space T ∗ (M × R) (with time included), with variables (x, t, p, p0 ), the variable p0 = −E opposite to the energy,8 being the dual variable of the time t = x 0 . Let θ˜ be the canonical differential form on T ∗ (M × R): θ˜ = θ + p0 dx 0 = pdx − Edt.

(2.36)

From a time-dependent Hamiltonian H , we define the submanifold H called a figuratrice (see [Mall]), of (odd) dimension 2n + 1, by F (x, x 0 ; p, p0 ) = H (x, t, p) − E = 0,

(x, t; p, −E) ∈ T ∗ (M × R).

(2.37)

We can consider this relation to be a state equation. Let θ˜ H be the differential form ˜ induced on H by θ: θ˜ H = θ − H dt = pdx − H (x, t, p)dt. A vector field X = X˜ H on T ∗ (M × R) is said to be Hamiltonian for the function H (x, t, p) of class C 1 if it satisfies i(X)d θ˜ = d(E − H ),

(2.38)

i(X˜ H )d θ˜ H = 0.

(2.39)

that is,

This is the Hamilton–Cartan equation, and X˜ H is a characteristic field for d θ˜ H , and its integral curves are said to be the characteristic curves. They are solutions of the equation dτ = X˜ H dt

p0 = − Ec , with x 0 = ct, c being the velocity of light, so that x 0 and p0 have the same units as x and p respectively.

8 Or

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in T ∗ (M × R) with τ˜ : τ˜ (t) = (x(t), t, p(t), −E(t)), with the time t as parameter. ∂ The vector field X˜ H on T ∗ (M × R) in the frame ( ∂x∂ j , ∂t∂ , ∂p∂ j , − ∂E ), j = 1, . . . , n, is given by X˜ H =

 ∂H ∂  ∂H ∂ ∂ ∂H ∂ − . + + ∂pj ∂x j ∂t ∂x j ∂pj ∂t ∂E

The velocity vector being given by X˜ P (t) = ( dx dt (t), 1 ; the following Hamilton equations: ∂H dx j = , dt ∂pj

dpj ∂H =− j, dt ∂x

dp dE dt (t), − dt (t)),

we obtain

∂H dE = , j = 1, . . . , n. dt ∂t

(2.40)

Note that i(X˜ H ) θ˜ H = < θ˜ H , X˜ H > =



pj .

 ∂H ∂H −E = pj . − H, ∂pj ∂pj

and thus < θ˜ H , X˜ H > = p.v − E = L, with L the Lagrangian of the system. Evolution in the Space T ∗ (M × R) × R. Hamilton–Jacobi Equation It is more interesting to use the space W = T ∗ (M × R) × R of odd dimension 2n + 3, with the supplementary variable s, called the action.9 Let ω = ds − θ˜ = ds −



pj dx j

j =0,...,n

be the basic differential form on W . Recall that the integral manifolds of the Pfaff equation ω = 0 are at most (n + 1)-dimensional, as seen in the first chapter. We can view the state equation (2.37) as an equation in W , thus giving a submanifold (of codimension 1) denoted by HW , and we denote by ωH the induced differential form on HW .

9 This name suggests an idea of control of the system through this variable. In some sense, it is a similar variable to the entropy in thermodynamics.

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Now let  be a hypersurface10 of (M × R) × R defined by S(x, t) = s,

(x, t) ∈ M × R, s ∈ R.

∂S Here we assume that S is of class C 1 . Let pS = (pj ), pi = ∂x i , i = 1, . . . , n, ∂S ∂S p0 = ∂t . We write pS = (∇S, ∂t ). We will look for such a hypersurface satisfying with (2.37) the so-called Hamilton–Jacobi equation

F (x, x 0 ; pS ) =

∂S + H (x, t, ∇S) = 0. ∂t

(2.41)

˜ be the set of regular contact elements of , that is, the set of elements Let  (x, t, s, X), where X is a tangent plane to  at (x, t, s) that is not parallel to the axis ˜ is identified with a submanifold Rs . If  is a submanifold of (M × R) × R, then  of T ∗ (M × R) × R.11 Then we have the following basic property (see [Mall, ch. II.5.5, Pb19]): ˜ A hypersurface  satisfies the Hamilton–Jacobi equation (2.41) if and only if  is an integral manifold of the Pfaff equation ω = 0. Such a hypersurface  may be constructed thanks to the characteristics of ω. The characteristic field X of this differential form is defined by [Mall, ch. II.5.5, Pb18] i(X)ω = 0,

i(X)dω = 0.

For a state equation F (x, s, p) = 0, the characteristic field XF is given by XF = (Pi ,



pj Pj , −Xi − pi Z),

with Pi =

∂F ∂F ∂F , Xi = , Z= . ∂pi ∂s ∂xi

Then with F given by (2.37), we have XF = (Pi ,



pj Pj , −Xi ),

with Pi =

∂H ∂H , P0 = 1, Z = 0, Xi = . ∂pi ∂xi

The integral curves of the vector field XF , thus the characteristic curves for ωH , are given by the Hamilton equations (2.40), with the supplementary equation   ∂H ds = pj Pj = −E + pj = L, dt ∂pi

(2.42)

with L the Lagrangian function.

10 This term is used because we usually do not have globally enough smoothness for  to be a manifold. 11 We can also identify  ˜ with {(x, t, s, X) ∈ T(x,t,s) ((M × R) × R), Xs = 1}.

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Remark 5 Let p˜ S be the map (x, t) ∈ M ×R → pS = (∇S, ∂S ∂t ), with the notation ∂S ∂S ∇S = ( ∂x i ) = (pi ), i = 1, . . . , n, and p0 = ∂t . Of course we can identify p˜S with dS ∈ T ∗ (M × R). We have p˜S∗ θ˜ = dS. Proof Let X =



∈ T (M × R) (with x0 = t). Then

i ∂ i=0,...,n X ∂x i



(p˜ S )∗ X =

(2.43)

Xi

i=0,...,n

∂ + ∂x i



Xi

i,k=0,...,n

∂pk ∂ , ∂x i ∂pk

and thus < p˜S∗ θ˜ , X > = < θ˜ , (p˜S )∗ X > =



pi Xi = < dS, X >,

i=0,...,n

which gives (2.43), and thus p˜S∗ θ˜ =



∂S i i=0,...,n ∂x i dx

= dS. This implies

d(p˜ S∗ θ˜ ) = p˜ S∗ (d θ˜ ) = 0. We also have S(x(t), t) = s(t),

since

 ∂S dx i ∂S ds dS = + = = L, dt ∂x i dt ∂t dt

(2.44)

so that we will determine the function S by going back to the time in the characteristics, as will be seen in the following examples. Integrating on a time T interval (0, T ), we obtain S(x(T ), T ) = S(x(0), 0)+ 0 Ldt, which may be viewed as searching for an infimum among the paths γ going from (x(0), 0) to (x(T ), T ): 

 (∇S.dx−H dt) = S(x(0), 0)+inf

S(x(T ), T ) = S(x(0), 0)+inf γ

γ

γ

(∇S.v−H )dt, γ

 that is, S(x(T ), T ) = S(x(0), 0) + infγ γ Ldt. This leads to the so-called Hamilton–Jacobi–Bellman equation, usually written with the variable V = −S in control theory (see, for instance, [Flem-Sto]), where the velocity v depends on a control u in a domain U to obtain a given goal: −

∂ V (t, x) + H (t, x, Dx V (t, x)) = 0, ∂t

with

H (t, x, p) = sup (−pv(, t, x, u) − L(t, x, u)). u∈U

(2.45) ¶

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1 2 Example 7 The case of a free particle. Let H (x, t, p) = 2m p be the Hamiltonian. Then the equation (of Hamilton) of the characteristics are given by

∂H p dp dx = = , = 0, dt ∂p m dt and thus the characteristics are given by p(t) = p0 ,

x(t) =

p0 t + x 0. m

Now we want to determine the action S with the initial condition S(x, 0) = S 0 =| x |

(2.46)

(in a domain | x |> a > 0). This is done by the following steps: (1) The first step is to determine the initial momentum p0 = ∇S 0 (x 0 ) =

x0 . | x0 |

(2.47)

(2) Then the characteristics are given by x(t) =

1 x0 t + x 0. m | x0 |

(2.48)

(3) We have yet to find x = (x(t)) from x 0 , so that we go back to the time in the 0 characteristics. Let r 0 =| x 0 |, α 0 = |xx 0 | . Then x 0 = r 0 α 0 , and let x = rα. The relation (2.48) is rα = α 0 (

t + r 0 ), m

thus r =

t + r 0, m

and thus r0 = r −

t m

if r −

t > 0. m

(2.49)

(4) We express S(x(t), t) as a function of x 0 thanks to the Lagrangian, then eliminate x 0 . We have  S(x(t), t) = S0 (x 0 ) +

t



t

Ldt = S0 (x 0 ) +

0

0

p2 dt, 2m

and thus with (2.47) and the initial condition (2.46), S(x(t), t) = S0 (x 0 ) +

(p0 )2 t t =| x 0 | + . 2m 2m

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Then with (2.49), we finally obtain S(x, t) = (r −

t t t )+ =| x | − . m 2m 2m

(2.50)

(5) Verification: the initial condition S(x, 0) =| x |= S0 (x) is satisfied. Then we x 1 have ∇S(x) = |x| , ∂S ∂t = − 2m and thus 1 ∂S + | ∇S |2 = 0, ∂t 2m which is the Hamilton–Jacobi equation. Of course the characteristics intersect at the origin, which must be excluded. Let us give briefly another example with S(x, 0) = S 0 =| x |2 . Then the corresponding initial momentum is p0 = ∇S 0 (x 0 ) = 2x 0 . The characteristic equation gives x(t) =

1 0 1 2t p t + x 0 = 2x 0 t + x 0 = x 0 (1 + ). m m m

Thus we simply have (with 1 +

2t m

= 0)

x 0 = x(t)(1 +

2t −1 ) . m

Then we obtain S(x(t), t) = S0 (x 0 ) +

(p0 )2 2t 2t t =| x 0 |2 (1 + ) =| x(t) |2 (1 + )−1 , 2m m m

so that we obtain the action by S(x, t) =| x |2 (1 + 2

t −1 ) . m

We verify that the initial condition is satisfied. We have ∇S(x) = 2x(1 +

2t −1 ) , m

∂S 2t 2 = − | x(t) |2 (1 + )−2 . , ∂t m m

and thus ∂S 1 + | ∇S |2 = 0, ∂t 2m which is the Hamilton–Jacobi equation. ¶

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2 Classical Mechanics

We will see below that the equation (2.39) implies that the integral curves of the vector field XH are extrema among the set of paths in H for the integral I (γ ) of the differential form θ˜ (with the same endpoints). We will see also the properties of minimizing linked with the action, and the foliation associated with the family of submanifolds s , s ∈ J , of the space (M × R) × R.

2.4.6 Fundamental Equation of Mechanics The modelling of the configuration space of a system leads one to define a Riemannian manifold (M, g), then a Riemannian connection or Levi-Civita connection, and the covariant derivative of (possibly time-dependent) fields of vectors. Here we give some notions; for developments, see the appendix.

Notions of Connection and Covariant Derivative To define a connection on a manifold M is equivalent to defining in the tangent space Tv (T M), for all v ∈ Tx M, x ∈ M, a subspace Hv (called horizontal), a supplement of the subspace Vv (called vertical), tangent space to the fibre Tx M, whence Tv (T M) = Hv ⊕ Vv (direct sum). Then by duality, we define forms ϕ on T M, called horizontal (respectively vertical), such that ϕ(X) = 0,

∀X ∈ Vv

(respectively X ∈ Hv ),

∀v ∈ Tx M.

(2.51)

Hence Tv∗ (T M) = Hv⊥ ⊕ Vv⊥ , with ϕ ∈ Hv⊥ if ϕ(X) = 0, ∀X ∈ Hv . Let v ∈ Tx M). We denote by qv the projector on the vertical space Vv in Tv (T M) along the horizontal space Hv ; thus ker qv = Hv . And we denote by τv the operator of identification of Vv with the tangent space Tx M. In a coordinate system (x j ) of an open set U ⊂ M, a coordinate system of −1 π (U ) ⊂ T M is (x j , v j ), and a frame of π −1 (U ) is given by ( ∂x∂ j ), so that every  v ∈ π −1 (U ) is given by v = v j ∂x∂ j . Then a frame in T (T M) (above π −1 (U )) is  given by (( ∂x∂ j , ∂v∂ j )), so that ∀w ∈ Vv , w = wj ∂v∂ j , with real components (wj ), we have τv (w) =



wj

∂ , ∂x j

with τv (

∂ ∂ )= j. ∂v j ∂x

We then define a map Qv = τv ◦ qv from Tv (T M) onto Tx M, called the connector (or the connection map). Definition of the covariant derivative of the velocity field v. ˜ We denote by τ the map t ∈ J → x(t) ∈ M, v˜ = τ˙ the map t ∈ J ⊂ R → v(t, x(t)) ∈ Tx(t )(M), w˜ = (τ, τ˙ ) the map of t ∈ J into T M.

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175

With the canonical projection π of T M onto M, we have π ◦ w(t) ˜ = x(t), and d d thus π ◦ w˜ = τ . Let w˜ ∗ ( dt ) be the image in Tv(t,x(t ))(T M) of dt on J by the derivative of the map w. ˜ Then for all smooth functions f on T M, we have w˜ ∗ (

 ∂f ∂x j  ∂f ∂v j d ∂v j ∂x k d )f = f (x(t), v(t, x(t)) = + + ), ( dt dt ∂x j ∂t ∂v j ∂t ∂x k ∂t

d and thus the vertical component of w˜ ∗ ( dt ) is

(w˜ ∗ (

 ∂v j  ∂v j ∂  d v˜ j ∂ d V )) = + vk k ) j = ( . dt ∂t ∂x ∂v dt ∂v j

d ) is a jet, since Notice that w˜ ∗ ( dt

π∗ w˜ ∗ (

dx d d d d ) = (π ◦ w) ˜ ∗ ( ) = xˆ∗ ( ) = (t) = v(t, x(t)) = πT M (w˜ ∗ ( )). dt dt dt dt dt

Then the covariant derivative of v˜ (in fact of w) ˜ along x(t), t ∈ J, is defined by (see [Dieud3, ch. XVII, 17.17]) ∇ d (v) ˜ = Qv (w˜ ∗ ( dt

d )) ∈ Tx(t )M, dt

(2.52)

D ˜ or by dt v(t, ˜ x(t)). In the coordinate system (x j ) which is often denoted by ∇v (v)  j j in U ⊂ M, with the Christoffel symbols l,k (with ∇Xl Xk = l,k Xj , Xj = ∂x∂ j )  ˜ = (∇ d (v)) ˜ j ∂x∂ j , of the connection, the covariant derivative is given by ∇ d (v) dt dt with

˜ j = (∇ d (v)) dt

 j dv j  j l k ∂v j  k ∂v j + l,k v v = ( + v )+ l,k v l v k . k dt ∂t ∂x

(2.53)

Then applying the metric operator G of a Riemannian manifold to a Riemannian connection, we obtain the covariant derivative in the form p˜ = G ◦ v˜ by ˜ = ∇ d (G ◦ v) ˜ = ∇v p. ˜ G(∇ d v) dt

dt

(2.54)

Remark 6 Transformation of vertical and horizontal vectors by G∗ . The map G∗ due to the Riemannian metric and connection transforms vertical (respectively horizontal) vectors of Tv (T M) into vertical (respectively horizontal) vectors of Tp (T ∗ M). Proof Let X = ((x, v); (w, γ )) ∈ Tx,v (T M) (see (2.24)). Then G∗ X is given by (2.25). Therefore, if the vector X is vertical, then (π being the projection from T M onto M) π∗ X = w = 0 implies π∗ G∗ X = w = 0. Thus G∗ transforms vertical vectors into vertical vectors.

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Furthermore, G∗ transforms horizontal vectors into horizontal vectors, which is a consequence of ∇g = 0, or still of the formula G∇ek ej = ∇ek Gej in a coordinate system (x j ), j = 1, . . . , n, of M, with ej =

∂ . ∂x j

¶

Remark 7 Reminder on the covariant derivatives. Let v˜ be a vector field that is a section of the canonical projection πM : T M → M. Then the covariant derivative of v˜ along a tangent vector field η is expressed in a coordinate system (x j ) thanks to the Christoffel symbols ji l for T M with Xi = ∂x∂ i as  ηi Xi , v i Xi , η =    ∇η v˜ = (∇η v) ˜ i Xi = (Lη v i )Xi + v i ∇η (Xi ),

v˜ =



(2.55)

with the Lie derivative Lη . Thus Lη v˜ i =



ηj

∂v i d = v˜ i (x(t)) ∂x j dt

and ∇η Xi =



j

ik ηk Xj .

Thus the covariant derivatives of v˜ and of the momentum field p˜ = the Christoffel symbols ˜ ji l for T ∗ M, are given by (∇η v) ˜ i=



ηj

∇η p˜ =

pi dx i , with

  ∂v i d + ji k v j ηk = v˜ i (x(t)) + ji k v j ηk , j ∂x dt j,k





(∇η p) ˜ i dx i =

 i,l





j,k

∂pl  ˜ k ηi i + il pk ηi dx l . ∂x

(2.56) ¶

k

Remark 8 Relation between the Christoffel symbols ji l and ˜ ilk . The metric G transforms the frame (Xj ) = ( ∂x∂ j ) of Tx M into the frame (j ) of ∗ Tx M, j = G Xj =



gj k dx k .

The metric being under the Riemannian covariant derivative, we deduce invariant i X the relation from ∇Xk Xj = kj i ∇Xk j = ∇Xk G Xj = G ∇Xk Xj =



i G kj Xi =

With the covariant derivation formula for T ∗ M, ∇Xj dx i =





i kj i .

˜ ji l dx l , we obtain

2.4 Mechanics of Newton, Lagrange, and Hamilton

∇Xi (



gj k dx k ) =



(LXi gj k )dx k +

177



k gj k ˜ im dx m =



ijk gkl dx l . (2.57)

Thus we have the relation between the Christoffel symbols 

k = gj k ˜ lm



But the Riemannian connection is such that whence 

ijk gkm − ∂i gj m . 

gmk ijk =

(2.58)

1 2 (∂i gmj +∂j gim −∂m gij ),

1 k = (∂j gim − ∂i gmj − ∂m gij ). gj k ˜ lm 2

¶

Fundamental Equation From a force field given by a function12 f : (x, v) ∈ T M → f (x, v) ∈ T ∗ M, the field of momenta (and of velocities v) ˜ of the system must be a solution of the fundamental equation of mechanics using the covariant derivative of momenta along an integral curve τ : t → x(t) ∈ M of v˜ (see (2.52) and (2.54)): ∇v p˜ = f,

with p˜ = mGv, ˜

also denoted by

D p˜ (t) = f (t), dt

(2.59)

with the metric tensor G and m the total mass of the system. We first notice the following: (i) We could also give the equation with the covariant derivative of the velocity instead of the momentum, but it is more natural to use momenta, with forces as differential forms. (ii) v˜ is not necessarily a section of the canonical projection πM : T M → M, since v˜ depends on the pair (t, x(t)), and not only on x(t); thus the usual covariant derivatives have to be enlarged (see below). Equation (2.59) is a differential equation in v˜ of first order. Thus the motion must be determined in solving the pair of equations (2.18), (2.59), with initial conditions (x0 , v0 ). Without forces, equation (2.59) is reduced to ∇v p˜ = 0, or ∇v v˜ = 0, and the integral curves of v˜ are the geodesics of the Riemannian manifold (M, g). Then the field of forces f is a differential form on the considered manifold. Equation (2.59) is seen as an “inhomogeneous equation” (that is, with a nonnull right-hand side) with respect to the geodesic equation. A weak formulation of the fundamental equation of mechanics will be given below.

12 Possibly

time-dependent.

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2 Classical Mechanics

We also notice that if the metric tensor g is multiplied by a positive number m, ˜ v, thus g˜ = mg, the Christoffel symbols ji l are unchanged, and thus if p˜ = G ˜ ˜ we have ∇η p˜ = G∇η v. ˜ Therefore, changing the total mass does not change these formulas. It is interesting to compare the system of equations (2.18), (2.59) with the secondorder equations. Remark 9 Comparison between the Hamiltonian system and the fundamental equation. In the Hamiltonian case H , we compare the formula dpj ∂H ∂U 1  ∂g ik =− j =− j − pi pk = f˜j , dt ∂x ∂x 2 ∂x j  with the new formula ∇η p˜ = F = fj dx j . The agreement between these two  formulas is due to (2.56) and to (2.59), using ∇η p˜ = ∇t ( pj dx j ): ∇η p˜ =

 dpj  dpj j dx j + pj ∇η dx j ) = dx j + pj ik v i dx k ). ( ( dt dt

We have ∂gik 1 ∂U 1 fj = f˜j − v i v k j = − j − j , with 2 ∂x ∂x 2 ik   ∂g ∂gik ∂g ik ∂gml j = (pi pk j + v i v k j ) = v m v l (gim gkl j + ). ∂x ∂x ∂x ∂x j We obtain that j = 0, since and (gml ) are inverses). Thus fj = −

∂U , ∂x j



pk

ml (gkl ∂g + g mp ∂g = 0 (the metric matrices (g pk ) ∂x j ∂x j

hence F =



fj dx j = −dU. ¶

2.4.7 Canonical Forms Decompositions We define the differential form α on T M, with m = 1, by α = G∗ θ,

(2.60)

the pullback of the canonical form θ by the map G; see (2.15). With π the canonical projection of T M on M, we notice that < αv , > = < θp , XP > = g(v, π∗ ),

∀ ∈ Tv (T M), v ∈ Tx M,

(2.61)

2.4 Mechanics of Newton, Lagrange, and Hamilton

179

with p = Gv, and XP = G∗ . Indeed, < αv , > = < (G∗ θ )v , > = < θp , G∗ >, and thus < αv , > = < π˜ p∗ Gv, G∗ > = < Gv, π˜ ∗ G∗ > . But π˜ ∗ G∗ = π∗ , since π˜ G = π. Then < αv , > = < Gv, π∗ >, and thus (2.61). With < πv∗ (p), > = < p, π∗ >, ∀ ∈ Tx,v (T M), we have αv = t π∗ (Gv) = (Gv) ◦ π∗ .

(2.62)

In a coordinate system (x j ), α is expressed by αv =



gij v j dx i .

(2.63)

The differential form θ , and thus α as well, are both horizontal for a Riemannian connection: indeed, every X ∈ Vv , is such that π∗ X = 0, and thus θp (X) = p(π∗ X) = 0, and it is the same for α. Let E(v) = g(v, v). We prove that the differential form dE is vertical, thus < dE, X > = 0, for all horizontal vectors X. In a coordinate system (x j ), the vector field Xj = ∂x∂ j has as horizontal lift (see, for example, [Kob-Nom, ch. III.7, p. 143]) (Xj∗ )v =

 ∂ ∂ − ji k v k j . ∂x j ∂v

(2.64)

But in a Riemannian connection, we have Xj g(v, v) = 2g(∇Xj v, v), thus (Xj∗ )v g(v, v) = 2g(∇Xj v, v) −



ji k v k (2gij v j ) = 0,

for all j , which implies that dE is a vertical form. ¶ Then we have two lemmas of decomposition of differential forms with a Riemannian connection: Lemma 1 For every jet section on T M, the differential form i( )dα on T M defined by (2.60) splits into a vertical form (−dE/2) and a horizontal form v → πv∗ γv , with γv = G(Qv ( )) ∈ Tx∗ M, where Qv is the connector: 1 (i( )dα)v = − (dE)v + πv∗ γv , 2

v ∈ Tx M,

with E defined by E(x, v) = gx (v, v). This decomposition is unique.

(2.65)

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2 Classical Mechanics

With respect to momenta, with the canonical differential form θ on T ∗ M, we have the following result. Lemma 2 Let XP be a mG-jet section13 of T (T ∗ M). The differential form i(XP )dθ on T ∗ M splits into a vertical form − 12 d E˜ and a horizontal form p → π˜ p∗ γ˜p with γ˜p = Qp (XP ) ∈ Tx∗ M, where Qp (XP ) is the connection map from XP onto Tx∗ M: 1 ˜ ˜ p∗ γ˜p , (i(XP )dθ )p = − (d E) p+ π 2

p ∈ Tx∗ M,

(2.66)

˜ with E(p) = mE(v) = < p, v >, p = mG(v), π(p) ˜ = π(v) = x. This decomposition is unique. Lemmas 1 and 2 are equivalent, thanks to the relation (with m = 1) G∗ (i(XP )dθ ) = i( )dα.

(2.67)

Proof of (2.67) Recall that if f is a differentiable map from a manifold M into another one N, and if ω is a 2-form on N, then the 2-form f ∗ ω on M is defined by (f ∗ ω)p (X1 , X2 ) = ωf (p) (f∗ X1 , f∗ X2 ), ∀X1 , X2 ∈ Tp M. Thus < i(X1 )f ∗ ω, X2 >=< i(f∗ X1 )ω, f∗ X2 >,

whence f ∗ (i(f∗ X1 )ω) = i(X1 )f ∗ ω.

We apply this formula with G = f, θ = ω, = X1 : G∗ (i(XP )dθ ) = G∗ (i(G∗ )dθ ) = i( )G∗ dθ = i( )dG∗ θ = i( )dα. Therefore in Lemmas 1 and 2, we have πv∗ f˜v = G∗ (π˜ p∗ fp ) = (π˜ ◦ G)∗ fp = π ∗ fp , thus fp = f˜v ∈ Tx∗ M, with p = mGv and Qp G∗ = GQv .

¶

Proof of Lemma 1 First step. Let τv be the identification of the vertical space of Tv (T M) with the fibre Tx M. Let ∧ be the exterior product. We have ∀ , ˜ ve ∈ Tv (T M), with vertical ˜ ve , < (dα)v , ˜ ve ∧ > = g(τ v ( ˜ ve ), π∗ ( )).

(2.68)

13 Notice that this hypothesis is not satisfied by all examples in mechanics, for instance when there is an electromagnetic field.

2.4 Mechanics of Newton, Lagrange, and Hamilton

181

Indeed, in the coordinate system (x j , v j ) of T M, ˜ ve and are given by ˜ ve =



γ˜ i

 ∂ ∂ ∂ and = (v˜ i i + γ i i ). ∂v i ∂x ∂v

But by (2.63), we have dα =



gij dv j ∧ dx i + v j

∂gij dx k ∧ dx i , ∂x k

(2.69)

and therefore   i( ˜ ve )dα = γ˜ j gij dx i , and < i( ˜ ve )dα, > = γ˜ j gij v˜ i = g(π∗ ( ), τ v ˜ ve ). Hence (2.68), which is also with (2.62), (i( ˜ ve )dα)v = Gw ◦ π∗ = (t πv )∗ Gw,

w = (τ v ˜ ve )

∀ ˜ ve ∈ Vv .

(2.70)

Second step. The spray Sˆ of the Riemannian connection is the horizontal lift on ˆ ˆ T M of the vector field X = v at (x, v), that is, S(v) = Xv∗ . Thus π∗ (S(v)) = v, and ˆ S is a jet section on T M. This field is such that (see [Dieud3, ch. XX.9 Ex. 2]) < α, Sˆ >v = E(v),

1 ˆ i(S)dα = − dE, 2

LSˆ α =

1 dE, 2

LSˆ dα = 0, (2.71)

and ˜ < θ, S˜ >p = E(p),

1 ˜ iS˜ dθ = − d E, 2

LS˜ θ =

1 ˜ d E, 2

LS˜ dθ = 0.

(2.72)

The relation < α, Sˆ > = E is a straightforward consequence of (2.62). The relation (2.68) implies for all ˜ ve ∈ Vv that 1 1 ˆ < i(S)dα, ˜ ve > = −g(v, τv ( ˜ ve )) = − ( ˜ ve )(E) = − < dE, ˜ ve >, 2 2 ˆ which implies the relation i(S)dα = − 12 dE. The two last relations are a consequence of the Cartan formula: LSˆ α = iSˆ dα + diSˆ α = − 12 dE + dE = 12 dE, and thus LSˆ dα = dLSˆ α = 0. Third step. The fields , Sˆ are jet sections: (v(t)) = (x(t), v(t)), (v(t), v(t)) ˙ ˆ and S(v) = ((x, v), (v, − x (v, v)). ˆ By subtraction, ˜ ve (v) = (v) − S(v) is a vertical vector. Then taking ˜ ve (v) into (2.70), we obtain (see the appendix, (2.513)) ˆ i( (v))dα − i(S(v))dα = πv∗ G(Qv (v)),

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therefore with (2.71), 1 i( (v))dα = − dE + πv∗ G(Qv (v)). 2 Hence (2.65).

¶

The application of the previous lemmas to a time-dependent vector field v˜ (or to p), ˜ knowing that the covariant derivative of v˜ is defined by ∇v v˜ = Qv ( ), and ∇v p˜ = Qv (XP ), implies the following expressions: 1 (i( ) dα)v˜ = − dEv˜ + πv∗ G∇v v, ˜ 2 1 (i(XP ) dθ )p˜ = − d E˜ p˜ + π˜ p∗ ∇v p. ˜ 2

(2.73)

Comparing (2.73) with the fundamental equation (2.59), we see that we can identify the term π˜ p∗ ∇v p˜ (or mπv∗ G∇v v) ˜ with the force field F , whence F = π˜ p∗ ∇v p˜ = π˜ p∗ γ˜p ,

(or F = mπv∗ G∇v v˜ = mπ˜ v∗ γv ).

This is another way to write the fundamental equation of mechanics, but with F ∈ T ∗ (T ∗ M) (or F ∈ T (T ∗ M)), to the second-order level, hence of accelerations, which is nearer to the Newton law. The comparison of (2.73) with (2.34) shows that in the conservative case, we have to identify the force field F = −dU with the covariant derivative of the momentum.

2.4.8 Virtual Power Principle The fundamental equation of mechanics is especially adapted to a weak formulation, linked to the notion of virtual work and power of forces. This equation is also equivalent to the relation (2.34), still valid when the force fields are not the derivative of a potential, so that 1 i(XP )dθ = − d E˜ + F = −dE kin + F. 2

(2.74)

The formula (2.74) may be expressed in many different ways, notably with a weak formulation,14 giving the virtual power principle:

14 Especially

adapted to the mechanics of continuous media.

2.4 Mechanics of Newton, Lagrange, and Hamilton

< i(XP )dθ, X > = −

183

1 ˜ X > + < F, X >, < d E, 2

∀X ∈ Tp (T ∗ M). (2.75)

Since we have for X = XP , < i(XP )dθ, XP > = < dθ, XP ∧ XP > = 0, ˜ XP > + < F, XP > = 0, or even the power balance we obtain − 12 < d E, dE kin = < F, XP > . dt

(2.76)

Integrating this formula on a time interval J = [0, T ], where τ is the path of the evolution x(t) in M of the system, we obtain  E

kin

(T ) − E

kin

T

(0) =

 < F, XP (t) > dt =

0

F.

(2.77)

τ

These formulas express the following balances: The time derivative of the kinetic energy is equal to the power of the force field F . The work of forces along the path τ is equal to the variation of the kinetic energy between the initial and final times. We also notice that the field of forces F is an exact differential form if and only if there exists a Hamiltonian field XF . Indeed, if F = −dUp , then we have 1 1 i(XF )dθ = −d( E˜ + U ◦ π) ˜ = −dH, with H = E˜ + U ◦ π. ˜ 2 2 Remark 10 Lie derivative of the differential forms θ and α with respect to jet sections. Let be a jet section on T (T M) and let X be an mG-jet section on T (T ∗ M). Using the relations ˜ E(p) = (i(X)θ )p = < θ, X >p = (i( )α)v = < α, >v = E(v), and the Cartan formula LX θ = di(X)θ + i(X)dθ , we obtain again LX θ =

1 ˜ d E + F = d E˜ cin + F, 2

L α =

1 dE + F = dE kin + F, 2

(2.78)

where we identify the force field F with a differential form on T ∗ (T ∗ M) or on T ∗ (T M). Compare these Lie derivatives to those of (2.72), (2.71) for a geodesic field. Notice that these formulas are similar to the following formula (where v is a section of T M): 1 Lv (Gv) = G(∇v v) + dEv . 2

(2.79)

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2 Classical Mechanics

˜ then Using the Cartan formula, we can verify that if ω is a differential form on M, we have L πv∗ ω = πv∗ Lv ω, and the choice ω = Gv indicates the relation between (2.79) and (2.65). The formula (2.79) is also verified when we apply it to an element w ∈ T M: < Lv (Gv), w > = Lv < Gv, w > − < (Gv, Lv w >= Lv g(v, w) − g(v, [v, w]). We also have the relations Lv g(v, w) = g(∇v v, w) + g(v, ∇v w),

[v, w] = ∇v w − ∇w v,

since the torsion of a Riemannian connection is null, and hence 1 < Lv (Gv), w > = g(∇v v, w) + g(v, ∇w v) = < G(∇v v), w > + w(g(v, v)), 2 which implies the relation (2.79).

¶

2.4.9 Special Relativity and the Lorentz Metric A basic point in relativity is the role of observers. An observer can give a model of the evolution of a point in a manifold thanks to a frame. The associated frame depends on the proper velocity of evolution of the observer in relativity. The relation between the frames of two observers is given by a Lorentz transformation that will be seen below. Therefore, we have to represent this by the bundle of Lorentz frames, that is, a principal fibre bundle P over M˜ with the Lorentz group (we refer to [Kob-Nom, ch. I.5] for this notion). Then the tangent space to M˜ is obtained as the bundle associated with P with standard fibre F = R 4 . But to take the special role of the time in F into account, it is usual to write it as R 1+3 , where the first variable stands for the time, with the projection p0 from R 1+3 onto R so that p0 (ξ ) = ξ 0 . In the theory of special relativity, the configuration space is identified with a Euclidean space E4 and thus with R 1+3 . ˜ = E4 Lorentz Metric on M The space M˜ is equipped with a pseudo-Riemannian metric called a Lorentz metric, i.e., a symmetric tensor field g of type (0, 2) that is a bilinear form on the tangent ˜ of index 3. It corresponds to the following bilinear map g0 on the space T (M), standard fibre F = R 1+3 : g0 (ξ˜ , η) ˜ = −ξ 0 η0 + ξ.η,

ξ.η = ξ 1 .η1 + ξ 2 .η2 + ξ 3 .η3 ,

(2.80)

2.4 Mechanics of Newton, Lagrange, and Hamilton

185

∀ξ˜ = (ξ 0 , ξ ), η˜ = (η0 , η) ∈ R 1+3 , that is, on the canonical frame  = (0 , 1 , 2 , 3 ) of F = R 1+3 : g0 (0 , 0 ) = −1, g0 (k , k ) = 1, g0 (i , j ) = 0, ∀i = j = 0, . . . , 4.

(2.81)

With I3 the identity matrix of R 3 , this metric is then given by the matrix  0

G =

−1 0 0 I3

(2.82)

.

In a chart c of M˜ that defines the coordinates (x 0 , x), with x = (x 1 , x 2 , x 3 ), and x 0 = ct, the metric g is given on M˜ by g = −(dx 0 )2 +



(dx k )2 = −c2 dt 2 +

k=1,2,3



(dx k )2 ,

(2.83)

k=1,2,3

which is also written g = −(dx 0 )2 + (dx)2 . Thus for u˜ ∈ Tx˜ M˜ a velocity vector, this pseudo-Riemannian metric is g(u) ˜ = −(u0 )2 + |u|2 ,

|u|2 = u.u =



(uk )2 , k = 1, 2, 3.

(2.84)

Lorentz Frames, Lorentz Group The frame associated with an observer in relativity depends on its velocity, that is, the new point with respect to the usual mechanics. The change of frame (between different observers) is an essential notion, the change between two affine (respectively linear) frames being given by the Poincaré (respectively by the Lorentz) transformation, preserving the Lorentz metric. First, a Lorentz frame U or ˜ such that UL is defined as a map from F = R 1+3 onto Tx˜ (M) g(U ξ, U ξ ) = g0 (ξ, ξ ),

∀ξ ∈ R 1+3 .

(2.85)

Moreover, the Lorentz frames must preserve the orientation of time and space. Thus ˜ a Lorentz frame is also defined by its basis U = (e0 , e1 , e2 , e3 ) in Tx˜ (M): g˜ (e0 , e0 ) = −1, g˜ (ei , ei ) = 1, i = 1, 2, 3, g˜ (ei , ej ) = 0, i = j, i, j = 0, . . . , 3.

(2.86)

The Lorentz group L = O(1, 3) is the set of transformations of F = R 4 that preserve the metric g0 :

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L = O(1, 3) = {a ∈ Gl(4, R), g0 (a ξ˜ , a η) ˜ = g0 (ξ˜ , η, ˜ ∀ξ˜ , η)} ˜ = {a ∈ Gl(4, R), t aG0 a = G0 }.

(2.87)

This implies det a = ±1. A Lorentz transformation may be defined by its line vectors or by its column vectors, and both may be identified with Lorentz frames.15 The components of e0 = a(0 ) ∈ R 1+3 are such that −(a00 )2 + (a01 )2 + (a02 )2 + (a03 )2 = −1, which implies a00 ≥ 1 or a00 ≤ −1. Therefore, the Lorentz group has four connected ↑ ↑ components. Let L+ be the connected component of identity; thus a ∈ L+ satisfies ↑ det a = 1, a00 ≥ 1, whence its first column vector a0 ∈ F+ ; L+ is an open subgroup of L, called the proper Lorentz group. (With such a transformation we can define a proper Lorentz frame in F and conversely; such a choice fixes an orientation of the space.) Of course the Lorentz transformations operate on the Lorentz frames on the right, that is, if UL is a Lorentz frame, then UL a is also a Lorentz frame, and we have for all ξ ∈ R 4 , (UL a).ξ = UL (aξ ). Some Subspaces of the Tangent Space and the Standard Space F ˜ and F with special Now we introduce some subspaces of the tangent space Tx˜ (M) ˜ + , respectively F+ , be the cone in T ˜ (M), ˜ respectively in properties. Let Tx˜ (M) M 1+3 F = R , defined by ˜ + = {u˜ ∈ Tx˜ (M), ˜ g(u) ˜ |u0 | > |u|} Tx˜ (M) ˜ < 0} = {u˜ = (u0 , u) ∈ Tx˜ (M), F+ = {ξ˜ ∈ F = R 1+3 , g0 (ξ˜ ) < 0} = {ξ˜ = (ξ 0 , ξ ) ∈ F, |ξ 0 | > |ξ |}. (2.88) ˜ + (respecOf course, they are stable under the Lorentz group L. The space Tx˜ (M) ˜ + , and Tx˜ (M) ˜ − (respectively F+ , tively F+ ) has two connected components Tx˜ (M) and F− ) its opposite: ˜ + = (u0 , u) ∈ Tx˜ (M), ˜ u0 > |u|} Tx˜ (M) F+ = {ξ˜ = (ξ 0 , ξ ) ∈ F, ξ 0 > |ξ |},

(2.89)

15 That the column vectors constitute a Lorentz frame may be deduced from the fact that if a is a Lorentz transformation, its transpose t a is also a Lorentz transformation (with the variable x 0 = ct), as we can see below.

2.4 Mechanics of Newton, Lagrange, and Hamilton

187 ↑

which are convex cones, stable with respect to the proper Lorentz group L+ . We define the spaces ˜ ◦ = {u˜ ∈ Tx˜ (M), ˜ g(u) Tx˜ (M) ˜ = −1, u0 ≥ 1}, F◦ = {ξ˜ ∈ F = R 1+3 , g0 (ξ˜ ) = −1}, ξ 0 ≥ 1}; ↑

(2.90)

↑

F◦ is stable with respect to L+ . Moreover, the action of L+ on F◦ is transitive. Let ˜ ◦ . Then u0 = (|u|2 + 1) 12 ≥ 1. Let ξ˜ ∈ F◦ . Then ξ 0 = (|ξ |2 + 1) 21 ≥ 1. u˜ ∈ Tx˜ (M) Thus ˜ + ⊂ Tx˜ (M) ˜ + ∪ Tx˜ (M) ˜ − , Tx˜ (M) ˜ ◦ ⊂ Tx˜ (M) ˜ +, Tx˜ (M) F+ ⊂ F+ ∪ F− ,

F◦ ⊂ F+ .

˜ ◦ and F◦ may be viewed as projective spaces of Tx˜ (M) ˜ + and F+ . Note that Tx˜ (M) ◦ ◦ ˜ Moreover, the spaces Tx˜ (M) and F are diffeomorphic to the open unit ball B1 = {η ∈ R 3 , | η |< 1} by the maps ˜ ◦ → v = u ∈ B1 , u˜ = (u0 , u) ∈ Tx˜ (M) u0 ξ ξ˜ = (ξ 0 , ξ ) ∈ F◦ → η = 0 ∈ B1 . ξ We now consider subspaces of the tangent space to F. The 3-dimensional tangent space Tξ˜ F◦ is the set of vectors z˜ ∈ R 4 such that g(ξ˜ , z˜ ) = ξ 0 z0 − ξ.z = 0 [Kob-Nom, ch. XI, 10, Ex. 10.2, p. 269]. This implies, with the Cauchy–Schwarz inequality, ξ 0 z0 ≤ |ξ |.|z| < ξ 0 |z|, and thus if z ∈ Tξ˜ F◦ , then z0 ≤ |z|. Furthermore, z0 is given from z by z0 = That brings us to define a metric on the tangent space to F◦ by g˜ (˜z) = −(z0 )2 + |z|2,

∀˜z = (z0 , z) ∈ Tξ˜ F,

which is positive definite, and thus is a Riemannian metric on F◦ .16

16 We

could also use this metric on F+ .

1 ξ.z. ξ0

(2.91)

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Principal Fibre Bundle of Lorentz Frames ˜ over M˜ with the A main notion is that of a principal fibre bundle17 P = L(M) ˜ Lorentz group L, which is the set of Lorentz frames over M. When the manifold M˜ is identified with E4 , the bundle P is trivial, and it may be identified with the set of pairs (x, ˜ U ), with U the Lorentz frame of an observer observing a particle at the point x. ˜ Thus with a choice of a particular Lorentz frame, the bundle P may be identified with the product M˜ × L. ˜ F) on M˜ with values in F such that ω−1 = Ux˜ A differential form ω ∈ (M, x˜ (for a field of frames) gives the value in F of the observation of the velocity u˜ of a particle by ω(u) ˜ = (u0 , u1 , u2 , u3 ) =



uj j ∈ F.

The exchange between two Lorentz frames U 1 , U 2 of two observers is given by a Lorentz transformation a, U 2 = U 1 a, so that u˜ = U 2 (ξ ) = (U 1 a)ξ = U 1 (aξ ). Thanks to the differential forms ω1 , ω2 (inverses of the field of the Lorentz frames), a comparison between the observations of the velocity of the particle is given through their components ξ2 = ω2 (u) ˜ = a −1 (ω1 (u)) ˜ = a −1 ξ1 ∈ F.

(2.92)

With the proper Lorentz frames, we define a reduced principal fibre bundle ↑ ˜ ˜ L↑+ ), or simply by P , which is the bundle of the L+ (M), also denoted by P (M, proper Lorentz frames. ˜ ◦ over M, ˜ with fibre Ex˜ = Tx˜ (M) ˜ ◦ defined by (2.90), Then the bundle E = T (M) ↑ ˜ F◦ , L+ , P ) the associated fibre bundle with P over the is identified with E = E(M, ˜ with standard fibre F◦ , and structure group G = L↑+ .18 base M, ↑

Let U ∈ P , ξ ∈ F◦ . Then U.ξ ∈ E with U a.ξ = U.aξ for all a ∈ L+ . Since ↑ ◦ F may be identified with the quotient space L+ /SO(3) (see below), E is also the ↑ associated bundle with P with standard fibre L+ /SO(3), and may be identified with the quotient bundle P /SO(3). Then the 7-dimensional bundle E replaces the 8-dimensional tangent space ˜ as a consequence of T M˜ in relativity; E may be viewed as a subbundle of T M,

17 There are many notations for this bundle, for instance P = P (F, G) with the Lorentz group G = L, with standard fibre F = R 4 . ↑ 18 It is also denoted by P × ◦ ◦ ↑ F , the fibre product of P and F above L+ . L +

2.4 Mechanics of Newton, Lagrange, and Hamilton

189

[Bour.var, 6.5.5]. We recall that the fibre Ex˜ may be identified with this open unit ball B1 by the map h such that h(u) ˜ = v = uu0 . ˜ a family (E ∗ ), λ ≥ 0 of leaves of a foliation of In the cotangent bundle T ∗ M, λ ∗ ˜ T M is defined by ˜ p) ˜ = −(p0 )2 + | p |2 = −λ2 < 0, Eλ∗ = {p˜ = (p0 , p), g−1 (p,

−p0 = E ≥ λ};

Eλ∗ is equipped with the differential form J ∗ θ if θ is the usual canonical differential ˜ and J is the canonical embedding of E ∗ into T ∗ M. ˜ The map u˜ → form on T ∗ M, λ ˜ w˜ > = λg(u, ˜ w). ˜ p˜ = λGu˜ is an isomorphism between E and Eλ∗ , so that < p, ˜ ˜ but Notice that a priori, the Lorentz group does not operate in T (M) or in M, ˜ ˜ since T (M) is an associated bundle with L(M), we can define, thanks to an arbitrary ˜ the action of any a ∈ L using normal coordinates or Lorentz frame U of Lx (M), 19 −1 ˜ (see [Kob-Nom, ch. I.5, p. 55]), and by duality vectorial charts U aU in Tx (M) ∗ ˜ ˜ in Tx (M), so that we can deduce an action in M. Some Other Properties of the Lorentz Group Notice that every proper Lorentz frame U has one vector in F◦ and only one. (Indeed, if ξ˜ = (ξ 0 , ξ ) ∈ F◦ , then every vector ζ˜ = (ζ 0 , ζ ) orthogonal to ξ˜ is such that −ξ 0 ζ 0 + ξ.ζ = 0; hence ζ 0 = ξ10 ξ.ζ < |ζ | by the Cauchy–Schwarz inequality.) The restriction of a proper Lorentz transformation a to F◦ is an isometry of F◦ . ↑ The isotropy subgroup of e0 ∈ F◦ , i.e., {a ∈ L+ , ae0 = e0 } is such that  a = aR =

1 0 , R ∈ SO(3), 0R

(2.93) ↑

identifiable with the rotation group. The quotient space L+ /SO(3) is identifiable with the set of special Lorentz transformations  aτ,α =

cosh τ t α sinh τ α sinh τ Bτ,α

,

(2.94)

with Bτ,α = (cosh τ − 1) α ⊗ t α + I3 , and τ ∈ R + , α ∈ S 2 . The matrix Bτ,α on R 3 is also given by Bτ,α = (cosh τ )Pα + Pα⊥ ,

19 See,

for instance, [Bour.var, 7.10.1].

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where Pα (respectively Pα⊥ ) is the orthogonal projection onto Rα (respectively onto its orthogonal space), and thus Pα + Pα⊥ = I3 . With α = e1 , then aτ,e1 is simply given by  aτ,e1 =

aτ 0 0 I2



 , aτ =

cosh τ sinh τ sinh τ cosh τ

(2.95)

.

↑

Since the quotient space L+ /SO(3) is diffeomorphic to F◦ , this implies that the set of special Lorentz transformations is diffeomorphic to F◦ . Thus it may be viewed as a Riemannian manifold, but this result may be obtained directly from the properties ↑ of the Lie algebra [Kob-Nom, Vol. 2, ch. XI.10, p. 270], since L+ /SO(3) is a Riemannian symmetric space. Note that the special Lorentz transformations are given by symmetric matrices (for the metric g0 ), whereas the rotations are such that t a = a −1 , whence the transpose of a Lorentz transformation is also a Lorentz transformation. Now we can give an interpretation of the change given by the proper Lorentz element a, see (2.92), between the observations by two observers of the velocity of a particle with two different frames, in the simple case that a is given by a = aτ,e1 ; see (2.95). Then we have 

ξ10 ξ11



 =

cosh τ sinh τ sinh τ cosh τ ξi , ξi0

with ξ12 = ξ22 , ξ13 = ξ23 . Let vi =



ξ20 ξ21

,

i = 1, 2. Then the relations between the

components of the velocities are given by v11 = v1i =

sinh τ + v21 cosh τ cosh τ + v21 sinh τ v2i

=

v21 + tanh τ v21 tanh τ + 1

, 1

cosh τ + v21 sinh τ

= (1 − tanh2 τ ) 2

v2i 1 + v21 tanh τ

,

with i = 2, 3. Thus we obtain the usual formula [Lan-Lif4, ch. I.5], with the speed between the frames of the two observers given by V = tanh τ .

Lie Algebra of the Lorentz Group ↑

Structure of a symmetric Lie algebra. The Lie algebra of L+ is given by the set o(1, 3) of 4 × 4 matrices A such that [Kob-Nom, Vol. 2, ch. XI, 10, Ex. 10.2] t

AG + GA = 0,

thus

t

A = −GAG−1 = −GAG.

(2.96)

2.4 Mechanics of Newton, Lagrange, and Hamilton

191

↑

An involutive automorphism σ of L+ is defined by σ (a) = Ga, G−1 = GaG. ↑

Thus σ 2 (a) = a, ∀a ∈ L+ , with σ (a) = a when a ∈ SO(3). This gives a ↑ structure of the symmetric space (L+ , SO(3), σ ), and a structure of the symmetric Lie algebra (o(1, 3), o(3), σ ), which implies the canonical decomposition o(1, 3) = o(3) + m∗

(2.97)

of o(1, 3) associated with the decomposition in the eigenspaces of σ : and σ (A) = −t A = −A, ∀A ∈ m∗ ,

σ (A) = −t A = A, ∀A ∈ o(3),

where o(3) is identified with a subalgebra of o(1, 3) (of skew-symmetric 3 × 3 matrices Aμ ); thus  Aμ =

0 0 0 Aμ

, Aμ ∈ o(3), μ ∈ R 3 ,

(2.98)

and m∗ is the subspace of symmetric matrices  Sv =

0 tv v 0

v ∈ R3 .

,

(2.99)

We can obtain this expression by taking the derivative of the special Lorentz transformations aτ,α (see (2.94)) with respect to τ at τ = 0; then v = α ∈ S 2 . Conversely, we obtain aτ,α from the geodesic (see below and [Kob-Nom]), so that with v = τ α, aτ,α = exp(τ Sα ) = exp Sv = I + sinh τ Sα + (cosh τ − 1) Sα2 , whereas the rotation with angle θ and axis μ ∈ S 2 is given by Rθ,μ = exp(θ Aμ ) = I + sin θ Aμ + (1 − cos θ ) A2μ . The matrices Sα and Aμ , with α, μ ∈ S 2 , satisfy  Sα2 =

0 0 0 Pα



 ,

A2μ =

0 0 0 −Pμ⊥

,

Sα3 = Sα ,

A3μ = −Aμ ,

with the projections Pα = α ⊗ t α respectively Pμ = μ ⊗ t μ onto the direction α, respectively μ, and Pμ⊥ = I − Pμ .

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Decomposition for the Lorentz group. We have the following property: for every ↑ g ∈ L+ there exist a special Lorentz transformation gS and a rotation R, unique up to a parametrization, such that g = gS R. We can write the Lorentz transformations gS and R as exponentials by gS = exp ξ = exp(τ Sα ), (τ, α) ∈ R × S 2 ,

R = exp(θ Aμ ), (θ, μ) ∈ 2πT × S 2 ,

so that g = exp(τ Sα ) exp(θ Aμ ).

(2.100)

↑

Proof We identify the quotient L+ /SO(3) with the space F◦ , so that we know that there exists a special Lorentz transformation gS = exp(τ Sα ) such that g.e0 = gS e0 ↑ ↑ (gS is identified with a lift of the projection π from L+ onto L+ /SO(3)). This implies the existence of a rotation R, which gives (2.100). Now if we have another decomposition g = gS R = gS R , by application to e0 we obtain gS e0 = gS e0 , and thus gS = gS , and then R = R . ¶ A matrix representation of a general element of o(1, 3) is given by  (SA)v,μ =

0 tv v Aμ

,

(2.101)

which we also write simply by (v, μ). With the decomposition (2.97), we have the following properties (see [Kob-Nom, Vol. 2, ch. XI.2, p. 226]): [o(3), o(3)] ⊂ o(3),

[o(3), m∗ ] ⊂ m∗ ,

[m∗ , m∗ ] ⊂ o(3).

(2.102)

The bracket operation of the Lie algebra o(1, 3) is given by [(SA)v1 ,μ1 , (SA)v2 ,μ2 ] = (SA)v,μ , with v = μ1 × v2 − μ2 × v1 ,

or [(v1 , μ1 ), (v2 , μ2 )] = (v, μ)

μ = v1 × v2 + μ1 × μ2

(2.103)

(we find it easily, looking for the symmetry or skew-symmetry of each term, in the decomposition of symmetric or skew-symmetric matrices), so that the adjoint representation of the Lie algebra is given by the matrix representation      v2 Aμ1 −Av1 v2 v v1 . = ad μ1 μ2 Av1 Aμ1 μ2 μ

(2.104)

This gives the Killing–Cartan form by   tr ad(SA)v1 ,μ1 ad(SA)v2 ,μ2 = 4[v1 .v2 − μ1 μ2 ].

(2.105)

2.4 Mechanics of Newton, Lagrange, and Hamilton

193

The adjoint representation on m∗ of the rotation group is given (with aR , see (2.93), and Av , see (2.99)) by −1 Ad aR (Sv ) = aR Sv aR = SRv ,

(2.106)

and the adjoint representation on m∗ of the Lie algebra o(3) is given (with Aμ given by (2.98)) by ad Aμ (Sv ) = [Aμ , Sv ] = SAμ v .

(2.107)

These formulas give the relation between the charts of two observers having the same velocity. From (2.103), we obtain the structure constants with the canonical frame (i ) of R 3 , [(0, i ), (0, j )] = (0, i × j ) = i,j,k (0, k ), [(i , 0), (j , 0)] = (0, i × j ) = i,j,k (0, k ),

(2.108)

[(i , 0), (0, j )] = i,j,k (k , 0), with i,j,k = +1 (respectively −1) if the permutation of i, j, k is even, respectively odd, and 0 if two subscripts are repeated. Complex structure. The Lie algebra o(1, 3) is also equipped with a complex structure20 thanks to the map ∗ defined21 by ∗SAv,μ = SA−μ,v ,

or ∗ (v, μ) = (−μ, v),

which satisfies ∗ ∗ (v, μ) = −(v, μ). Then the complex structure is obvious if we define (v, μ) = (v, 0) + (0, μ) = ∗(0, −v) + (0, μ) = −i(0, v) + (0, μ) = (0, μ − iv). (2.109) The Lie algebra o(1, 3) is isomorphic to the dual of o(3 + 1) = o(3) + m, the Lie algebra of the Lie group S0(3 + 1), that is, to o(3) + im (see [Kob-Nom, Vol. 2, ch. XI.10, p. 270, T2], then to the set of 4 × 4 skew-symmetric matrices given by  Av,μ =

0 −t v v Aμ

.

20 This 21 We

(2.110)

may be viewed as a consequence of the Hodge transformation of the differential forms. can also give a different definition by ∗SAv,μ = SAμ,−v ,

or ∗ (v, μ) = (μ, −v).

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We can identify o(1, 3) with itself by the duality < A, B > = tr (AB) where A, B ∈ o(1, 3), ˜ v,μ is but we can also identify its dual with the skew-symmetric matrices, since GA a matrix of o(1, 3). We easily verify that (SA)v,μ .A−μ,−v = 4v.μI,

(2.111)

so that we write ˜ −μ,−v ) = 4 v.μ ((SA)v,μ , A−μ,−v ) = tr ((SA)v,μ GA

(2.112)

as a duality formula. Notice that using the complex structure, we also have (SA)v,−μ (SA)μ,v = 4v.μI. We obtain two invariants for the Lorentz group on applying the trace to the two products: tr ((SA)v,μ (SA)μ,−v ) = 4v.μ, tr ((SA)v,μ (SA)v,μ ) = 2(| v |2 − | μ |2 ).

(2.113)

↑

A Riemannian Structure on F◦ and on L+ /SO(3) On the tangent space of the standard manifold F◦ , or on the homogeneous ↑ space L+ /SO(3), from the properties of the symmetric spaces we can define a Riemannian metric. We first define the subbundle of the tangent space T (F◦ ): Q = {˜z = (z0 , z) ∈ Tu˜ F◦ ,

z0 =

1 u.z, u0

u˜ ∈ F◦ }.

From the Riemannian metric on F◦ given by (2.91), we define the map F˜ , also called a metric, by F˜ (˜z) = (|z|2 − (z0 )2 ) 2 , 1

z˜ = (z0 , z) ∈ Q.

This metric is positively homogeneous, i.e., F˜ (a z˜ ) = a F˜ (˜z), ∀a > 0.

(2.114)

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195

Then we define the Hilbert form w˜ on Q by22 w˜ z˜ =

1 F˜ (˜z)

(



zk duk − z0 du0 ).

(2.115)

↑

↑

Geodesics in L+ /SO(3) and in F◦ . Let π be the canonical projection from L+ ↑ ↑ onto the quotient space L+ /SO(3). The geodesics of L+ /SO(3) starting from the 23 ∗ origin are given by π(exp (τ ξ )) with ξ ∈ m identified with R 3 . If we identify ↑ ↑ L+ /SO(3) with F◦ , by the diffeomorphism f such that f (π(a)) = a.e0, a ∈ L+ , we obtain the geodesic γα , γα (τ ) = π(exp(τ α)) in F◦ , starting from e0 in the direction α ∈ S 2 , by γα (τ ) = (u0 (τ ), u(τ )) = (cosh τ, α sinh τ ) = cosh τ e0 + sinh τ



with τ ∈ R + . Then the corresponding unit geodesic field is Au˜ =

 ∂ ∂ dγα = sinh τ 0 + cosh τ αk k = (sinh τ, α cosh τ ). dτ ∂u ∂u

α k ek , (2.116)

(2.117)

Notice that γα (τ ), as a vector, is identical to the first column vector of the matrix of the special Lorentz transformation aτ,α given by (2.94). This transformation applied to the vector γα (τ  ) gives aτ,α (γα (τ  )) = γα (τ  + τ ).

(2.118)

Furthermore, we see that this matrix aτ,α is determined by its first column vector, so that there is a diffeomorphism between the set of geodesic fields and the set of special Lorentz transformations. The section of Q ⊂ T (F◦ ) → F◦ defined by ϕ(u) ˜ = Au˜ allows us to define the Hilbert form λ on F◦ by λu˜ = (ϕ ∗ (w)) ˜ u˜ = cosh τ α.du − sinh τ du0 ,

(2.119)

 ˜ thanks to (2.117). with α.du = αk duk . This Hilbert form λ is identified with GA, ∗ ∗ ∗ Notice that we also have π λ = π ϕ (w) ˜ = w. The geodesic field A is such that < λ, A > = F (A) = 1, and (see [Mall, ch. IV.3.2]) i(A)dλ = 0. Let e0 = {u˜ = (u0 , u) = (cosh τ, α sinh τ ) ∈ F◦ , (τ, α) ∈ R∗+ × S 2 }.

(2.120)

the following definition, we use the simple notation duk , du0 , for π ∗ duk , π ∗ du0 with π the canonical projection from Q onto F◦ . 23 According to [Kob-Nom, Vol. 2, ch. XI, 10, Ex. 10.2]. 22 In

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Then we see that e0 = F◦∗ = F◦ \{1, 0} is identifiable with R∗3 in polar coordinates, and that each geodesic has only one common point with S 2 , and at each point of S 2 , there is one geodesic. Furthermore, two distinct geodesics have no common point. With a parametrization of the sphere given by ξ : (θ, ϕ) ∈]0, π[×[0, 2π] → α = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) ∈ S 2 , the corresponding tangent vectors to the sphere are identified with ∂α = (cos θ cos ϕ, cos θ sin ϕ, − sin θ ), ∂θ ∂α = (− sin θ sin ϕ, cos θ sin ϕ, 0). ∂ϕ

(2.121)

∂α ϕ Thus the vector fields on F◦ , given by V θ = (0, ∂α ∂θ ), V = (0, ∂ϕ ), satisfy

g˜ (V θ , V θ ) = 1, g˜ (V ϕ , V ϕ ) = sin2 ϕ, g˜ (V θ , V ϕ ) = 0 and g˜ (Au˜ , V θ ) = g˜ (Au˜ , V ϕ ) = 0, with the geodesic field Au˜ given by (2.117). Then in the domain e0 (see (2.120)), the differential form λ is exact, i.e., there is a function S, identified with the action,24 that is also a wave function (see below) on e0 ⊂ F◦ such that λ = dS = dτ.

(2.122)

Proof We have from (2.119) that λu˜ = d(cosh τ α.u − sinh τ u0 ) −



uk d(cosh τ αk ) + u0 d sinh τ = dJ0 + J1 ,

with u˜ = (cosh τ, α sinh τ ). We verify that J0 = 0, and since α.α = 1, α.dα = 0, we have J1 = − sinh τ d cosh τ + cosh τ d sinh τ = dτ.

¶

↑

We have thus a foliation of F◦∗ (and thus of L+ /SO(3)), with the leaves S −1 (τ ) = {u0 } × |u|.S 2 ,

τ > 0,

1

u0 = cosh τ = (1 + |u|2 ) 2 ,

identifiable with S 2 , transverse to the geodesics. We also have a foliation of F+ , with the leaves R + .S −1 (τ ), each leaf being a Riemannian manifold.

24 From (2.119), with  ∂S 2 ∂S 2 ( ∂uk ) = ( ∂u 0) −

independent variables u0 and u, S satisfies the Hamilton–Jacobi equation −1.

2.4 Mechanics of Newton, Lagrange, and Hamilton

197

Since every geodesic of F◦ can be extended to arbitrarily large values of the parameter τ , F◦ is a complete Riemannian manifold, and thus any two points of F◦ can be joined by a minimizing geodesic [Kob-Nom, ch. IV, 4]. Then there is a distance function d(u˜ 0 , u˜ 1 ) on F◦ , given by the infimum of the length of all curves (of class C 1 ) joining u˜ 0 , u˜ 1 . If these two points are on the geodesic γα with parameters τ0 and τ1 (so that they are the endpoints of the curve γα,u˜ 0 ,u˜ 1 ), then the distance between these points is given by  d(u˜ 0 , u˜ 1 ) =

τ1

˙˜ dτ = F (u)



τ0

τ1 τ0

 F (Au˜ )dτ =

τ1

dτ = τ1 − τ0 ,

(2.123)

τ0

using the geodesic field Au˜ given by (2.117), which satisfies F (Au˜ ) = 1. Since exp τ = cosh τ + sinh τ = u0 + |u|, we have τ = log (u0 + |u|), that is (with c = 1), with

u u0

with (u0 )2 − |u|2 = 1,

= vc , and β = (1 −

τ = log (β(1 − which is the inverse of the relation without unit.

|v| c

v 2 − 12 ) c2

(2.124)

(thus u0 = β), by

1+ 1 |v| )) = log c 2 1−

|v| c |v| c

,

(2.125)

= tanh τ . Notice that this distance in F◦ is

Remark 11 On the space F+ . Let r ∈ R + with r 2 = (u0 )2 − |u|2 , u˜ ∈ F+ . Then we can choose the coordinates r, τ, θ, ϕ such that u0 = r cosh τ, u = rα sinh τ . We can verify that the pseudo-Riemannian metric g˜ = −(du0 )2 + (du)2 on F+ is given in these coordinates by g˜ = −dr 2 + r 2 dτ 2 + r 2 sinh2 τ (dα)2 ,

with (dα)2 = dθ 2 + sin2 θ dϕ 2 .

Thus we verify that the submanifolds of F+ with constant r have a Riemannian structure. ¶ Remark 12 We can define a metric on F+ by F (ξ˜ ) = ((ξ 0 )2 − |ξ |2 ) 2 . 1

(2.126)

This map F has the property of positive homogeneity, and Fˆ = −F is both positive homogeneous and strictly convex, that is, for every linearly independent pair ξ, ξ  ∈ F◦ , we have Fˆ (ξ˜ + ξ˜  ) < Fˆ (ξ˜ ) + Fˆ (ξ˜  ).

(2.127)

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2 Classical Mechanics

Proof We prove that F (ξ˜ + ξ˜  ) > F (ξ˜ ) + F (ξ˜  ).

(2.128)

Let ξ˜ = (ξ 0 , ξ ), ξ 0 = r cosh θ . Then |ξ | = r sinh θ, r > 0, with similar relations for ξ˜  . Then F (ξ˜ ) = r, F (ξ˜  ) = r  . Taking the square of F (ξ˜ + ξ˜  ), we obtain with ξ.ξ  = rr  α sinh θ sinh θ  and −1 ≤ α ≤ 1, (F (ξ˜ + ξ˜  ))2 =r 2 + (r  )2 + 2rr  (cosh θ cosh θ  − α sinh θ sinh θ  ≥ r 2 + (r  )2 + 2rr  cosh(θ − θ  ).

(2.129)

Since cosh(θ − θ  ) > 1 with the hypothesis of independence, we have proved the inequality (2.128). This property will imply that for any “extremal curve” that may be defined, its length will not be a minimum, but a maximum.25 ¶ ˜ Consider a particle of mass m evolving Relativistic evolution of a free particle in M. in free space with a model of special relativity. Here we will use the phase space Eλ∗ ⊂ T ∗ M˜ with λ2 = m2 c2 for c = 1 (or λ2 = m2 with c = 1): (Eλ∗ )x˜ = {p˜ = (p0 , p) = λGu, ˜ u˜ ∈ Ex˜ } = {p˜ = (p0 , p),

p02 − p2 = λ2 ,

p0 = −

E < 0}, c

(2.130)

which is also stable under Lorentz transformation. This corresponds to the energy E of the particle: 1

E = H (t, x, p) = c(p2 + λ2 ) 2 ,

(2.131)

where H is the free Hamiltonian (without exterior forces). Let θ be the canonical form on T ∗ M˜ expressed (with x 0 = ct) by θ = p0 dx 0 + p.dx = −E dt + p.dx.

(2.132)

Since the dimension of Eλ∗ is odd, the evolution of the particle will be obtained by   the characteristic equations. Let A = a j ∂x∂ j + bj ∂p∂ j be a characteristic vector on Eλ∗ , that is, i(A)d(θ |Eλ∗ ) = 0.

25 See

the proof of [Mall, ch.!IV, 4.3, Thm. 4.3.3]; see also [Lan-Lif4, ch. I, 3].

(2.133)

2.4 Mechanics of Newton, Lagrange, and Hamilton

199

Then we have i(A)dθ |Eλ∗ ) = −a 0 dp0 − =−



a k dpk +



bk dx k

 a0  ( pk + a k )dpk + bk dx k = 0, p0

(2.134)

0

since dp0 = p10 p.dp. Thus pa 0 pk + a k = 0, bk = 0, and the equation of the characteristic is given, with k = 1, 2, 3, by dx k pk ak = = − , a0 dx 0 p0

dpk = 0, dt

dp0 = 0. dt

(2.135)

Let p0 = −mc cosh τ, p = mc α sinh τ, with α ∈ S 2 , τ ≥ 0, both constant. Then dx u v = 0 = = α tanh τ, dx 0 u c which gives the velocity vector u˜ = (u0 , u) = (cosh τ, α sinh τ ) ∈ Ex˜ .

(2.136)

˜ so that with the Then the evolution x(s) ˜ of the particle in M˜ is obtained by ddsx˜ = u, initial condition x(0) ˜ = 0, the characteristic curve 0 is given by x(s) ˜ = s u˜ = s(cosh τ, α sinh τ ).

(2.137)

This implies x 0 (s) = s cosh τ = sβ. With the variable x 0 = ct, the characteristic curve is given by x(x ˜ 0 ) = x 0 (1, α tanh τ ) =

x0 x0 u˜ = (cosh τ, α sinh τ ), cosh τ cosh τ

(2.138)

with the corresponding velocity vector, Ax(x ˜ 0) =

1 d x˜ 1 u˜ = (cosh τ, α sinh τ ). = 0 dx cosh τ cosh τ

(2.139)

We see that x(s) is, respective to the initial condition, in the future cone ˜ x 0 ≥ |x|}, + = {x˜ = (x 0 , x) ∈ M, whence |x 0 (s)|2 −|x(s)|2 ≥ 0. Now we know that the restriction to the characteristic curves of the canonical form θ is exact. We can find this result in another way, using 0 0 ˜ the Hilbert form w = F (1u) ˜ (u dx − u.dx) on T M, which defines (through its ˜ ˜ also denoted by w, pullback by the section u˜ of T M) the differential form on M, ∗ ∗ 0 w ∈ Eλ ⊂ T M, so that with x = 0,

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2 Classical Mechanics

w = cosh τ dx 0 − sinh τ α.dx =

cosh τ 0 0 (x dx − x.dx). x0

(2.140)

With respect to the unit characteristic vector u, ˜ notice that we have < dS, u˜ > = < w, u˜ > = 1. From (2.138) we also have 1 cosh τ = (|x 0 (s)|2 − |x(s)|2 )− 2 , x0

(2.141)

and thus if we define the function S on + by 1

S(x) ˜ = (|x 0 (s)|2 − |x(s)|2) 2 ,

(2.142)

then (2.140) gives w=

1 1 0 (|x (s)|2 − |x(s)|2 )− 2 d[(x 0)2 − x 2 ] = dS. 2

With β = cosh τ = (1 − tanh2 τ )− 2 = (1 − ,

v 2 − 12 ) , c2

1

w | =

(2.143)

we also have, along the curve

1 c dx 0 = dt. cosh τ β

(2.144)

The length of the curve from x˜ 0 = x(t ˜ 0 ) to x˜ 1 = x(t ˜ 1 ) is given by 

 dS =

I ( ) =

t1 t0

 F (Ax(t ˜ ) ) dt =

t1 t0

c c dt = (t1 − t0 ). cosh τ β

(2.145)

Proposition 1 The curve = x˜ 0 ,x˜ 1 is an extremal of the integral I (τ ) in the sense that if τ is a (smooth) curve with endpoints x˜ 0 , x˜ 1 , then its length I (τ ) is less than the length I ( ) of , which is a maximum. Proof Let x˜ 0 = 0, and let x(s) ˜ be a parametric representation of τ , and A = ddsx˜ , with F (A) = 1. s The length of τ is I (τ ) = − 0 1 F (A)ds = s1 . Let u˜ = (u0 , u) ∈ I be the velocity vector of . Then we have  I ( ) = − 0

s1



s1

g(A, u) ˜ ds = 0

β β (c2 u0 − v.u) ds. c c

2.4 Mechanics of Newton, Lagrange, and Hamilton

201

0 Let β = ch θ0 , β |v| c = sh θ0 , cu = chθ, |u| = sh θ , and ξ =



s1

I ( ) =

v u |v| . |u| .

Then

(ch θ0 ch θ − ξ sh θ0 sh θ ) ds.

0

Now ξ ≤ 1, and thus 

s1

I ( ) ≥



s1

(ch θ0 ch θ − sh θ0 sh θ ) ds =

0



0

whence I ( ) ≥ I (τ ).

s1

ch (θ0 − θ ) ds ≥

ds = I (τ ),

0

¶

Remark 13 Let us replace the differential form θ of (2.132) by θ − d sˆ, with a function sˆ (the action), for the characteristic equation. Then with equation (2.133), we have < A, θ − d sˆ > = 0, and thus we have on , d sˆ = w = mcds. This gives the relation between the metric and the action in this case. ¶ Evolution of a wave in a dispersive medium, in optics. We again consider the canonical form θ on T ∗ M˜ expressed by (2.132), but with the idea of applying this to optics. Then we divide this form by h¯ , the Planck constant, so that we obtain the right units, and we define k = ph¯ ∈ R 3 , with | k | the wave number, ω = Eh¯ the frequency. Thus ω θ = k.dx − dx 0 = k.dx − ωdt. h¯ c

(2.146)

Here we prefer to use the following form with the action s: θˆ = k.dx −

ω 0 dx − ds = k.dx − ωdt − ds. c

(2.147)

In a medium with permittivity  and permeability μ (that depend a priori on the √ frequency ω), thus with index n = n(ω) = c μ, we have the so-called dispersion relation between the wave number and the frequency: n | k |= ω , c

thus λ =

ω2 ω2 − | k |2 = 2 (1 − n2 ) < 0. 2 c c

(2.148)

We recall that in the free space, we have n = 1 with μc2 = 1, and in a dispersive medium, n ≥ 1. More generally we will write the dispersion relation (that is, a state equation) by | k |= ϕ(ω),

or ω = ψ(| k |)

(2.149)

(if we assume that the function ϕ is invertible), and we denote by  the manifold in T ∗ M˜ on which this relation is satisfied. Let Aˆ be a characteristic vector for θˆ | :

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2 Classical Mechanics

Aˆ =



Aj

 ∂ ∂ ∂ ∂ ∂ +ν , + + Bj +γ ∂x j ∂x 0 ∂kj ∂ω ∂s

so that the equations of the characteristic curves are dx = A, dx 0

dk = B, dx 0

dω = γ, dx 0

ds = ν. dx 0

By definition, Aˆ must satisfy ˆ θˆ | ) = 0. i(A)d(

ˆ θˆ | > = 0, < A,

(2.150)

The second equation with the previous coordinates is ˆ θˆ | = − i(A)d



 1 1 Bj dx j − γ dx 0 = 0. Aj dkj + dω + c c

(2.151)

Thus B = (Bj ) = 0, γ = 0. Then we have dk = d(| k | α) with α ∈ S 2 , and with the usual spherical variables θ, φ, we have dα = eθ dθ + eφ dφ,

with eθ =

∂α ∂α , eφ = , ∂θ ∂φ

both orthogonal to α, so that the tangential component of A along the sphere k is null, and thus A = λ |k| (with a real number λ). Then (2.151) is reduced to

−λd(| k |) + 1c dω = 0, so that

λ=

1 ∂ϕ −1 1 ∂ψ ( ) = . ; c ∂ω c ∂|k|

λc = vg is usually called the group speed, whereas characteristic curves are straight lines, so that x(t) = x0 + cλ

ω |k|

is the phase speed. The

k k t = x0 + vg t. |k| |k|

(2.152)

From the first equation of (2.150), we have ˆ θˆ | > = A.k − ω − ds = 0, < A, c dx 0 and thus ds = cλ | k | −ω = vg | k | −ω = vg ϕ(ω) − ω. dt

(2.153)

2.4 Mechanics of Newton, Lagrange, and Hamilton

203

∂s From θˆ = 0 on , we have ∂x j = kj , thus | ∇s |=| k |= ϕ(ω), and the elimination of ω between these two last equations gives a Hamilton–Jacobi equation. Notice that if we replace the energy in (2.132) by Eh¯ = nc ω instead of Eh¯ = ω, we obtain similar results, but with λ = 1, then ds dt = 0, and we obtain the eikonal equation

| ∇s |2 −

n2 2 ω = 0. c2

2.4.10 Evolution with an Electromagnetic Field Here we consider (under special relativity assumptions) the action of the electro˜ and we see how it changes the usual symplectic magnetic field on the spacetime M, structure. We first use units such that the speed of light is c = 1. The canonical differential form θ on T ∗ M˜  is given (using dual variables k , and the usual exterior ˜ x˜ = p0 dx 0 + pk dx  (p0 , pk , k = 1, 2, 3)) by θ = p.d 0 differential form ω = dθ is ω = d p˜ ∧ d x˜ = dp0 ∧ dx + dpk ∧ dx k .

The Maxwell Equations The electromagnetic field may be defined from a field A˜ on M˜ (called the ˜ denoted by θ ˜ in a electromagnetic potential), which is a differential form on M, A i coordinate system (x ), i = 0, . . . , 3: θA˜ = A0 dx 0 +



Ak dx k .

(2.154)

The electromagnetic field F is given by the exterior differential form ωF = dθA˜ , ωF = θE ∧ dx 0 + ωB ,

with θE =



Ek dx k , (2.155)

ωB = B1 dx 2 ∧ dx 3 + B2 dx 3 ∧ dx 1 + B3 dx 1 ∧ dx 2 , the electric field E being given by the differential form θE and the magnetic induction B by the 2-form ωB .26 Now the Maxwell equations are given by dωF = 0.

(2.156)

course, if the electromagnetic potential is changed into θA˜ + d, then the electromagnetic field ωF does not change. This is a gauge transformation.

26 Of

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2 Classical Mechanics

In the coordinate system (x i ), the electromagnetic field is also given by its components as a 4 × 4 skew-symmetric tensor (Fμ,ν ), that is, F = AE,B with the notation of (2.110), since it would be an element of o(1 + 3) or the dual of the Lie algebra o(1, 3). By changing the basis dx k ∧ dx 0 , dx i ∧ dx j , i = j, of the two-forms, for the canonical basis27 of the Lie algebra o(1, 3), we replace the tensor F by the element (SA)E,B of o(1, 3), see (2.101), so that the element (SA)v,μ is given by (SA)v,μ =



v i ei0 +



μk e k

with ei0 = i0 + 0i , e1 = 23 − 32 , . . .

j

with i the canonical basis of R 4 . Thus (SA)E,B is given, with c = 1, by the matrix  (SA)E,B =

0 tE E AB

.

(2.157)

˜ for a tensor of Here we change the 2-form ωF , which is a tensor in the space T2 (M), ˜ (respectively type (1, 1), denoted by Fˆ (respectively F˜ ), such that Fˆ ∈ L(Tx˜ (M)) ˜ for every x˜ ∈ M, ˜ which is obtained from F via the metric G by F˜ ∈ L(Tx˜∗ (M)) ˜ these tensors Fˆ = G−1 F = F G−1 . With a Lorentz frame U ∈ L(R 1+3 , Tx˜ (M)), 0 1+3 0 ˆ ˜ are obtained from a tensor F ∈ L(R ), respectively F ∈ L((R 1+3 )∗ ), so that U Fˆ 0 U −1 = Fˆ ,

U ∗ F˜ 0 (U ∗ )−1 = F˜ ,

thus t Fˆ = −F˜ ,

(2.158)

˜ the contragredient frame of U , that is, U ∗ = t U −1 , with U ∗ ∈ L((R 1+3 )∗ , Tx˜∗ (M)) ∗ and thus U satisfies < U ξ, U ∗ η > = < ξ, η >,

thus U ∗ = GU (G0 )−1 .

The space of the tensors Fˆ (respectively F˜ ) is an associated bundle with the ↑ ˜ ˜ o(1, 3), L↑+ , P ) which is the fibre bundle E = E(M, principal bundle P = L+ (M), ˜ with standard fibre the Lie algebra o(1, 3), and with the structure over the base M, ↑ group G = L+ . We can identify Fˆ 0 or F˜ 0 with (SA)E,B , so that the action of a Lorentz frame U on (SA)E,B ∈ o(1, 3) is given by (2.158). This will be used to define a connection form. Remark 14 Invariant of the Lorentz group. We will use a natural duality between the p-forms given by the Hodge transformation ∗, which is an isomorphism from the r-forms onto the (4−r)-forms by contraction with the 4-form μc = dx 0 ∧dx 1 ∧ dx 2 ∧ dx 3 , so that

27 We

can also change the basis of the 2-forms for the basis of the tangent space to the Lorentz frames, that is, ( ∂ k , ∂ i ), with the components of a Lorentz frame (a0 , a1 , a2 , a3 ). ∂a0

∂aj

2.4 Mechanics of Newton, Lagrange, and Hamilton

∗(dx 1 ∧ dx 0 ) = dx 2 ∧ dx 3 ,

205

∗(dx 2 ∧ dx 3 ) = −dx 1 ∧ dx 0 ,

and dx 1 ∧ dx 0 ∧ ∗(dx 1 ∧ dx 0 ) = −μc . Thus a natural duality for ωF , thus for F , is given by the 2-form ∗ωF = ω∗F , so that ∗(ωF ) ∧ ωF = (B 2 − E 2 ) μc and thus ∗F = A−B,E . We also have ωF ∧ ωF = −2(B.E) μc . Since the Hodge transformation is invariant under the Lorentz group, we have obtained in this way the two invariants of the Lorentz group, as in (2.113): κ2 = (B 2 − E 2 ),

κ1 = B.E,

E2 ), c2

or κ2 = (B 2 −

κ1 = B.

E , c

(2.159) with c = 1. ¶

Remark 15 On the constitutive relations with the Lorentz invariance. With the fields E and B, there are also the fields D (the electric induction) and H the magnetic field that must be the solution of the Maxwell–Ampère law with the Gauss electric law: −

∂D + curl H = J, ∂t

div D = ρ,

(2.160)

with a charge density ρ and a current density J . A priori from (−D, H ), a contravariant tensor G of degree 2 is defined, so that G = (−D, H ) ∈ o∗ (1, 3). In a similar way as in (2.156), we can write equation (2.160) with the codifferential δ = ∗d∗ (with the Hodge transformation ∗) as δωG = θJ˜ ,

J˜ = (ρ, J ).

(2.161)

With the hypothesis that G is a contravariant tensor, this implies that there are two invariants with respect to the Lorentz transformations: κ1 = −D.H,

κ2 = D 2 − H 2 ,

(2.162)

or with c = 1, c2 D 2 − H 2 = κ2 , −cD.H = κ1 . When the current J˜ is unknown, the following constitutive relations are often adopted: D = E,

B = μH,

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2 Classical Mechanics

with the permittivity  and the permeability μ of the medium. These relations are in contradiction to the transformation of the tensor G by the Lorentz group, except in the free space. Indeed, taking these constitutive relations in (2.162), we obtain  2 μ2 E 2 − B 2 = μ2 κ2 , which is in contradiction to the invariance (2.159), except if  2 μ2 = 1, which is the case of the free space. We can find other constitutive relations that depend on the velocity (see [Jones, ch. 2.13, p. 114], with c = 1), and a priori on the chosen frame (i) D + v × H = (E + v × B), (ii) H − v × D = μ−1 (B − v × E).

(2.163)

Then if we multiply these relations by u0 , with v = uu0 , we can write them in the following form, with u˜ = (u0 , u) (in a Lorentz frame U ): (i) u(SA) ˜ ˜ (D,H ) =  u(SA) (E,B) , −1 (ii) u(SA) ˜ ˜ (H,−D) = μ u(SA) (B,−E) ,

(2.164)

which takes explicitly the relations u.D = u.E and u.B = μu.H into account. Taking the transpose of these relations, we obtain (i)(SA)(E,−B) u˜ =  −1 (SA)(D,−H ) u, ˜ (ii)(SA)(B,E)u˜ = μ(SA)(H,D)u. ˜

(2.165)

Using p˜ = (p0 , p) = bf Gu, ˜ we also have (i)(SA)(E,B) p˜ =  −1 (SA)(D,H ) p, ˜ (ii)(SA)(B,−E)p˜ = μ(SA)(H,−D)p. ˜

(2.166)

Thus using the relations (2.164), (2.165), we obtain u(SA) ˜ ˜ = μ−1 u(SA) ˜ ˜ (D,H ) (SA)(H,D) u (E,B) (SA)(B,E) u,

(2.167)

which gives the following relation, which is independent of u: ˜ D.H = μ−1 E.B.

(2.168)

2.4 Mechanics of Newton, Lagrange, and Hamilton

207

Yet we also have ˜ =  2 u(SA) ˜ ˜ (i) u(SA) ˜ (D,H ) (SA)(D,−H ) u (E,B) (SA)(E,−B) u, (ii) u(SA) ˜ ˜ = μ−2 u(SA) ˜ ˜ (H,−D) (SA)(H,D) u (B,−E) (SA)(B,E) u,

(2.169)

which gives H 2 − D 2 = μ−2 (u0 )2 [B 2 −  2 μ2 E 2 ] + μ−2 (u)2 [E 2 −  2 μ2 B 2 ] + ( 2 − μ−2 )[(u.B)2 + (u.E)2 ]. Yet if we have μ = 1, then this relation is simply reduced to H 2 − D 2 = μ−2 [(u0 )2 − (u)2 ][B 2 − E 2 ] = μ−1 [B 2 − E 2 ],

(2.170)

which is also independent of the velocity u˜ in this case. Furthermore, the two invariants of (E, B) and (D, H ) are proportional (see (2.168) and (2.170)) with the same coefficient Z = μ−1 when μ = 1 (which is the case of the free space).

Hamiltonian Evolution of a Particle in an Electromagnetic Field Let a particle (with charge e and mass m) in M˜ be in the presence of an electromagnetic field. The evolution equation of this particle will be obtained through a symplectic structure and a Hamiltonian, as follows. Let π be the canonical ˜ onto M. ˜ Then we define on T ∗ (M) ˜ the differential form projection from T ∗ (M) ˜ x).d θe,A˜ = θ + eπ ∗ (θA˜ ) = p.d ˜ x˜ + eA( ˜ x, ˜

(2.171)

ωe,F = dθe,A˜ = ω + eπ ∗ (ωF ).

(2.172)

and the 2-form

˜ Here, This 2-form is not degenerate, and thus defines a symplectic structure on M. ∗ ˜ ˜ as a variant of (2.131), we call the function H on T (M) given (with the Lorentz metric, and with m = 1, c = 1) by  1 1 ˜ p) ˜ = (−p02 + H˜ (p) = g(p, pk2 ) (2.173) 2 2  Hamiltonian, and thus d H˜ = −p0 dp0 + pk dpk . The Hamiltonian vector X = XH˜ on T ∗ M associated with H˜ is defined by i(X)ωe,F = −d H˜ .

(2.174)

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In the coordinates (x i , pi ), the vector X is expressed by X=



(a i

∂ ∂ + bi ), ∂x i ∂pi

i = 0, . . . , 3. i

i Then X will generate an evolution in T ∗ M given by a i = dx bi = dp dτ , dτ , the evolution variable being denoted by τ (here τ is the proper time). Thus the evolution equations of the particle along X are given by

dx 0 dx k = pk , = −p0 , dτ dτ  0  dpk dx (ii) −e Ek − (i(X)ωB )k = 0, dτ dτ (i)

 dx k dp0 +e Ek = 0. dτ dτ (2.175) Then from (i), we have pˆ = (p0 , p) = X˜ = Gu, ˆ and (ii) becomes  dp0 +e pk Ek = 0, dτ

dpk + e [p0 Ek − (p × B)k ] = 0, dτ

(2.176)

which is also in matrix form with p˜ = (p0 , p): d p˜ + e(SA)E,B .p˜ = 0. dτ

(2.177)

 dp0 d H˜ k From these equations, we obtain pk dp dτ − p0 dτ = 0, that is, dτ = 0. We verify that H˜ is constant along the motion. We can write H˜ (p) = 12 g(p, ˜ p) ˜ = − 12 (with 1

m = 1, c = 1).28 Let β = (1 − v 2 )− 2 , with v = uu0 = − pp0 , β dτ = dt. We obtain the equation of evolution of the particle along with the time t, using the cross product × for the Lorentz force dp − e [E + v × B] = 0, dt

(2.178)

0

which implies dp dt + eE.v = 0. Change of the symplectic structure. The change of the symplectic structure gives ˜ We prove this the same result as changing the momentum p˜ for P˜ = p˜ + eA. following [Sternberg]. Let ϕe,A˜ be the map in T ∗ M˜ given by ˜ x). ϕe,A˜ (p, ˜ x) ˜ = (p˜ + eA, ˜

28 With

c, m = 1, then β = (1 −

v 2 − 12 ) , c2

βm dτ = dt, and p0 = βmc, pk = −βmv k = −mcuk .

2.4 Mechanics of Newton, Lagrange, and Hamilton

209

Then we have ∗ ϕe, θ = θe,A˜ , A˜

∗ ϕe, ω = ωe,F , A˜

(2.179)

and ˜ x), H˜ e,A˜ (p, ˜ x) ˜ = H˜ (p−e ˜ A( ˜ x) ˜ = H˜ (ϕ −1˜ (p, ˜ x)) ˜ = ((ϕ −1˜ )∗ H˜ )(p, ˜ x). ˜ e,A

e,A

(2.180)

From equation (2.174), we obtain i((ϕe,A˜ )∗ (X))ω = −d H˜ e,A˜ .

(2.181)

Then the evolution curves, solutions of (2.175), are the images under ϕe,A˜ of the ˜ we have curves whose generator is (ϕe,A˜ )∗ (X). By projection on (M) π∗ (ϕe,A˜ )∗ (X) = (πϕe,A˜ ))∗ (X) = π∗ X, ˜ from since πϕe,A˜ (p, ˜ x) ˜ = π(p, ˜ x) ˜ = x. ˜ Thus we have the same trajectories on (M) one system as from the other. But we have to note that a priori, P˜ does not satisfy the properties of the momentum p, ˜ that is, g(P˜ , P˜ ) is not necessarily negative, so that P˜ is not a section ∗ + ˜ of T M , and −P˜ 0 is not necessarily positive. But the requirement that P˜ be a ˜ = 0, that is, if p0 A0 − p.A = 0, and thus section of T ∗ M˜ + is satisfied if g(p, ˜ A) ∗ ˜ ˜ A) ˜ ≥ 0, and A ∈ Tp˜ Eλ , so that A0 ≤| A |, whence g(A, ˜ A) ˜ = −λ2 + e2 g(A, ˜ A) ˜ ≤ 0, g(P˜ , P˜ ) = g(p, ˜ p) ˜ + e2 g(A,

2 ˜ A) ˜ ≤λ . if g(A, e2

˜ A) ˜ = 0, then g(P˜ , P˜ ) = −λ2 . ¶ Notice furthermore that if g(A, Contact structure with the action. Now let θ˜ = θe,A˜ + ds be a new differential ˜ form on T ∗ (M)⊕R, with a new variable s (the action), so that XH˜ is a characteristic of θ˜ (with XH˜ the Hamiltonian vector field associated with H˜ ; see (2.174)), that is, < θe,A˜ + ds, XH˜ > = 0. We have < θ, XH˜ > =



pi

dx i = −p02 + pˆ 2 = g(p, ˜ p) ˜ = −m2 , dτ

and < θA˜ , π∗ XH˜ > =



Ai

 dx i = −p0 A0 + pk Ak . dτ

(2.182)

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2 Classical Mechanics

Then (2.182) becomes, with S(x(τ ˜ ), τ ) = s(τ ),  dS pk Ak ). = m2 + e(p0 A0 − dτ

(2.183)

Since p0 dτ = dt, we still have, up to changing e into ec and s into cs , the usual formula with the Lagrangian function L (see [Lan-Lif4, ch. III, 16]), dS = mcβ −1 + e(A0 + v.A) = L. dt

(2.184)

It would be equivalent to define a metric (in the sense of [Mall]) on T M, using ˆ by the relation p˜ = mGu˜ for u˜ = (u0 , u), ˜ ˜ A). F (ξ ) = F (x, ˜ p) ˜ = (−g(p, ˜ p)) ˜ 2 − eg(p, 1

(2.185)

The function F is positively homogeneous, ˜ ∀a > 0, ξ = (x, ˜ p) ˜ ∈ T ∗ M.

F (aξ ) = F (x, ˜ a p) ˜ = aF (ξ ) = aF (x, ˜ p), ˜

The function Fˆ = −F is also strictly convex, see (2.127), as it is without an electromagnetic field. Then we have the same properties on the curves with or without the electromagnetic field. Let be an integral curve of XH whose endpoints’ projections on M˜ are x˜0 , x˜1 . Then by integrating (2.183) along this curve, we have  S(x˜1 ) − S(x˜0 ) =

 m dτ + 2

[τ0 ,τ1 ]



eθA˜ ,

(2.186)

or by integrating over time, with L the Lagrangian,  S(t1 ) − S(t0 ) =

t1 t0

 Ldt =

t1

[mβ −1 + e(A0 + v.A)]dt.

(2.187)

t0

Then we can define the action as a function on spacetime from a field of noncrossing characteristics (or a field of normals), as seen previously. Notice that if the electromagnetic potential is changed into θA˜ + d, then the action S is changed into S + e, which is a gauge transformation.

A Variant We can also obtain the previous results with additional variables, in the following equivalent way. We introduce a new variable λ, dual to the proper time τ , and the variable s (the action) in the new differential form ϕ:

2.4 Mechanics of Newton, Lagrange, and Hamilton

211

ϕ = θe,A˜ − λdτ + ds.

(2.188)

We consider the surface , a submanifold of T ∗ (M˜ × R), given by29 H˜ (p) ˜ + λ2 = 0.

(2.189)

We look for the integral manifolds of the Pfaff equation ϕ | = 0,

(2.190)

the restriction of ϕ to . Thus the differential of ϕ | is dϕ | = ωe,F − d H˜ ∧ dτ.

(2.191)

Let X be a characteristic vector for this 2-form: X=



Pi

 ∂ ∂ ∂ ∂ ∂ + Xi + αs , + +γ i ∂x ∂τ ∂pi ∂λ ∂s

giving the characteristic curve through dx i = Pi , dτ

dpi = Xi , dτ

d H˜ dλ = = γ, dτ dτ

ds = αs . dτ

(2.192)

Then X must satisfy the equations i(X)(ϕ | ) = 0,

i(X)d(ϕ | ) = 0.

We successively obtain from the first relation 

(pj + eAj )Pj − H˜ + αs = 0,

(2.193)

then from the second relation, first for the coefficients of dpi , then those of dx i , (i) − P0 + p0 = 0, −Pk − pk = 0,  (ii) X0 + e Pk Ek = 0, Xk − e[P0 Ek + (P ∧ B)k ] = 0.

(2.194)

Thus we obtain again equation (2.176). Finally, taking H˜ = −λ2 = − 12 m2 a constant, we have the equation for the action (which is the result (2.183) up to a constant):

29 Instead of using the “Hamiltonian” H ˜ , we could have chosen to work directly with p˜ in the space Eλ∗ , with λ = m2 (or m2 c2 ).

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2 Classical Mechanics

 ds = αs = −(p02 − p2 ) + H˜ − e(A0 p0 − Ak pk ) dτ  1 = − (m2 ) − e(A0 p0 − Ak pk ). 2

(2.195)

From the Pfaff equation (2.190), we also have ∂s + (Pi + eAi ) = 0. ∂x i Then (2.195) gives the following (linear!) Hamilton–Jacobi equation:  ds 1 ∂s ∂s = αs = − (m2 c2 ) − e[A0 ( 0 + eA0 ) − Ak ( k + eAk )]. dτ 2 ∂x ∂x

(2.196)

From equation (2.189), we also have a Hamilton–Jacobi equation: (

 ∂s ∂s + eA0 )2 − ( k + eAk )2 − m2 = 0. 0 ∂x ∂x

(2.197)

With the condition of having a field of noncrossing characteristics, we determine an action function S such that S(x(τ ˜ ), τ ) = s, for a given initial condition S(x, ˜ 0) = S0 (x) ˜ satisfying equation (2.197). Note that the characteristic curves depend on the electromagnetic field (E, B) only, and not on the specific electromagnetic potential A. By a gauge transformation ∂ ˜ (A˜ j = Aj + ∂x j ), S = S − e is a solution of (2.197), and it is also a solution of (2.196), since we have d  ∂ dx i = dτ ∂x i dτ

with (τ ) = (x(τ ˜ )).

This result is obvious from the relations (2.190) and (2.188). If we can choose (at least locally) S − e = 0, then the electromagnetic potential is such that g(A, A) = m2 . e2 A Covariant Point of View. An Electromagnetic Connection Form We recall the notion of the covariant differential ∇σ of a (smooth) section σ of a k-dimensional vector bundle  E over a manifold M, given in a local frame of E (σ1 , . . . , σk ) such that σ = fi σi , with (smooth) functions fi on M, by ∇σ =



dfi σi +



fi ωi,j ⊗ σj ,

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213

with (ωi,j ) a matrix of differential forms on M, which is called a connection form [Gilk, ch.!2.1, p. 89]. Then the covariant derivative of σ in the direction of a vector X ∈ Tx M is given by ∇X σ =



< X, dfi > σi +



fi < ωi,j , X > σj .

We now define on M˜ the connection form ωe˜ due to the electromagnetic field, M with values in the Lie algebra o(1, 3), using the (Hilbert) form w = dS given by (2.143), (2.144), also previously denoted by dτ , with the proper time τ , by e e ωM ˜ = ((ωM˜ )i,j ) = ew.(SA)E,−B .

(2.198)

This implies, with p˜ = (p0 , p) = (p0 , p1 , p2 , p3 ), e pω ˜ M ˜ = ew(



pi ((SA)E,−B )i,j ) = ew(SA)E,B p˜

(2.199)

= ew(E.p, p0 E + B × p) = −ewu0 (−E.v, E + v × B). Let X = u˜ be the unit characteristic vector given by30 (2.136), which satisfies < d d w, X > = 1 and can be identified with the derivative ds (or dτ ), since S∗ = dS ⊗

d , ds

S∗ X =< dS, X >

d d = . ds ds

Thus X is transverse to the oriented foliation of the future cone + with the leaves s+ = {x˜ ∈ + , S(x) ˜ = s}, so that the tangent space is Tx˜ s+ = ker S∗ = ker dS. Observe that S is an increasing function of the time x 0 = ct. ˜ the covariant derivative of the Then in aLorentz frame U ∗ = (σi ) of T ∗ M, section p˜ = pi σi of the bundle T ∗ M˜ in the direction of the vector X is given by e ∇X p˜ =

 dpi dτ

σi + e



pi ((SA)E,−B )i,j σj .

(2.200)

By comparison with equation (2.177) or (2.176), we see that equation (2.200) is simply e ∇X p˜ = 0,

(2.201)

which looks like a geodesic equation with respect to the Lorentz connection. 30 But

in this relation (2.136), τ is not the proper time; it is related to S by (2.141).

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i From the relation (2.198), we deduce the Christoffel symbols mk by i = −e mk

∂S ((SA)E,B )ik ∂x m

∂S x 0 ∂S xj , = =− , j 0 S ∂x S ∂x

with

(2.202)

and ((SA)E,B )i0 = ((SA)E,B )0i = Ei , i = 0, ((SA)E,B )ik = −ikl Bl , i, k, l = 0, with (ikl ) the structure constants of o(3), the Lie algebra of the rotation group (see also (2.108)), that is, ikl = 1 (respectively −1) if the permutation of i, j, k, is even (respectively odd), and 0 if two subscripts are equal. ˜ π(U ) = x. Let π be the projection from P onto M, ˜ Then Fˆ = (SA)E,B is the 1 ˜ section of T1 (M) corresponding to the electromagnetic field Fˆ at the point x˜ (see (2.158)). Thus 0 −1 Fˆx˜0 = Fˆπ(U ) = U (SA)E,B U

is the map from P to o(1, 3) associated with this section, and thus Fˆ 0 satisfies the property Ra∗ Fˆ 0 = ad a −1 Fˆ 0 . ↑ ˜ Then the connection form31 ωe on the principal bundle P = L+ (M), which e ˜ is defined at the proper Lorentz corresponds to the connection form ω ˜ on M, M frame U by e 0 ωU = e dS.Fˆπ(U ) + θU ,

(2.203)

j

where θU is defined as follows. Let (Ei ) be the canonical base of the Lie algebra o(1, 3), and let (Xjk ) (respectively (Yki )) be the components of the frame U ˜ then (respectively of U −1 ) in a frame (Xj = ∂x∂ j ) with coordinates (x j ) in M; θU is defined by θU = U −1 dU =



j

Yki dXjk Ei .

↑

Let σU be the map at ∈ L+ → U.at ∈ PU , and σU−1 its inverse: σU−1 (U  ) = ↑ U −1 U  = a. Let ωˆ be the Maurer–Cartan form of L+ , which is the left invariant form such that ωˆ a = (la −1 )∗ . Then θU is also defined by ˆ U; θU  = (σU−1 )∗ = ((σU−1 )∗ ω)

31 See,

for instance, [Kob-Nom, ch. II.1, p. 64].

(2.204)

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215

θ = (σU−1 )∗ ωˆ is the Darboux differential of (σU−1 ) and does not depend on U . Proof Let X ∈ T (PU  ). Then ((σU−1 )∗ ω) ˆ U  (X) = ωˆ a (σU−1 )∗ (X) = (la −1 )∗ (σU−1 )∗ (X). Since la −1 σU−1 (U  ) = a −1 U −1 U  = (U  )−1 U  = σU−1 (U  ), ˆ U  (X) = (σU−1 )∗ (X), that implies (2.204). it follows that ((σU−1 )∗ ω)

¶

Now we also have (Ra )∗ θU = θU a (Ra )∗ = (σU−1a )∗ (Ra )∗ = (σU−1a Ra )∗ = ad a −1 θU ,

(2.205)

since σU−1a Ra (U  ) = (U a)−1 U  a = a −1 U −1 U  a = ad a −1 σU−1 (U  ). Let A∗ be the fundamental vector field corresponding to A ∈ o(1, 3), which is defined as follows. Let (at = exp tA) be a 1-parameter group of proper Lorentz transformations. Then we have A∗U = (σU )∗ A = U.A ∈ TU P . Then from (2.204), we have < θU , A∗U > = A. We can obtain this relation using coordinates. As a matrix, U is represented by (Xji )  (that is, U.ξ = Xji ξ j ), and its inverse by (Yki ). Then A∗U is represented by A∗U =



(A∗U )ik

∂ ∂Xki

=



∂

j

Xji Ak

∂Xki

.

Thus we have < θU , A∗U > =
= Aij Ei = A.

Then the connection form ωe satisfies the properties of a connection form on P , that is, [Kob-Nom, ch. II.1, p. 64]: ωe (A∗ ) = A,

Ra∗ ωe = ad a −1 ωe ,

(2.206) ↑

which means that for every X ∈ TU P and Ra U = U.a, a ∈ L+ , we have Ra∗ ωe (X) = ωe ((Ra )∗ X) = ad a −1 ωe (X). These properties are a simple consequence of the relations for Fˆ 0 and θU .

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2 Classical Mechanics

˜ Let J be a local section of the bundle Horizontal lift X∗ on P of a vector X on M. P on M˜ on an open set O. Then from X we define a vector J∗ X on P , which we also denote by X, since it is written with the same notation in a coordinate system. Then X is decomposed into a horizontal vector X∗ and a vertical vector A∗ on P , respectively called horizontal lift and vertical lift, so that X = X∗ + A∗ . Then we apply the connection form ωe , and we have, since the connection form is null on the horizontal vectors, < ωe , X > = e < dS, X > Fˆ 0 = < ωe , A∗ > = A. Thus A∗U = U A = e < dS, X > U Fˆ 0 , and the horizontal lift of X is given by ∗ = X − e < dS, X > U Fˆ 0 . XU

Remark 16 Of course this is a different point of view with respect to the connection given by the differential form θA , which is R-valued, linked with a conformal group (according to Weyl), so that dθA = ωF appears as a curvature form. ¶

2.5 Paths, Curves, and Geodesics 2.5.1 On the Curve Parametrization Here we specify some useful notions. A differentiable path in a differentiable manifold M is a map of class C k , with k ≥ 1, of an open interval J of R into M. A curve in M is a 1-dimensional submanifold of M. The curves that we will use will be oriented curves. We can associate various paths with a curve. Conversely, the image τ (J ) of a path τ in V is not necessarily a curve, and a priori the map τ is not injective. A parametric representation of a curve γ (submanifold of M) is a path τ : J → M, τ = (xt ), t ∈ J , that is a map of class C k , k ≥ 1, of an open interval J on γ , which is an embedding of J into M. Recall that a map τ of class C 1 is an embedding from J into M if it is (i) injective, (ii) regular, i.e., τ˙ (t) = 0, ∀t ∈ J, and (iii) homeomorphic from J onto τ (J ), i.e., τ and its inverse are continuous. A change of parametric representation τ˜ = (x˜s ), s ∈ J˜, of an oriented curve is linked to τ by an increasing function f (t) such that s = f (t), and thus we have x˜s = x˜f (t ) = xt , and by derivation, the corresponding velocity vectors are linked by vt = x(t) ˙ = f  (t).v˜s . Also recall that if τ is an injective regular map at t0 , then there exists a neighborhood of t0 such that τ is an embedding on this neighborhood.32 32 This is a consequence

of the following properties [Bour.alg1, TG.I.10.2]: Every continuous map of a quasicompact space into a separate space is proper. Every proper immersion is an embedding.

2.5 Paths, Curves, and Geodesics

217

Lifts in T M and in T ∗ M of Curves in M Let x0 and x1 be two points of M, with x0 = x1 . We consider the set of oriented curves in M, of class C k , k ≥ 1, containing x0 , x1 . Let Cxk0 ,x1 (M) be the set of restrictions of these curves to their parts with endpoints x0 , x1 . Let γ be such a restriction of a curve, and let τ be a parametric representation of γ , that is, an embedding of an interval J of R onto γ in M; thus τ is the map t ∈ J → τ (t) = xt ∈ M. By time derivation of τ in M, we obtain a path, denoted by τ˙ , in T M of class C k−1 ; τ˙ is also an embedding from J into T M, called a lift of γ in T M. With τ∗ (λ) = Tt τ (λ) = (xt , λx˙t ), we have that τ˙ is the map t ∈ J → τ˙ (t) = τ∗ (1) = (xt , x˙t ) = (xt , vt ) ∈ T M, and τ˙∗ = Tt τ˙ is the map λ ∈ R → ((xt , x˙t ), λ(x˙t , x¨t )) ∈ T(xt ,x˙t ) (T M), which then defines τ¨ (t) = Tt τ˙ (1) = τ˙∗ (1). The map v = τ˙ ◦ τ −1 then is a section of the bundle T γ on γ . There are as many lifts of γ and of sections v of T γ as parametric representations of γ . Let M be a Riemannian manifold. Applying the Riemannian metric G to τ , we obtain a path Gτ˙ in T ∗ M, which is of class C k−1 from J into T ∗ M; Gτ˙ is also an embedding from J into T ∗ M, called a lift of γ in T ∗ M.

Curve Integrals Let ω be a differential form on a manifold M. Then the integral of ω along a differentiable path τ : J → M is expressed by 



τ ∗ω =

ω=

I (τ ) = τ

J

 < ω, τ˙ (t) > dt.

(2.207)

J

Let γ be an oriented curve in M with τ as a parametric representation. Then we can verify that the integral I (τ ) is invariant by changing its parametric representation, thus denoted by I (γ ). On the other hand, the situation is not the same for lifts of curves in T M or in T ∗ M, which we can verify in the following example. Let θ = p.dx be the canonical form on T ∗ M; the integral Gτ˙ θ depends on the lift τ and not only on γ . Indeed, we have, for J = [0, T ],  I (τ ) =

 Gτ˙

T

θ= 0



T

< p(t), v(t) > dt =

g(v(t), v(t)) dt, 0

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2 Classical Mechanics

whereas if τ˜ is another parametric representation of γ , with J  = [0, T  ], then we have  I (τ˜ ) =



Gτ˙˜

T

θ=

g(v(t ˜  ), v(t ˜  )) dt  =



T

f  (f −1 (t))g(v(t), v(t)) dt.

0

0

Canonical Parametrization of Geodesics In a Riemannian manifold, special curves, namely geodesics, have a particular importance, for which we can give several equivalent definitions. A natural mechanics definition for a Riemannian manifold is that the geodesics are extremal regular curves for the metric (see [Mall, ch. IV, 2.1]) that will be specified below. The notion of covariant derivative allows us to express the geodesic equation by ∇v v = 0 (or ∇t,v v = 0 to indicate the time parametrization), with vt = x˙t the velocity vector also identified with a section of T (γ ) for a curve γ . The parametrization for which s is the length of the geodesic is called the canonical parametrization. Then the corresponding velocity vs is of unit norm for all s. Let τ be an oriented curve in a Riemannian manifold M with its metric g, with a time parametric representation of τ , giving the velocity vt = dτ dt , and the measure ds = (g(vt , vt ))1/2 dt = |vt |dt.

(2.208)

In this relation vt = x(t) ˙ = f  (t).v˜s , we have |v˜s | = 1, |vt | = f  (t) = g(vt , vt )1/2 . Therefore, for a curve (τ (t)) with endpoints x0 , x1 with τ (0) = x0 , τ (T ) = x1 , the length L of the curve is expressed by 

L

L= 0



T

ds =

g(vt , vt )1/2 dt.

(2.209)

0

Now let τ be the map s → xs from the interval J˜ = [0, L] onto γ with |vs | = 1. Then the pullback of the differential form Gvs on γ by τ is the differential ds of the ∂ length on J ⊂ R, ds = τ ∗ (Gvs ). Indeed, we have τ∗ ( ∂s ) = vs , since τ∗ (

 j ∂f ∂ ∂ )f = f (τ (s)) = vs j (τ (s)) = vs f, ∂s ∂s ∂x

for all functions f on M (of class C 1 ); we deduce ∂ ∂ > = < Gvs , τ∗ ( ) > = < Gvs , vs > = 1. ¶ ∂s ∂s  By integration, we obtain the length L of the curve γ : L = γ τ ∗ (Gvs ). < τ ∗ (Gvs ),

2.5 Paths, Curves, and Geodesics

219

2.5.2 Exponentials Recall the following essential property on geodesics: For every (x0 , X0 ) ∈ T M, there exists a unique geodesic (xt ), t ∈ J = [0, T ] in M, satisfying the initial conditions x0 = x0 , x  (0) = X0 . We assume that M is a connected complete Riemannian manifold, i.e., that every geodesic can be extended to a geodesic (xt ) defined for all values of t ∈ R. For each X ∈ Tx M, let xt , t ∈ R + , be the geodesic with initial condition (x, X). Then the exponential map is defined by33 exp tX = xt ∈ M. Thus exp is a map from Tx M into M such that exp X = x1 . Recall the main property on the domain of the exponential map: For every point x ∈ M, there is a neighborhood Nx of 0 in Tx M such that the exponential is a diffeomorphism from Nx onto a neighborhood Ux ∈ M. Let Sx = {X ∈ Tx M, |X| = 1}, and for X ∈ Sx , we define μ(X) = sup t > 0, such that τ = xs , 0 ≤ s < t is a minimizing geodesic between x and xt (that is, the shortest curve between x and xt ). The point xt with t = μ(X) = ∞ is called the cut point of x along the geodesic τ . Then34 μ is a continuous function from Sx into R + ∪ ∞. Let Ex = {tX ∈ Tx M, 0 ≤ t < μ(X), X ∈ Sx }. Then Ex is an open set in Tx M, and exp is a diffeomorphism from Ex onto an open set Mx of M; M is a disjoint union: M = Mx ∪ C(x), with C(x) the set of cut points of x, called the cut locus of x. Let xr ∈ C(x). Then there are two possibilities: (i) there exist at least two minimizing geodesics from x to xr ; (ii) xr is the first conjugate point of x along the geodesic, that is, with xr = exp rX, the Jacobian matrix of exp X is singular at X ∈ Tx M.

33 We refer, for instance, to [Kob-Nom, Vol. 2, ch. VIII.1, 2], [Lang2], for the main properties of this map, notably for the Jacobi fields. 34 See [Kob-Nom, Vol. 2, ch. VIII, 7].

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2.5.3 Wave Function of a Geodesic Field The family of geodesics with initial point x is such that there is no common point between two geodesics in Mx \ {x}. Therefore, there exists a unit vector field A, on Mx \ {x}, called the geodesic field, such that Ay ∈ Ty M, ∀y ∈ Mx \ {x}, is the tangent vector to the geodesic τx,y joining y to x; A is defined with the covariant derivative by ∇A A = 0 and is invariant by parallel displacements along the geodesics; A is unique up to its orientation. Thus Axt =

d (exp tX)|t . dt

Let ωA = GA be the differential form on M corresponding to the geodesic field by the metric. Now the question is whether the differential form ωA = GA is closed, i.e., whether it satisfies dωA = 0, which is equivalent to knowing when there exists locally a function S on M (at least on a domain V ⊂ M) such that ωA |V = GA|V = dS,

(2.210)

or also whether A = G−1 dS = grad S is a gradient. Let μ0 = inf μ(X), X ∈ Sx0 , 0 < t < μ0 , and tx0 = {x ∈ Mx0 , d(x, x0 ) = t} = {x ∈ Mx0 , }. x

Since t < μ0 , for each x ∈ t 0 , there exists X ∈ Sx0 such that x = exp tX. Let Bμ0 (x0 ) = {x ∈ M, d(x, x0 ) < μ0 }, and let the function S be defined on the domain V = Mx0 ∩ Bμ0 (x0 ) by S(x) = t,

for x ∈ tx0 .

Proof of the Relation (2.210) With x ∈ tx0 , the tangent space Tx M splits into Tx M = Tx tx0 ⊕ RAx , the two spaces being orthogonal:35 For all Y ∈ Tx t , we have < ωA , Y > = g(Ax , Y ) = 0, < dS, Y > = 0. For Y = Ax , we have < ωA , Ax > = 1, and < dS, Ax > = dS dt = 1. This implies (2.210). For t ∈ (μ0 , μ1 ) with μ1 = sup μ(X), X ∈ Sx0 , we can also define the function S, but we would have to restrict the domain of definition. Huygens Property. Let X ∈ Tx0 , | X |= 1. Then let Xs = τ0s X ∈ Txs M, the unit tangent vector at xs to the geodesic curve τ , with initial conditions (x0 , X), parametrized along the length. Then we have (with t + s < μ0 )

35 This

is due to the fact that the geodesics are minimizing in Mx0 ; see also [Kob-Nom, ch. IV, 3, Prop. 3.2] with the metric in spherical coordinates.

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221

exp ((t + s)X) = exp (tXs ). Let Bx0 = {X ∈ Tx0 M, | X |≤ 1}, and Bxs = {Y ∈ Txs M, | Y |≤ 1. This implies exp (tBxs ) ⊂ exp ((t + s)Bx0 ). Proof Let y ∈ exp (tBxs ), whence d(y, xs ) = t, and then d(y, x0 ) ≤ d(y, xs ) + d(xs , x0 ) ≤ d(xt , xs ) + d(xs , x0 ) = d(xt , x0 ). ¶ Instead of the previous definition of the family of geodesics coming from a single point x0 , we can define36 a priori a family of geodesics in a domain V , with the following properties: There exists37 a hypersurface 0 that is transverse38 to the geodesic field A of , so that (i) at every x0 ∈ 0 , there is one and only one geodesic τx0 of , (ii) every x ∈ V belongs to one and only one geodesic of . Then for every x ∈ τx0 (geodesic of containing x0 ), there is one geodesic vector Ax0 tangent to the geodesic τx0 , thus a geodesic field on V , and we define on  V the function S from the canonical parametrization of the geodesics by S(x) = τx0 ,x ds, with τx0 ,x the geodesic curve going from x0 ∈ 0 to x. The function S is called the wave function of the geodesics, or of the field of “normals ” (see [Mall, ch. IV, 4.2]). This function indicates the distance from a point of V to the hypersurface 0 transverse to the geodesic field A. We still have dS = GA, A= grad S, and dS = < vs , dS > = < A, dS > = < A, GA > = g(A, A) = 1, ds i.e., the parameter s identified with S is used as a coordinate in V . Remark 17 On the terminology of metric. It is essential to note for the applications that for the metric of a Riemannian manifold we can substitute a positively homogeneous function as in [Mall, ch. IV, 1.1.2] (also called a metric) that is not necessarily always a positive function. Keeping the same terminology of geodesic (which may be confusing), we can develop a similar theory with the notion of field of normals of the wave function of the geodesics with the notion of extremum, which is no longer necessarily a minimum. This is notably useful for the notion of action, which explains the notation S for the action. ¶ 36 See

[Mall, ch. IV, 4.2]. can assert the existence of such hypersurface thanks to a section of the geodesics with the same origin x0 . 38 Here we assume that the tangent space at every point of  is orthogonal to the geodesic field A. 0 37 We

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2.5.4 Foliation and Eikonal Equation The map x ∈ V → S(x) ∈ R defines a geodesic foliation of the domain V of S, the leaves being defined by s = S −1 (s), which is called the wave front. Thus V = ∪s∈J s . Since the geodesic field A is such that |A| = 1, notice also that S satisfies the so-called eikonal equation in V , |∇S| = 1, that is, with coordinates (x j ): 

g ij

∂S ∂S = 1. ∂x i ∂x j

(2.211)

Relation with the canonical form θ = p.dx on T ∗ M, or in an intrinsic way < θx,p , X > = < p, π∗ X > with π the canonical projection from T ∗ M onto M, and X ∈ Tx,p M. Since dS is a differential form on M, we have to lift it on T ∗ M by π ∗ dS = dS ◦ π∗ , and we define ϕ = π ∗ dS − θ. Then for all X ∈ Tx,p (T ∗ M), we have < ϕ, X > = < dS, π∗ X > − < p, π∗ X > .

(2.212)

Let A be the previous geodesic field. With p = ωAx , we write p.dx = ωA , and then we have < ϕ, X > = 0. Therefore, taking ωA as a section of the bundle T ∗ M over M, the differential form ϕ is null on the domain Im ωA ⊂ T ∗ M; we have dS − pdx = dS − ωA = 0 on this domain. We can find this in the following way. The dS = GA on V  also  property relation ∂S i = j dx i , and thus p = ∂S = j ; p = (p ) is said implies dx g A g A ij i ij i ∂x i ∂x i to be the normal vector of slowness of the wave front [Arn1, ch. 9, 46]. Example 8 In the case of the unit sphere S 2 , we can choose either to start with the first method, from the north pole as point x0 , or with the method of the remark, with the equator as 0 , and with s as parallel lines, with V = S 2 \{N, S}. Using the first method and spherical coordinates, we have, with 0 < θ < π, 0 ≤ ϕ ≤ 2π, x1 = sin θ cos ϕ,

x2 = sin θ sin ϕ,

x3 = cos θ.

Then the metric is given by gθ,ϕ = ds 2 = dθ 2 + sin2 θ dϕ 2 . We notice that the chosen chart is such that the metric degenerates at the poles. At every point x ∈ V , with coordinates θ, ϕ we have an orthonormal frame given by 1 2 uθ,ϕ = (Xθ,ϕ , Xθ,ϕ )=(

1 ∂ ∂ , ). ∂θ sin θ ∂ϕ

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223

∂ 1 1 2 has no limit. Then the vector field As θ → 0, Xθ,ϕ → X0,ϕ = ( ∂θ )ϕ , and Xθ,ϕ 1 on V is the geodesic field A, tangent to the meridians, and the distance from Xθ,ϕ a point xθ,ϕ ∈ S 2 to the north pole N is given by θ with θ ∈ [0, π], and the wave function S N = S is simply S(xθ,ϕ ) = θ . (Notice that the latitude corresponds to l = π2 − θ.) The set of the cut locus C(N) is reduced to the south pole S. ¶

2.5.5 Extremum Among Path Integrals The evolution equations of mechanics are linked to searching for the extrema of an integral on a family of paths in a connected manifold M. That extremum notion will be specified, following [Mall, ch. IV, 2.3]). In mechanics, we have to consider many problems of minimization or of searching for extrema with or without constraints. For instance: (i) To find a curve of minimum length between two points in a Riemannian manifold M, which leads to the definition of geodesics. (ii) Fermat’s problem (minimum time to go from one point to another) in the previous frame, with constraints on the velocity or on the energy. (iii) Extremum of a Lagrangian among a family of curves in T M. This problem is a generalization of the following problem (next to (i)): For a “free system,” to find a path of minimum kinetic energy in going from one point of M to another. (iv) Extremum of the action (also with the idea of control of the system, as in the Hamilton–Jacobi–Bellman equation).

Extremum Among Paths in M with Fixed Endpoints Let X be a vector field on M,39 that generates a one-parameter group of transformations of M, denoted by (Uλ ), λ ∈ R.40 Let τλ be the path defined by t ∈ J = [0, T ] → τλ (t) = (Uλ ◦ τ )(t) ∈ V . With I (τ ) defined by (2.207), let fX (λ) = I (τλ ). Observe that τλ is defined on the same domain J as that of τ . Requiring that τ be an extremum for I among the set of paths having the same time interval is not a constraint, thanks to the invariance of I (τ ) under changing the parametric representation. Then we have

39 Or

on a neighborhood of the curve γ . can make a weaker hypothesis, such as that Uλ (z) exists for all z ∈ τ (J ), for all λ in a neighborhood of 0.

40 We

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fX (λ) = I (τλ ) =



ω= τλ

ω= Uλ (τ )

τ

Uλ∗ ω.

(2.213)

Using the Lie derivative LX and the Cartan formula LX ω = d i(X)ω + i(X)dω, with the interior  product i(X) by X, we obtain, on taking the derivative with respect to λ, fX (0) = τ LX ω. Thus if x0 , x1 are the endpoints of the path τ , we have fX (0) = < X, ω >x1 − < X, ω >x0 +

 i(X)dω. τ

To say that the path τ is an extremum for the integral (2.207) among the set of differentiable paths in M, with endpoints x0 , x1 , implies that fX (0) = 0 for all vector field X such that Xx0 = Xx1 = 0,

(2.214)

which implies Uλ (xi ) = xi , i = 0, 1, and the path τλ also has x0 and x1 as endpoints for all λ, and then < X, ω >x1 =< X, ω >x0 = 0. Hence, with (2.214) fX (0) =

 < i(X)dω, τ˙ (t) > dt. J

Now < i(X)dω, τ˙ (t) > = < dω, X ∧ τ˙ (t) > = − < i(τ˙ (t)) dω, X > . Then the condition “τ is an extremum path” is equivalent to  < i(τ˙ (t))dω, X > dt = 0, J

for all vector fields X on M satisfying (2.214). Thus i(τ˙ (t)) dω = 0,

∀t ∈ J.

(2.215)

If the extremum path is a curve γ , let v be a vector field on γ = (τ (t)), t ∈ J , such that vx(t ) = τ˙ (t), ∀t ∈ J . Then (2.215) is written i(v) dω = 0. Minimization among curve integrals, geodesics. Let ωA = GA be the differential form associated with a geodesic field A on V ⊂ M. The extremum condition (2.215) gives i(A)dωA = 0. If A is a geodesic field on the subdomain V of M such that ωA = GA = dS on V (and |A| = 1), then the extremum condition is trivially satisfied (for all v ∈ T V ), since dωA = 0 on V . Using the wave function S allows us to prove that geodesics correspond to minima.

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225

Let τ0 be an oriented geodesic with endpoints x0 , x1 . The length of the path τ0 then is given by 



L0 = I (τ0 ) =

ωA = τ0

dS = S(x1 ) − S(x0 ) > 0. τ0

For every path τ with endpoints x0 , x1 , we also have L0 = exact form. Using the Cauchy–Schwarz inequality, we obtain 



T

ωA = τ



T

< ωA , vt > dt =

0





T

g(A, vt ) dt ≤

0

τ

ωA , since ωA is an

|A|.|vt | dt.

0

Since |A| = 1, this implies the inequality L0 = I (τ0 ) ≤ L = I (τ ), where I (τ ) denotes the expression (2.209). Under the existence hypothesis of the wave function S in a domain V ⊂ M for the geodesic field A, the geodesic corresponds to the shortest curve from x0 to x1 in V . In a general setting, we prove (see [Kob-Nom, ch. IV, 3, 4]) the following properties: (i) Every point x0 of a Riemannian manifold M has a convex neighborhood U in the sense that every pair of points x, y in U can be joined by a unique minimizing geodesic (its length is equal to the distance d(x, y)) in U . (ii) If M is a connected complete Riemannian manifold, then all pair of points x, y in M can be joined by a minimizing geodesic. In order to study the problem of extrema among paths in V = T M or T ∗ M, we have to recall some lifts of vector fields. Lifts on T M and on T ∗ M of Vector Fields on M (i) Lift on T M of a vector field X on M. Let X be a vector field on M generating a 1parameter group41 (Uλ ) of transformations on M. Let U˜ λ = (Uλ )∗ = T Uλ , λ ∈ R be the corresponding transformation group on T M, and let X˜ be its generator, which is said to be a canonical lift of X on T M. The two groups are naturally related through π ◦ U˜ λ = Uλ ◦ π, ∀λ. Hence the corresponding vector fields satisfy π∗ ◦ X˜ = X ◦ π,

41 Here

that is π∗ (X˜ v ) = Xπ(v), v ∈ T M.

(2.216)

we assume for simplicity that X is the generator of a group (then X is said to be complete).

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We say that X˜ is π-related to X, and that X˜ is the lift of the field 42 X (see [Kob-Nom, ch. I, 1, p. 10] and [Mall,ch. IV, 2.3] for these notions). If X is given in a coordinate system (x j ) by Xx = a j (x) ∂x∂ j , then X˜ is given by X˜ x,v =



a j (x)

 ∂a j ∂ ∂ + v i i (x) j ∈ Tx,v (T M). ∂x j ∂x ∂v

(2.217)

Let τˆ be a differentiable path in the space T M with endpoints z0 , z1 with zi = (xi , vi ), i = 0, 1. Let τˆλ = U˜ λ τˆ be the path defined by composition with U˜ λ : t ∈ J → τˆλ (t) = U˜ λ (τˆ (t)). Assume that X is null at x0 and at x1 : Xxi = 0,

xi = π(zi ) ∈ M, i = 0, 1.

(2.218)

Then the paths τˆ and τˆλ have the same projections x0 and x1 in M of the endpoints, but not the same endpoints in T M. (ii) Lift on T ∗ M of a vector field X on M. From a 1-parameter group (Uλ ) of transformations on M, we define the group (Uˆ λ ) of transformations on T ∗ M, with t (U ) the transpose of the derivative (U ) of U : λ ∗ λ ∗ λ ∗ Uˆ λ (x, p) = (Uλ (x), t (Uλ )−1 ∗ (p)) ∈ T M.

(2.219)

Let π˜ be the canonical projection of T ∗ M. We have π˜ ◦ Uˆ λ = Uλ ◦ π, ˜ ∀λ ∈ R. The generator Xˆ of (Uˆ λ ) then is such that π˜ ∗ ◦ Xˆ = X ◦ π, ˜

that is π˜ ∗ (Xˆ p ) = Xπ(p) . ˜

ˆ the canonical lift of X on T ∗ M, is π-related ˜ to X. Thus the vector field X,

(2.220) ¶

Extremum Among Paths in V = T M Let ωˆ be a differential form on T M that is the pullback of a differential form ω on M by the projection π: ωˆ = π ∗ ω. Let τˆ be a differentiable path in the space T M. Here we want to express the extremum of the integral  I (τˆ ) =

τˆ

ωˆ

(2.221)

in a set of paths in V = T M whose projections in M of the endpoints are fixed. (We can also consider only the lifts in V = T M of curves in M whose endpoints are

lift is related to the derivative X∗ = T (X) by X˜ = J ◦ X∗ , where J is the canonical involution of T (T M), expressed by (2.20); see [Dieud3, ch. 18.6, Pb3, ch. 16.20, Pb2].

42 This

2.5 Paths, Curves, and Geodesics

227

fixed.) In fact, this integral boils down to the previous one (2.207), since ωˆ = π ∗ ω. Indeed, if τ = π ◦ τˆ is the path in M that is the projection of the path τˆ in T M, then  I (τˆ ) =

τˆ

π ∗ω =



 π◦τˆ

ω=

ω. τ

Afterward, we will verify the correspondence of extrema between these two problems. For the extremum problem (2.221) with respect to paths τˆλ in T M, we are led to use instead of (2.213),    ωˆ = ωˆ = ˆ (2.222) fX˜ (λ) = I (τˆλ ) = U˜ λ∗ ω. U˜ λ (τˆ )

τˆλ

τˆ

Taking the derivative of this expression with respect to λ at λ = 0, we obtain ˜ ωˆ >z1 − < X, ˜ ωˆ >z0 + fX˜ (0) = < X,



˜ ω, < i(X)d ˆ τ˙ˆ (t) > dt.

(2.223)

J

Stating that the path is an extremum for (2.221) among the set of differentiable paths whose projections of endpoints on M, x0 , x1 , are fixed implies that f ˜ (0) = 0 X for all vector fields X with (2.218). Now since ωˆ = π ∗ ω, (2.218) implies that ˜ ωˆ >zi = < π∗ (X), ˜ ω >xi = < X, ω >xi = 0, whence < X, 

< i(τ˙ˆ (t))d ω, ˆ X˜ > dt = −

J



˜ ω, < i(X)d ˆ τ˙ˆ (t) > dt = 0,

(2.224)

J

for all vector fields X on M satisfying (2.218). Now with A = τ˙ˆ (t), we have the relation ˜ > = < dω, (π∗ A, π∗ X) ˜ >. < i(A)d ω, ˆ X˜ > = < π ∗ dω, (A, X) ˜ >. Then with X = π∗ X, ˜ τ˙ (t) = Thus < i(A)d ω, ˆ X˜ > = < i(π∗ A)dω, π∗ X) ˙ π∗ τˆ (t), (2.224) becomes  < i(τ˙ (t))dω, X > dt = 0, (2.225) J

for all vector fields X on M satisfying (2.218). We deduce the relation [Mall, ch. IV, 2.4] i(τ˙ (t))dω = 0,

and thus i(τ˙ˆ (t)) d ωˆ = 0,

∀t ∈ J.

(2.226)

Now we study the following case, in which ωˆ is not the pullback by π of a differential form on M.

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2.5.6 Euler–Cartan Equation and Some Other Extrema Let F be a metric on a space T M defined from a Riemannian metric g on M as F (v) = g(v, v)1/2 ,

v ∈ T M.

(2.227)

Thus F is positively homogeneous, that is, F (av) = aF (v), ∀v ∈ Tx M, a > 0. Consider the following submanifolds43 of T M: T 0 M = {(x, v) ∈ T M, v = 0} ,

SM = {(x, v) ∈ T M, |v| = 1}.

(2.228)

Let ωˆ be the Hilbert differential form on T 0 M, defined from the metric F on T 0 M by < ωˆ v , > = lim

→0

1 {F (v + π∗ ( )) − F (v)]} , ∀ ∈ Tv (T 0 M), v ∈ Tx0 M. 

Thus with α given by (2.60), (2.62), we have the relation ωˆ v =

1 αv , F (v)

v ∈ T 0 M,

(2.229)

whence on SM, ωˆ = α. Let h be the Hamiltonian map defined from T 0 M into T ∗ M by p = hv =

1 Gv ∈ T ∗ M; F (v)

(2.230)

thus |p| = 1, and h(av) = h(v), ∀a ∈ R∗+ . The Hilbert form on T 0 M is the pullback by h of the canonical form θ on T ∗ M: ωˆ v = (h∗ θ )v = θp ◦ h∗ = p ◦ π∗ h∗ = p ◦ π˜ ∗ = t π∗ p,

with p = hv,

(2.231)

with π˜ the canonical projection from T ∗ M onto M. Let A be a unit geodesic field on M; ωA = GA is a differential form on M, and it is identified with the map ψ : x ∈ M → (ωA )x ∈ Tx∗ M, so that the pullback of θ by this map is the differential form ωA : ψ ∗ θ = ωA . The vector field A is identified with the map ϕ : x ∈ M → Ax ∈ Tx M. Then ϕ ∗ ωˆ = ωA , i.e., ωA is the pullback of the Hilbert form ωˆ by the map ϕ. Indeed, < ϕ ∗ ω, ˆ X > = < ω, ˆ ϕ∗ X > = < hv, π∗ ϕ∗ X > = < hv, X > = < ωA , X > .

43 See

[Mall, ch. IV, 2.2].

2.5 Paths, Curves, and Geodesics

229

Let γ be an oriented curve in M, with origin x0 , and with end x1 . Then let τ : t ∈ J = [0, T ] → x(t) ∈ M be a parametric representation of γ , and let τ˙ (t) = (x(t), x(t)) ˙ = (x(t), v(t)), and π∗ τ¨ (t) = τ˙ (t); here the extremum problem is relative to the functional    I (τ˙ ) = ωˆ = < ωˆ v , τ¨ (t) > dt = < t π∗ (hvt ), τ¨ (t) > dt, τ˙

J

J

with (2.231), and thus    I (τ˙ ) = ωˆ = < hvt , τ˙ (t) > dt = |vt |dt. τ˙

J

(2.232)

J

Notice that < ωˆ v , τ¨ (t) > is positive for all τ , and I (τ˙ ) is still the length of the curve τ from x0 to x1 . The extremum condition of I (τ˙ ) for τ is expressed by (2.224). We will prove that the relation (2.226) is still satisfied. The tangent vector (t) = (v(t), v(t)) ˙ to T γ at τ˙ (t) = (x(t), v(t)), is a jet. Lemma 3 For all jet sections , the differential form i( ) d ωˆ is horizontal;44 there exists a covector zt ∈ Tx∗t (M) such that i( ) d ωˆ vt = t π∗ (zt ). Proof Since (t) is a jet for all t, is written = S(v) + ve , with S the spray (see (2.71)) (S(v) is also a jet), and ve is a vertical vector. Moreover, from (2.229), we have, with F (v) = (E(v))1/2 , d ωˆ v = d(

1 1 ) ∧ αv + dαv F (v) F (v)

1 1 dαv . = − E(v))−3/2 dE(v) ∧ αv + 2 F (v) Using (2.71)) and (2.70), we have < , αv > = < S(v), αv > + < ve , αv >= E(v), 1 i( )dαv = i(S)dαv + i( ve )dαv = − dE(v) + i( ve )dαv . 2 Then the interior product of d ωˆ v by = S(v) + ve is i( )d ωˆ v = < , d( = < , d(

44 Or

1 1 1 ) > αv − < , αv > d( )+ i( )dαv F (v) F (v) F (v) 1 ) > αv + i( ve )dαv . F (v)

basic (see [Mall, ch. IV, 1.4]).

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Now these two last terms are horizontal, see (2.70) (2.62), and thus there exists a covector zt ∈ T ∗ M such that i( )d ωˆ v = zt ◦ π∗ . ¶  Hence, as in the previous case, we obtain J < zt , X > dt = 0, for all vector fields X on M satisfying (2.218), and then we verify that zt = 0, and thus i(τ¨ (t))d ωˆ v = 0. We obtain the following result. Proposition 2 Every extremum τ˙0 of I (τ˙ ) defined by (2.232) among the set of lifts on T M of oriented curves of Cxk0 ,x1 (M) satisfies the Euler–Cartan equation i( 0 (t)) d ωˆ = 0,

0 (t) = τ¨ (t), ∀t ∈ J.

(2.233)

This extremum problem is still associated with the minimization problem relative to the distance between two fixed points x0 , x1 of M. Extremum Among Paths in V = T ∗ M The extremum problem among paths is naturally posed in a phase space T ∗ M thanks to the existence of the canonical form θ on T ∗ M. The extremum problem is also posed among the set of lifts in V = T ∗ M of curves in M. (i) Lifts of geodesics in T ∗ M. With the functional (2.232) and (2.231), we have     ωˆ = h∗ θ = θ= θ. (2.234) I (τ˙ ) = τ˙

τ˙

h◦τ˙

τˆ

Let τˆ be a path in T ∗ M, and let τˆλ = Uˆ λ τˆ be the path obtained by applying Uˆ λ on τˆ . With hypotheses (2.218), we are led to the relations (2.222), (2.223), changing X˜ for the corresponding field Xˆ on T ∗ M. The Hamiltonian map h is a diffeomorphism from SM onto h(T 0 M) = h(SM). Then we can transport the Euler–Cartan equation (2.233) by h. Thus the extremum condition I (τˆ ) for a path τˆ in T ∗ M among the set of paths whose endpoints’ projections x0 , x1 ∈ M are fixed still reads i(XP ) dθ |h(SM) = 0,

with XP =

d τˆ . dt

(2.235)

(ii) Hamilton–Cartan equation for a conservative system. Let H = Ekin + U be a time-independent Hamiltonian on T ∗ M. Let Zx0 ,x1 be the set of lifts τˆ in T ∗ M of curves in Cxk0 ,x1 (M).  We will express the extremum condition relative to the integral I (τˆ ) = τˆ θ of the canonical form θ of T ∗ M on paths τˆ ∈ Zx0 ,x1 . dp ˙ ∈ T(x(t ),p(t ))(T ∗ M), v = x˙ = dx Let XP (t) = (v(t), p(t)) dt , and p˙ = dt . Let τ (J ) be a path in Cxk0 ,x1 (M); we have

2.5 Paths, Curves, and Geodesics

 I (τˆ ) =

 τˆ

231



θ=

< θ, XP (t) > dt = J

< p(t), v(t) > dt.

(2.236)

J

Proposition 3 The requirement that τˆ 0 be an extremum of I (τˆ ) defined by (2.236) among the set of paths τˆ ∈ Zx0 ,x1 is given by the Hamilton–Cartan equation i(XP0 (t))dθ = 0,

∀t ∈ J,

(2.237)

with XP0 (t) = (v 0 (t), p˙ 0 (t)) ∈ Tz0 (t ) τˆ 0 ⊂ Tz0 (t )(T ∗ M), z0 (t) = (x 0 (t), p0 (t)), and then τˆ 0 is contained in the manifold E = {(x, p) ∈ T ∗ M, H (x, p) = E} if there exists t0 ∈ J such that τˆ 0 (t0 ) ∈ E . Proof (i) The proof of equation (2.237) is similar to that of (2.226). Let X be a vector field on M such that Xx0 = Xx1 = 0 (see (2.218)), generating a transformation group (Uλ ) of M, with lifts Xˆ and (Uˆ λ ) in T ∗ M. Let τλ = Uλ (τ ), and τˆλ = Uˆ λ (τˆ ). (τˆλ ) The condition that τˆ is an extremum of I (τˆλ ) is dIdλ |λ=0 = 0, which is also  given by τˆ LXˆ θ = 0. Now at each end of the curve τˆ 0 , thus for j = 0, 1, we have ˆ θ >pj = < π∗ X, ˆ pj = < X, ˆ pj > = < Xxj , pj > = 0. i(X)θ Thus  τˆ

 τˆ

ˆ i(X)dθ = 0. But

ˆ i(X)dθ =

 J

ˆ < i(X)dθ, XP0 (t) > dt = −

 J

< i(XP0 (t))dθ, Xˆ > dt,

 whence for all X with (2.218), we have J < i(XP0 )dθ, Xˆ > = 0, which gives (2.237). (ii) Let XH be the Hamiltonian vector field. Then i(XH )dθ = −dH , and thus < dθ, XP0 ∧ XH > = − < i(XH )dθ, XP0 > = < i(XP0 (t))dθ, XH > = 0. Thus we have obtained < dH, XP0 > = 0, which is the last part of the proposition. ¶ Since XP0 (t) is an mG-jet, i.e., p0 (t) = mGv 0 (t), ∀t ∈ J , it follows that I (τˆ ) is positive, which allows the extremum to be a minimum. Extremum Among Lifts of Paths in R × (T M) and in T ∗ (R × M) Let τ be a path in R × M such that τ (t) = (t, x(t)), and let τ˙ (t) = (t, x(t), x(t)) ˙ and τˆ (t) = (t, x(t), p(t)) with p(t) = mGx(t) ˙ be the lifts of τ in R × T M and

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in R × T ∗ M. We first express the extremum condition for the Lagrangian: I (τ ) = T ˙ dt on the set of paths τ such that τ (0) = x0 , τ (T ) = x1 . Let X˜ 0 L(t, x(t), x(t)) be the lift in T (R×T M)) of a vector field X generating a transformation group (Uλ ) on M. Applying the group to the path τ gives the path τλ (with the same variable T t), and then τ˙λ (t) = (t, xλ (t), x˙λ (t)), whence I (τλ ) = 0 L(t, xλ (t), x˙λ (t)) dt = T dI 0 L(τ˙λ (t)) dt. Then the extremum condition is dλ (τλ )|λ=0 = 0, thus with v(t) = x(t): ˙ 

T 0

j

[

j

∂L dv ∂L dxλ + j . λ ] dt|λ=0 = . j ∂x dλ ∂v dλ dx



T 0

[

∂L d ∂L j ∂L − ].a dt + [ j a j ]T0 = 0, j j ∂x dt ∂v ∂v

j

∂L j T 45 Thus the with a j = dλλ |λ=0 , by integration by parts, and with [ ∂v j a ]0 = 0. extremum must satisfy the Euler–Lagrange equations (2.14). Recall that the Hamiltonian is obtained from the Lagrangian by the Legendre transformation, giving the relation L(t, x, v) = p.v − H (t, x, p), and thus

 0

T



T

L(t, x(t), x(t)) ˙ dt =

(p(t).v(t) − H (t, x(t), p(t)))dt.

0

Let θ˜ = θ − Edt be the differential form on T ∗ (R × M), and  the submanifold of T ∗ (R × M) such that E − H (t,x, p) = 0. To write that I (τ ) is an extremum is equivalent to writing that I (τ ) = τˆ θ˜ is an extremum, which gives the Hamilton equations (2.40). We can also consider the path τˆ in T ∗ (R × M) such that τ˜ (t) = (t, x(t);E(t), p(t)) with E(t) − H (t, x(t), p(t)) = 0, and express that the integral I (τˆ ) = τˆ θ˜ is an extremum, which gives equation (2.38).

2.6 Some Simple Examples in Mechanics 2.6.1 Mechanics Problem on a Circle Definition of Bundles T S, T ∗ S, T (T S), T (T ∗ S) with S ⊂ R 2 Let S = Sr be the circle with radius r in the Euclidean space E2 identified with R 2 ; thus S is defined, with x = (x 1 , x 2 ), by φ(x) = g(x, x) = (x 1 )2 + (x 2 )2 = r 2 . The tangent bundle T S is defined by   T S = (x, v), x ∈ S, v ∈ R 2 , φ  (x)(v) = 2x.v = 0 .

45 Since

the endpoints are fixed. We would also obtain this result thanks to the Lie derivative LX˜ T ˜ along the vector field X˜ by 0 LX˜ L(τ˙λ (t)) dt = 0 with LX˜ L = X(L).

2.6 Some Simple Examples in Mechanics

233

The “bitangent” space T (T S) is defined by   T (T S) = ((x, v), (x, ˙ v)), ˙ x ∈ S, v, x, ˙ v˙ ∈ R 2 , x.v = 0, x.x˙ = 0, x.v ˙ + x.v˙ = 0 , which is obtained taking the derivative of the expression 2φ  (x)(v) = 2x.v = 0. Thus we see that the space T (T S) is the 4-dimensional submanifold of T (T R 2 ) (identified with (R 2 )4 ). The tangent space Tx,v (T S) at (x, v) is thus defined by   Tx,v (T S) = (x, ˙ v) ˙ ∈ R 2 × R 2 , x.x˙ = 0, x.v ˙ + x.v˙ = 0 . The tangent space of T S at (x, v) splits into Tx,v (T S) = Qx,v ⊕ Hx,v , with (i) the vertical space Qx,v defined by   ˙ ∈ R 2 × R 2 , x.v˙ = 0 , Qx,v = (0, v) (ii) the horizontal space Hx,v defined by 

Hx,v

 2 |v| = (x, ˙ v) ˙ ∈ R 2 × R 2 , (x, ˙ v) ˙ = λ(v, − 2 .x), λ ∈ R . r

Use of the Internal Variable ϕ The circle is previously viewed as a submanifold of R 2 . It is advisable to use an angle variable ϕ, especially for generalization (see [Kob-Nom, Vol. 2, ch. VII, 3]). With r = 1, we define x = α = αϕ = (cos ϕ, sin ϕ) ∈ S1 ,

τ = τϕ = (− sin ϕ, cos ϕ) = ∂αϕ /∂ϕ.

The evolution of position, velocity, and acceleration of a material point on the circle Sr is given by the time-dependent vector functions, with αϕ(t ) and τϕ(t ) denoted by α(t) and τ (t), x(t) = rα(t) = r(cos ϕ(t), sin ϕ(t)), x(t) ˙ = v(t) = r α(t) ˙ = r ϕ(t)τ ˙ (t) = r ϕ(t)(− ˙ sin ϕ, cos ϕ),

(2.238)

2 x(t) ¨ = v(t) ˙ = r[ϕ(t)τ ¨ (t) + ϕ(t) ˙ τ˙ (t)] = r[ϕ(t)τ ¨ (t) − (ϕ(t)) ˙ α(t)].

We can write the acceleration as x(t) ¨ = v(t) ˙ = r ϕ(t)τ ¨ (t) −

|v(t)|2 .α(t). r

(2.239)

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The metric g = dr 2 + r 2 dϕ 2 of the space R∗2 induces the metric g|Sr = r 2 dϕ 2 on the circle Sr . Therefore, for a point of the circle given by s = rϕ and for the velocity v = r ϕ, ˙ we have g|Sr (v, v) = r 2 (ϕ) ˙ 2. The covariant derivative of the velocity in R 2 (respectively Sr , a submanifold of 2  v (respectively ∇ v), we obviously have ∇  v = v˙ R ) being here denoted by ∇t,v t,v t,v and ∇t,v v = r ϕτ ¨ . Then the Gauss formula is  ∇t,v v − ∇t,v v = −

|v|2 .α, r

(2.240)

this last term is orthogonal to Sr (it belongs to the normal bundle T (Sr )⊥ of Sr ). Notice the relations with the metric g in R 2 : g(α, .) = Gα = dr,

g(τϕ , .) = Gτϕ = rdϕ.

Force. Let F be a differential form modelling a given force field in R 2 on a material point, with the constraint to move on the circle Sr by an a priori unknown link force Rr dr; F is expressed by F = f1 dx 1 + f2 dx 2 = Fr dr + Fϕ rdϕ, with Fr = f1 cos ϕ + f2 sin ϕ,

Fϕ = −f1 sin ϕ + f2 cos ϕ.

Let J be the canonical injection of Sr into R 2 ; the force field F induces a force field Fτ = J ∗ F = Fϕ rdϕ on the circle Sr . If F is the derivative of a potential, i.e., F = −dU , then Fτ = J ∗ F = −d(U ◦ J ) is the derivative of a potential too. The fundamental equation (in Sr ) is mG∇t,v v = Fϕ rdϕ,

thus mϕ¨ = Fϕ .

(2.241)

In order that the material point move on the circle Sr , it is necessary that the normal component Fr of the force F , and the link force Rr , be such that Rr + Fr + m

|v|2 = 0. r

(2.242)

Indeed, we have  m∇t,v Gv = F + Rr dr = (Fr + Rr )dr + Fϕ rdϕ = (Fr + Rr )dr + mG∇t,v v,

2.6 Some Simple Examples in Mechanics

235

which gives (2.242), using (2.240). We thus obtain the link force Rr . The term 2 m |v|r .α is called the centrifugal force. Example of the Pendulum In the case of uniform gravity directed along 1 in the canonical frame (1 , 2 ) of R 2 , this force field is given by F = mgdx 1 ,

(2.243)

g being the positive gravitational constant.46 Then F = −d U,

U (x) = U (x 1 , x 2 ) = −mgx 1 = −mgr cos ϕ.

The force field Fτ = J ∗ F is also the derivative of a potential: Fτ = −J ∗ dU = −dU ◦ J = −mgr sin ϕdϕ. Then the equation of motion of the pendulum of mass m submitted to the gravitational force field is mv˙ d(rϕ) = mr ϕ¨ d(rϕ) = −mgr sin ϕ dϕ, and thus r ϕ¨ + g sin ϕ = 0.

(2.244)

Thus the system depends on a Hamiltonian H , which is (on the circle Sr ) H (ϕ, ϕ) ˙ =

1 2 2 mr ϕ˙ − mgr cos ϕ. 2

(2.245)

The energy conservation of the system is given by the first integral 1 2 1 r ϕ˙ − g cos ϕ = r ϕ˙ 02 − g cos ϕ0 = constant, 2 2

(2.246)

where (ϕ0 , ϕ˙ 0 ) are the initial conditions. This relation is equivalent to equation (2.244), since this is a 1-dimensional problem. Then the normal component of the gravitational force is Fr = mg cos ϕ, and the link force must be such that Rr + mg cos ϕ + m

|v|2 = 0, r

46 Here we assume that the gravitational force is in the plane of the circle. If that is not the case, we have to change the term mg to mg sin ϕ0 with ϕ0 the angle of the gravitational force with the normal to the plane.

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2 Classical Mechanics

hence with (2.246) and C0 = − m2 |vr0 | + mg cos ϕ0 , C1 = −m |vr0 | − 2mg cos ϕ0 , 2

3 |v|2 + C0 = −3mg cos ϕ + C1 . Rr = − m 2 r

2

¶

2.6.2 Two Material Points with a Link Tangent and Bitangent Configuration Spaces Consider an elementary system with two material points in the Euclidean space E3 (identified with R 3 ), each with position xi ∈ R 3 , mass mi , and with the constraint to be at a fixed distance from each other (for instance, they are linked with a rod of ˜ of the negligible mass). The configuration space (here denoted by M instead of M) system is given by (with fixed r > 0)   M = (x1 , x2 ) ∈ R 3 × R 3 , |x1 − x2 | = r . This is a 5-dimensional submanifold of R 3 × R 3 , which can be identified with the product R 3 × S 2 , using the coordinates of the center of mass xG = (m1 x1 + m2 x2 )/(m1 + m2) and the direction α of the vector x1 − x2 . This example looks like a generalization of the previous example. We also define the tangent bundle and the bitangent bundle. The tangent bundle T M is defined as a submanifold of T (R 3 × R 3 ) (identified with the product (R 3 × R 3 ) × (R 3 × R 3 )) by  TM =

 ((x1 , x2 ), (v1 , v2 )) ∈ (R 3 × R 3 ) × (R 3 × R 3 ), |x1 − x2 | = r, . (x1 − x2 ).(v1 − v2 ) = 0

This is verified as in the previous example, by deriving the expression φ(x) = φ(x1 , x2 ) = (x1 − x2 )2 − r 2 . The tangent bundle T M is also identified with T (R 3 ) ⊕ T S 2 (using the coordinates (xG , α)). The tangent vector space at (x, v) ∈ T M to the tangent bundle T M is defined by  Tx,v (T M) =

 (w, γ ) = ((w1 , w2 ), (γ1 , γ2 )), (x1 − x2 ).(w1 − w2 ) = 0, , (w1 − w2 ).(v1 − v2 ) + (x1 − x2 ).(γ1 − γ2 ) = 0,

with xi , vi , wi , γi ∈ R 3 , i = 1, 2. (This is verified by the differential of the expression ψ(x, v) = dφ(x).v = 2(x1 − x2 ).(v1 − v2 ).)

2.6 Some Simple Examples in Mechanics

237

The set of jets in this space is thus the set of pairs (w, γ ) such that w = v satisfying (v1 − v2 )2 + (x1 − x2 )(γ1 − γ2 ) = 0. Using the coordinates (xG , α) and denoting by ((xG , vG ), (α, v)) an element of T M, we can identify the tangent space Tx,v (T M) with the product T((xG ,vG ),(α,v))(T M) = T(xG ,vG ) (T R 3 ) × T(α,v)(T S 2 ),   T(α,v)(T S 2 ) = (α, ˙ v) ˙ ∈ R 3 × R 3 , α.α˙ = 0, α.v ˙ + α.v˙ = 0 , with |α| = 1, α.v = 0. We will use the variables (xG , α), with xG for the position of the center of mass, 2 and α = x1 −x ∈ S 2 to specify the relative position of the two material points with r positions x1 , x2 . These variables are linked to (x1 , x2 ) by mxG = m1 x1 + m2 x2 ,

rα = x1 − x2 , m = m1 + m2 .

(2.247)

Let 1 1 1 m1 m2 = , m2 = . + , and m1 = μ m1 m2 m m Thus μ =

m1 m2 m ,

and m1 m2 = m2 m1 = μ. Then the inverse formulas of (2.247) are x1 = xG + rm2 α, x2 = xG − rm1 α.

(2.248)

Deriving the formulas (2.247), we have mvG = m1 v1 + m2 v2 ,

v = dα/dt = (v1 − v2 )/r,

(2.249)

and thus v1 = vG + m2 rv, v2 = vG − m1 rv.

(2.250)

Then twice the kinetic energy of the system, 2E(vt ot ), is given by 2 2E(vt ot ) = m1 v12 + m2 v22 = mvG + μr 2 v 2 .

Thus the Riemannian metric of the unit sphere must be multiplied by μr 2 . Force field. Let fi (xi ) be a given exterior force field in R 3 on the material point of position xi with i = 1, 2. Then the corresponding force field F on the system, F = f1 dx1 + f2 dx2 , is split, with (2.248), into F = FG + FS 2 + FS⊥2 = fG dxG + rfS 2 .dα + fr dr,

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with fG = (f1 + f2 ),

fS 2 = m2 f1 − m1 f2 ,

fr = fS 2 .α,

fS 2 .α, which corresponds to the orthogonal projection of fS 2 in R 3 onto the sphere Sr , and fr is the component of fS 2 that is normal to the sphere. Use of Internal Coordinates ϕ, θ of the Unit Sphere We use a chart of the unit sphere with spherical coordinates (ϕ, θ ) such that 0 < ϕ < 2π, 0 < θ < π, given by α = αϕ,θ = (cos ϕ sin θ, sin ϕ sin θ, cos θ ).

(2.251)

On taking the derivative with respect to these variables ϕ, θ , we obtain a basis of Tα S 2 : e1 = eϕ =

∂α = sin θ (− sin ϕ, cos ϕ, 0), ∂ϕ

∂α e2 = eθ = = (cos ϕ cos θ, sin ϕ cos θ, − sin θ ). ∂θ We denote these vectors simply by eϕ =

∂ ∂ϕ

and eθ =

∂ ∂θ . The vector α

(2.252)

is orthogonal

∂ to the sphere and is also denoted by e3 , ξ , or ∂r . The Riemannian metric of the unit sphere induced by the metric of R 3 is given by

gϕ,θ = dα.dα = sin2 θ dϕ 2 + dθ 2 , the metric of R∗3 in spherical coordinates being given by gr,ϕ,θ = dr 2 + r 2 (sin2 θ dϕ 2 + dθ 2 ). The vector eθ has unit norm with this metric, but not the vector eϕ . Thus we define an orthonormal basis of Tx (S 2 ) by47 e1 =

1 1 ∂ e1 = , sin θ sin θ ∂ϕ

e2 = e2 =

∂ . ∂θ

that the frame (e1 , e2 , α) (with orientation of the normal going out of the sphere) has the reversed orientation.

47 Note

2.6 Some Simple Examples in Mechanics

239

Then (e1 , e2 , e3 ) is an orthonormal basis of Tx (R 3 ) with x = α. We also define the forms σ1 = σϕ = sin θ dϕ and σ2 = σθ = dθ , giving a dual basis of (e1 , e2 ), σi (ej ) = δij , and then the Riemannian metric is gϕ,θ = σϕ ⊗ σϕ + σθ ⊗ σθ .

Covariant Derivative on the Sphere The covariant derivative on S 2 relative to this Riemannian structure is simply obtained from the covariant derivative in R 3 using the fact that S 2 is a submanifold  of R 3 , with induced metric. Let ∇  be the covariant derivative in R 3 ; ∇∂/∂ϕ ej and  ∇∂/∂θ ej are also the derivatives of ej along ϕ and θ . We thus obtain ∇e 1 e1 =

∂ 2α ∂e1 = = − sin θ (cos ϕ, sin ϕ, 0), ∂ϕ ∂ϕ 2

∇e 1 e2 = ∇e 2 e1 = ∇e 2 e2 =

∂e2 ∂ 2α ∂e1 = = = cotg θ e1 , ∂θ ∂ϕ ∂ϕ∂θ

∂ 2α ∂e2 = = −α. ∂θ ∂θ 2

Note that α sin θ + e2 cos θ = (cos ϕ, sin ϕ, 0). Then the covariant derivatives in S 2 are simply 

∇e 1 e1 = − sin θ cos θ e2 − sin2 θ α, ∇e 1 e2 = cotg θ e1 , ∇e 2 e1 = cotg θ e1 , ∇e 2 e2 = −α,

and the Christoffel symbols of the sphere in the frame (e1 , e2 ) (see (2.519)) are 2 11 = − sin θ cos θ,

1 1 12 = 21 = cotg θ,

the others ijk being null. Thus, setting x 1 = ϕ, x 2 = θ , with i, j = 1, 2, we have (see [Kob-Nom, Vol. 2, ch. VII, 1.3, p. 18]) ∇e i ej =

 ∂ej k = i,j ek + hi,j ξ, i ∂x

with ξ = α, h11 = − sin2 θ, h12 = h21 = 0, h22 = −1. Let v(t) =

∂α ∂α dα = ϕ˙ + θ˙ = e1 ϕ˙ + e2 θ˙ ∈ Tx S 2 . dt ∂ϕ ∂θ

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Then the covariant derivative of the velocity (for S 2 ) is given by (∇t,v v)i =

dv i  i j k + j k v v , dt

 v. We obtain whereas the covariant derivative of the velocity (for R 3 ) is given by ∇t,v   ˙  ∇t,v v =(ϕe ¨ 1 + θ¨e2 ) + (ϕ∇ ˙ v(t ) e1 + θ ∇v(t ) e2 ),

= ϕe ¨ 1 + θ¨e2 + ϕ˙ 2 [− sin θ cos θ e2 − sin2 θ ξ ] + 2ϕ˙ θ˙ cotg θ e1 − θ˙ 2 ξ. Thus  ∇t,v v = e1 [ϕ¨ + 2ϕ˙ θ˙ cotg θ ] + e2 [θ¨ − ϕ˙ 2 sin θ cos θ ] − [θ˙ 2 + ϕ˙ 2 sin2 θ ]ξ,

= ∇t,v v − [θ˙ 2 + ϕ˙ 2 sin2 θ ] ξ, which is the Gauss formula, and the second fundamental form is given by h(v, v) = θ˙ 2 + ϕ˙ 2 sin2 θ = |v|2 = g(v, v). Remark 18 Use of the orthonormal frame (see [Gilk, ch. II, 2, p. 102]). We can also use the orthonormal frame (e1 , e2 ) to obtain the covariant derivative of the velocity. In a Euclidean space, the covariant differentials ∇ej are identified with the differentials dej . Then we obtain the covariant derivatives on the sphere by projection onto T S 2 : ∇t,v v = v˙ 1 e1 + v˙ 2 e2 − cotg θ v 1 (v 1 e2 − v 2 e1 ),

(2.253)

or (∇t,v v)1 = (∇t,v v)ϕ = v˙ 1 + cotg θ v 1 v 2 ,

(2.254)

(∇t,v v)2 = (∇t,v v)θ = v˙ 2 − cotg θ v 1 v 1 .

 v), is such that The covariant derivative of the velocity in R 3 , denoted by (∇t,v (Gauss formula)  ∇t,v v = ∇t,v v − [(v 1 )2 + (v 2 )2 ]e3 .

Note that we thus obtain ∇ej =  ω=

ω11 ω21 ω12 ω22



 =



ωji ei with the connection form

0 −cotg θ σ1 cotg θ σ1 0



 =

0 −cos θ dϕ cos θ dϕ 0

.¶

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241

Remark 19 The great circles of the sphere are geodesics; indeed, the path θ → αϕ,θ ∈ S 2 with fixed ϕ is also the canonical parametrization, ds = dθ , and the velocity vector v is given by v = dx/dt = ω.dα/dθ = ω.e2 , with ω = dθ/dt = ds/dt constant. Thus v 1 = v ϕ = 0, v 2 = v θ = ω, v˙ = 0, and Qv (x, ˙ v) ˙ = ∇v v = 0, which proves that the path is a geodesic.¶ We define the momenta by pG = p1 + p2 = (m1 v1 + m2 v2 ) = mvG , (with components) p = μr 2 gϕ,θ (v, .) = μr 2 Gϕ,θ v = μr 2 (ϕ˙ sin2 θ dϕ + θ˙ dθ ).

(2.255)

Evolution Equations The equations of mechanics on the momenta are (i)

dpG = fG , dt

(ii) ∇t,v p = FS 2 .

(2.256)

∂ ∂ ) = μr 2 sin2 θ dϕ, G( ∂θ ) = μr 2 dθ, we have Since G( ∂ϕ

∇t,v p = G(∇t,v v) = μr 2 sin2 θ [ϕ¨ + 2ϕ˙ θ˙ cotg θ ] dϕ + μr 2 [θ¨ − ϕ˙ 2 sin θ cos θ ] dθ = (fS 2 .e1 ) dϕ + (fS 2 .e2 ) dθ. Thus we have the equations (i) (∇t,v p)ϕ = μr 2 [sin2 θ ϕ¨ + 2ϕ˙ θ˙ sin θ cos θ ] = fS 2 .e1 , (ii) (∇t,v p)θ = μr 2 [θ¨ − ϕ˙ 2 sin θ cos θ ] = fS 2 .e2 .

(2.257)

Furthermore, as in the example of motion of a material point on the circle, from the Gauss formula, the link force Rr between the two material points must be Rr + fr + μ

v2 = 0. r

Power balance. Then the power balance of the system (see (2.76)) reads v2 d v2 (m G + μ r 2 ) = fG .vG + fS 2 .v dt 2 2 = (f1 + f2 ).vG + r(m2 f1 − m1 f2 ).v.

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Remark 20 “Euclidean” presentation according to the d’Alembert principle. Let Ri be the link force at xi ; we can define Ri from Rr dr by Ri = < Rr dr,

∂ >. ∂xi

Note that dr = α.dx1 −α.dx2; we have < dr, ∂x∂ 1 > = α, < dr, ∂x∂ 2 > = −α, thus R1 = < Rr dr, ∂x∂ 1 > = αRr , R2 = < Rr dr, ∂x∂ 2 > = −αRr , and thus we verify the relation R1 + R2 = 0. In the following we adopt a more traditional presentation of the link forces. The fundamental equation of mechanics in R 3 for each material point is mi

d 2 xi = fi + Ri , dt 2

i = 1, 2,

(2.258)

2

or also mi x¨i − fi = Ri , with x¨i = ddtx2i . The elimination of the link forces is usually by projection of these equations on the tangent space to the configuration space, using the d’Alembert principle: 

(mi x¨i − fi )vi = 0,

∀(v1 , v2 ) ∈ Tx M.

(2.259)

This also means that the work of link forces is null, or that the pair of link forces R = (R1 , R2 ) is such that R1 v1 + R2 v2 = 0, for all (v1 , v2 ) ∈ Tx M ⊂ R 3 × R 3 , i.e., satisfying (x1 − x2 )(v1 − v2 ) = 0. We deduce that there exists a real number λ (dependent on x and v) such that R1 = −R2 = λ(x1 − x2 ) = λrα. The d’Alembert principle, which states that the link forces do not work, may also be written as WR |T M = R1 dx1 + R2 dx2 = λ(x1 − x2 )dx1 − λ(x1 − x2 )dx2 =

λ d(x1 − x2 )2 = 0. 2

Using this property allows us to obtain the equation of motion. Adding the two equations (2.258), we obtain the center of mass equation m

d 2 xG = fG . dt 2

(2.260)

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243

We obtain the equation for the direction of the straight line x1 − x2 by dividing each equation (2.258) by mi and then subtracting the relation for i = 2 from the relation for i = 1, and finally projecting onto the tangent space to the sphere: pτ (

d 2α 1 1 1 1 pτ fS 2 , ) = pτ ( f1 − f2 ) = dt 2 r m1 m2 μr

(2.261)

where pτ is the projection that is defined from the frame (ei ) by pτ (y) =  (y, ei ) ei . Thus, we substitute the vector equations (2.260) and (2.261) with known right-hand side into the two vector equations (2.258) with unknown link forces. The pair of “exterior” forces (f1 , f2 ) has been changed into the pair of forces (fG , fS 2 ) following the law of transformation of momenta (2.255). When the force fields fi are the derivative of a potential Ui , with i = 1, 2, and are time-independent, then the energy of the system is preserved (the system is conservative), the Hamiltonian H being constant: H (x, p) = T + U =

 i

(

1 2 p + Ui (xi )) = constant. ¶ 2mi i

Remark 21 The generalization of the system to a finite number of material points, with the constraint to be at fixed distances from one another, allows one to deal with a situation in which the configuration space is identifiable with that of the rigid body, which is the displacement group SO(3) × R 3 . This is the case with a system with three unaligned points. The framework of differential geometry used here is also that of an axially symmetric rigid body, the main difference being the “inertia tensor,” which is due to the mass distribution. ¶ Remark 22 It is interesting to compare this situation with that of two material points that are not necessarily at a fixed distance from each other, and with an interaction force f (x1 , x2 ) depending on the distance only (for instance due to an elastic thread), f (x1 , x2 ) =

x1 − x2 φ(|x1 − x2 |), |x1 − x2 |

with φ(z) = k(z − r) when z ≥ r, and φ(z) = 0 otherwise. Then this force is a potential derivative f (x) = grad U (x), with U (x) = k2 (|x| − r)2 if |x| ≥ r, and U (x) = 0 otherwise. This potential expresses the internal energy of the system. ¶ Remark 23 On holonomic links. Let M be a Riemannian manifold with a link family ωλ , λ = 1, . . . , n − p, that builds up independent differential forms on M. They define a differential form ω = (ωλ ), with vector values R n−p , and hence < ω, v > = (< ωλ , v >), ∀v ∈ T M. Let Px be its kernel, Px = ker ωx = ∩ ker(ωλ )x . The link family (ωλ ) is holonomic if P is a (completely) integrable

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(n − p)-field. This is achieved if the Frobenius condition is satisfied (see [Mall, ch. II, 5.4] and the appendix): ker ω ∧ ker ω ⊂ ker dω. Then at each point x ∈ M, there is a p-dimensional submanifold V of M (called a maximal integral manifold) such that Tx V = Px , which defines a foliation of M. The links’ action on the system, modeled by V , is given by a family of functions (R λ ) such that the basic mechanics equation in M is  p=F+ ∇t,v



R λ ωλ ,

(2.262)

 its covariant F being the known force field on the system, p its momentum, and ∇t,v derivative relative to M. Then the basic mechanics equation in the manifold V is

∇t,v p = F |T V .

(2.263)

The difference between the two covariant derivatives is given by the Gauss formula, which allows us to obtain the link forces R λ . If X and Y are vector fields on V ,  Y) let αx (X, Y ) be the normal component of the covariant derivative in M, (∇X x  at x ∈ V , hence (∇X Y )x − (∇X Y )x = αx (X, Y ). If the force field F is split into F = F |T V + F |T V ⊥ , then (2.262) and (2.263) imply Gα(v, v) =



R λ ωλ + F |T V ⊥ ,

(2.264)

which determines the link forces R λ , λ = 1, . . . , n − p, as of v and F . a function The linking forces do not perform any work, since < R λ ωλ , v > = 0, ∀v ∈ T V . Let (x j ), j = 1, . . . , n, be a coordinate system  of M in a neighborhood of a point x ∈ V ; let Rjλ = < ωλ , ∂x∂ j >. Then for all v = v j ∂x∂ j ∈ Tx V , we have < ωλ , v > =



Rjλ v j = 0.

The d’Alembert principle is thus implied. Applications on the previous examples. The case of holonomic links. (i) for the pendulum: n = 2, p = 1, M = R∗2 = ∪r>0 Sr1 , ω1 = dr, and V = Sr1 . (ii) for two material points at a fixed distance, n = 6, p = 5: M = R 3 × ∪r>0 Sr2 ,

ω1 = rdr = (x1 − x2 ).d(x1 − x2 ),

V = R 3 × Sr2 . ¶

2.7 Rigid Bodies

245

2.7 Rigid Bodies 2.7.1 Mechanics Modelling of a Rigid Body Configuration Space. Euclidean Frames A Euclidean space En is an affine space that is identified with an n-dimensional vector space V with a specified point of En chosen as the origin of the vector space. It has a manifold structure with global charts given by frames, and V may be identified with R n by a choice of chart. The space En is equipped with a Euclidean metric, and hence a distance function d(O1 , O2 ) = |a| if τa O1 = O1 +a = O2 , a ∈ V , with a scalar product on V . A rigid body is a system of material points in the Euclidean space E3 such that the distances between the points are fixed in time. Let Mt be a connected domain in E3 occupied, at time t, by the rigid body. Its motion is associated with the motion of a linked Euclidean frame. Thus the modelling of the system leads to the configuration space M˜ = E3 , which is the set of Euclidean frames in E3 . A Euclidean frame u˜ = (u, O), also denoted by u˜ O , in E3 , is defined by giving an origin O ∈ E3 and a direct orthonormal basis of E3 , u = (e1 , e2 , e3 ). We identify u with the linear map x = (x 1 , x 2 , x 3 ) ∈ R 3 → u(x) =



x j ej ∈ E 3 ,

and we identify u˜ with the (affine) isometric map from R 3 into E3 : u(x) ˜ = O + u(x), thus u(0) ˜ = O. Let ˜ = (, 0) = ((1 , 2 , 3 ), 0) be the natural frame of R 3 . Then the Euclidean frame u˜ is the image of ˜ under the affine map u. ˜ We denote by u˜ 0 = (u0 , O0 ), a Euclidean frame that is used as a reference and that transfers the evolution in E3 into R 3 . Since the change of Euclidean frames is made by the displacement group (rotations and translations), we can also use this group as configuration space if we specify a particular frame. We can use the displacement group either as the transformation group D(E3 ) on E3 , which seams natural from the mechanics point of view, or as the transformation group D(3) = D(R 3 ) on R 3 ; in the first case D(E3 ) acts on the left on the frames, while in the second case D(R 3 ) acts on the right independently of the frames. We will develop this point of view. Of course the displacements of D(E3 ) are transformed into displacements of D(3), thanks to a Euclidean frame u˜ 0 . The action of D(3) on the Euclidean frames is given by u˜ ◦ g = (u ◦ R, u(a) + O),

g = (R, a) = τa ◦ R ∈ D(3).

The action of the group D(3), has the following properties.

(2.265)

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2 Classical Mechanics

This action is transitive: for every Euclidean frame u˜ ∈ E3 , there exists a unique displacement g = (R, a) in R 3 that transforms u˜ 0 into u˜ by u˜ = u˜ 0 ◦ g

thus u = u0 ◦ R,

O = O0 + u0 (a).

(2.266)

This action is free: the relation u˜ = u˜ ◦ g for some u˜ implies g = e, the unit of D(3). Thus the space E3 has the structure of a principal affine fibre bundle over E3 , with group D(3). Let u˜ 0 ∈ E3 = (u0 , O0 ) be a reference frame. Then E3 may be identified with the displacement group D(3) through the map φ0 : −−→ u˜ = (u, O) ∈ E3 → g = (u˜ 0 )−1 u˜ = ((u0 )−1 u, (u0 )−1 (O0 O)) ∈ D(3), (2.267) with inverse φ0−1 , so that φ0−1 (g) = u˜ 0 ◦ g. The evolution of the rigid body is given by the evolution of a Euclidean frame u˜ t ∈ E3 (linked to the rigid body) by u˜ t = (ut , Ot ) = u˜ 0 ◦ gt , thus ut = u0 ◦ Rt ,

Ot = O0 + u0 (at ),

(2.268)

with the time-dependent displacement gt = (Rt , at ), g0 = (R0 , a0 ) = (I, 0), and the initial Euclidean frame u˜ 0 = (u0 , O0 ), which may be taken as reference frame, so that we can exchange the evolution in E3 for the evolution in D(3) with gt = (u˜ 0 )−1 u˜ t . Then u˜ 0 is said to be fixed, linked to the space. Euler coordinates. The coordinates in the frame u˜ 0 of a material point of the rigid body in evolution are given by xt = gt (x) = Rt x + at ∈ R 3 .

(2.269)

and the evolution in the Euclidean space E3 is given by xMt = u˜ t (x) = u˜ 0 (gt x) = u˜ 0 (xt ) ∈ E3 .

(2.270)

Then the evolution of the set Mt ⊂ E3 of points of the rigid body is Mt = u˜ 0 (t ) = u˜ t () ⊂ E3 . In the frame u˜ 0 , the coordinates of points of the rigid body, called Euler coordinates, are time-dependent, and they correspond to points in t ⊂ R 3 , the domain at time t. Lagrangian coordinates. Sometimes we exchange the roles of the frames: the frame u˜ is fixed, and the frame u˜ 0 evolves in time, so that gt = (u˜ 0t )−1 u˜ = (u˜ 0 )−1 u˜ t ,

2.7 Rigid Bodies

247

which gives u˜ t = u˜ 0 ◦ gt ,

u˜ 0t = u˜ ◦ gt−1 .

Thus the evolution of the frame u˜ t is forward, whereas the evolution of the frame u˜ 0t is backward. In the frame u, ˜ the coordinates of points of the rigid body, called Lagrangian coordinates, are fixed and correspond to points in the domain  ⊂ R 3 . ¶

2.7.2 Lie Group and Lie Algebra of Displacements Lie Group of Displacements Recall that every displacement preserves the Euclidean metric, and that every direct displacement is the product of a rotation and a translation. Let R be a rotation and let τa be the translation of the vector a. The successive action in R 3 of these two displacements is the displacement g = τa ◦ R, also denoted by (R, a), given for all x ∈ R 3 by gx = τa Rx = R.x + a.

(2.271)

The group D(3), also denoted by SO(3)×R 3 or ASO(3), is the (semidirect) product of the rotation group SO(3) and the translation group R 3 , which is made explicit by the group operation: (R1 , a1 ).(R2 , a2 ) = (R1 R2 , a1 + R1 a2 ).

(2.272)

Thus we have (R, a) = (I, a).(R, 0). The identity element e of the group is e = (I, 0), and the inverse of an element (R, a) is given by (R, a)−1 = (R−1 , −R−1 a).

(2.273)

Thus D(3) is a subgroup of the group A3 = AGL(3) = GL(3) × R 3 of affine transformations of R 3 , the (semidirect) product of the translation group and the group GL(3) of nonsingular real matrices. Remark 24 In a frame where a point O of the rotational axis is given by its coordinates x0 ∈ R 3 , the rotation with respect to the μ-axis is given by Rθμ,x0 = τx0 Rθμ τ−x0 = (Rθμ , x0 − Rθμ x0 ), since for all λ ∈ R, we have g(x0 + λμ) = (x0 + λμ).

¶

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Lie Algebra of Displacements Let us specify the Lie algebra (denoted by d3 ) of the displacement group D(3) by specifying the generators of the translation and rotation groups with one parameter. The translation group with vectors λα, λ ∈ R (for fixed α ∈ R 3 ), such that τλ α a = a + λ α, λ ∈ R, has α as generator. The rotation group (Rθμ ) θ ∈ R, with axis μ ∈ S 2 , which preserves the Euclidean scalar product, has the skew-symmetric matrix Aμ as generator: Aμ x = lim

Rθμ x − x θ

θ→0

= μ × x,

x ∈ R3 ,

(2.274)

with the cross product ×. More generally for μ ∈ R 3 , the action of Aμ is given by ⎛

Aμ x = μ × x ∈ R 3 ,

⎞ 0 −μ3 μ2 thus Aμ = ⎝ μ3 0 −μ1 ⎠ . −μ2 μ1 0

(2.275)

The Lie algebra d3 is identifiable with the vector space M3a × R 3 (where M3a is the space of skew-symmetric matrices of order 3, which is the Lie algebra o(3) of the group SO(3)), also identifiable with the semidirect sum o(3) ⊕ R 3 , and thus d3 = M3a × R 3 = {(Aμ , α),

μ, α ∈ R 3 } = o(3) ⊕ R 3 ,

or with R 3 × R 3 (if we identify (Aμ , α) with (μ, α)), equipped with the bracket operation 

 (μ, α), (μ , α  ) = (μ×μ , μ×α  −μ ×α),

∀(μ, α), (μ , α  ) ∈ d3 .

(2.276)

Thus the structure constants of d3 , with the symbol i,j,k (as in (2.108)) and the basis (i , 0), (0, j ), are [(i , 0), (j , 0)] = i,j,k k ,

[(0, i ), (0, j )] = 0,

[(i , 0), (0, j )] = (0, i,j,k k ) = i,j,k (0, k ).

(2.277)

We emphasize the following properties: o(3) is a subalgebra of d3 , and the translation space R 3 is an ideal of d3 , that is, 

 (μ, α), (0, α  ) = (0, μ × α  ).

The differential operators (by the Lie derivative) corresponding to the translations and rotations (2.274) are given, with α ∈ R 3 and μ ∈ S 2 , by

2.7 Rigid Bodies

249

α.∇f (x) = lim

λ→0

μ.Lf (x) = lim

f (τλ α x) − f (x)  j ∂f = α (x), λ ∂x j f (Rθμ x) − f (x)

θ→0

θ

=

 ∂f (μ × x)j j (x), ∂x

∀f ∈ C k (R 3 ), k ≥ 1. We then deduce the commutation relations [α.∇, μ.L] =



αj

∂ ∂ , (μ × x)i i ∂x j ∂x

 =

 ∂ (μ × α)j j = (μ × α).∇, ∂x

according to the general expression of the bracket (2.276) on d3 . Let the action of D(3) in R 3 be given by the map σx from D(3) into R 3 such that σx (R, a) = Rx + a. Then (σx )∗ is a map from d3 = Te D(3) into Tx R 3 (identified with R 3 by identifying ( ∂x∂ j ) with the canonical basis (j )) of R 3 such that (σx )∗ (μ, α) =



(μ × x + α)j j .

(2.278)

Exponential mapping and frames. The exponential for the rotation group is a mapping from the Lie algebra o(3), thus from R 3 , into the group SO(3), such that with μ ∈ R 3 , Aμ ∈ o(3), we have exp Aμ = Rθν = exp(θ Aν ),

θ = |μ| (mod 2π),

ν=

μ , |μ|

(2.279)

with exp A0 = I for μ = 0. We verify this formula thanks to the Taylor development, with Pν ⊥ the orthogonal projection on the orthogonal space to ν, and with the relations Aμ = |μ|.Aν ,

−(Aν )2 x = x − (x.ν)ν = Pν ⊥ x,

−(Aν )3 x = Aν x. (2.280) With Pν (x) = (ν.x)ν, the orthogonal projection on Rν, we have exp Aμ = I + (1 − cos θ )A2ν + sin θ Aν = I cos θ + (1 − cos θ )Pν + sin θ Aν .

(2.281)

In order to eliminate the periodicity, we can restrict the exponential to the ball Bπ = {μ ∈ R 3 , θ = |μ| ≤ π} = [0, π] × S 2 . From (2.281), we see that (θ, ν) and (π − θ, −ν) give the same rotation exp Aμ . This corresponds to the fact that SO(3) is homeomorphic to the real projective space RP 3 , thus to the quotient space of S 3 by the equivalence μ ∼ −μ, thus with antipodal points identified, and to Bπ with antipodal boundary points identified (see [Hatcher, ch. 3.3D, p. 293]).

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The map ((θ, ν), a) = (μ, a) → τa exp(θ Aν ) is a diffeomorphism from (]0, π[×S 2 ) × R 3 into (SO(3)\{I }) × R 3 ⊂ D(3). With the Euclidean frame u˜ 0 , the map U0 : (μ, a) ∈ (]0, π[×S 2 ) × R 3 → u˜ = u˜ 0 ◦ τa exp Aμ ∈ E3 gives a coordinate system in E3 .

(2.282)

¶

Dual Lie Algebra of Displacements The dual space d3∗ = Te∗ D(3) is identified with the vector space M3a × R 3 , and thus with R 3 × R 3 , by the duality (where tr B denotes the trace of the matrix B): 1 < p, v > = − tr (AM Aμ ) + p0 .α = M.μ + p0 .α, 2

(2.283)

for all p ∈ d3∗ , and v ∈ d3 , represented by (M, p0 ) ∈ R 3 ×R 3 for p, and by (μ, α) ∈ R 3 × R 3 for v. A force F transported in d3∗ is represented by (MF , RF ) ∈ R 3 × R 3 . In mechanics, such a force is called a wrench (see, for instance, [Germ], [D-L19, Appendice mécanique]).

2.7.3 Adjoint Representation of D(3) Let Int be the automorphism of D(3): g ∈ D(3) → Int g ∈ Aut D(3), with Int g(s) = gsg −1 , ∀s ∈ D(3).

(2.284)

Then we define the adjoint representation of D(3) in its Lie algebra d3 : g ∈ D(3) → Adg = Te (Int g) ∈ L (d3 ), d3 = Te D(3). With g = (R, a) ∈ D(3), for all (μ , α  ) ∈ d3 , (R, a) ∈ D(3), we have Ad(R,a)(μ , α  ) = (Rμ , −(Rμ ) × a + Rα  ),

(2.285)

and for this adjoint representation we have the following matrix formulas in Rμ3˜ × Rα3  :  Ad(R,a) = Ad(I,a).Ad(R,0) =

R 0 Aa R R



 =

I 0 Aa I

R.

(2.286)

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251

The derivative map of g ∈ D(3) → Adg ∈ L (d3 ), is identified (if we identify R 3 × R 3 with d3 ) with the map (μ, α) ∈ R 3 × R 3 → ad(μ,α) ∈ L (d3 ). This map is such that (see (2.276))   ad(μ,α)(μ , α  ) = (μ, α), (μ , α  ) ,

(2.287)

and also given in matrix form (in Rμ3  × Rα3  ) by  ad(μ,α) =

Aμ 0 Aα Aμ

.

(2.288)

Proofs of (2.285) (i) Let φ(t) = (R˜ t , a˜ t ) be a differentiable family of displacements such that φ(0) = e = (I, 0), with generator A˜ = (μ , α  ). Then by definition of the adjoint mapping, we have     d d  ˜ ((Int s) φ(t)) = Te (Int s)( ( φ(t))) = Ads A. dt dt t =0 t =0 Now with s = (R, a), we easily verify with (2.273) that (Int s)φ(t) = sφ(t)s −1 = (R, a)(R˜ t , a˜ t )(R−1 , −R−1 a) = (RR˜ t R−1 , a + Ra˜ t − RR˜ t R−1 a).

(2.289)

By time derivation, for all x ∈ R 3 , we have   d (RR˜ t R−1 )x  = RAμ R−1 x = R(μ × R−1 x) = (Rμ ) × x, dt t =0 which applied to (2.289) gives (2.285). (ii) We can also give another proof using the matrix representation (of order 4) of affine transformations:    Ra x Rx + a s = (R, a) → ∈ M4 , so that s = . (2.290) 0 1 1 1 Then the matrix representation of the Lie algebra as an affine transformation is A˜ = (μ , α  ) →



Aμ α  0 1



  x Aμ x + α  ˜ so that A . = 1 1

(2.291)

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The matrix representation (of order 4) of Ads A˜ as an affine transformation is then expressed by 

RAμ R−1 Rα  − RAμ R−1 a 0 1

since RAμ R−1 Ad(R,a). ¶

=



 =

ARμ Rα  − (Rμ ) × a 0 1

,

ARμ , which gives the matrix representation of

Proof of (2.287) for the mapping ad. Let s(t) = (Rt , at ) be a differentiable family of displacements with generator A = (μ, α) such that s(0) = e = (I, 0). By time derivation, at t = 0,     d ˜ ˜ = A, A˜ , ∀A˜ ∈ d3 , (Ads(t )A) = (ade A)(A) dt t =0 and thus by (2.276), with a0 = 0,   d    (Rt μ , Rt α − (Rt μ ) × at ) = (μ × μ , μ × α  − μ × α). dt t =0 Therefore, by time derivation at t = 0 of the adjoint representation formula, we obtain again the bracket (2.287). ¶ The map Adg is a Lie algebra homomorphism:     Adg (μ, α), (μ , α  ) = Adg (μ, α), Adg (μ , α  ) ,

(μ, α), (μ , α  ) ∈ d3 , (2.292)

which is also written Adg ad(μ,α) = adAdg (μ,α) Adg .

(2.293)

Notice an important property associated with the split d3 = o(3)⊕R 3: the adjoint representation of the rotation group keeps invariant the Lie algebra of translations AdSO(3)(R 3 ) = R 3 (see [Kob-Nom, Vol 2, ch. X, 2, p. 190]). We can also define the Killing–Cartan form of the displacement group. This is a bilinear symmetric form ϕ on d3 defined by ϕ((μ, α), (μ , α  )) = tr (ad(μ,α)ad(μ ,α  ) ) = 2 tr (Aμ Aμ ) = −4 μ.μ . Thus −ϕ(X, X) is a positive degenerate quadratic form on d3 that induces a Riemannian metric on SO(3).48 But we will define another Riemannian metric on SO(3) for a rigid body. 48 See

[Kob-Nom] for developments.

2.7 Rigid Bodies

253

2.7.4 Coadjoint Representation of D(3) The matrix representation of the coadjoint representation of the displacement group: g ∈ D(3) → t Adg ∈ L (d3∗ ), also denoted by Adg∗ , is given with (2.286), for g = (R, a), by Ad(∗R,a)

=

t

R −t RAa tR 0



= R t

I −Aa 0 I

.

(2.294)

∗ Then its derivative map is given by (μ, α) → t ad(μ,α) ∈ L (d3∗ ), denoted by ad(μ,α) 3 3 with matrix representation in Rμ × Rα  : ∗ ad(μ,α)

 =−

Aμ Aα 0 Aμ

.

(2.295)

Then we can give the coadjoint orbit of a point p = (M, p0 ) ∈ d3∗ , which is the set {p = Ad(∗R,a)p, R, a) ∈ D(3)}. With p = (M, p0 ) we obtain p = (M  , p0 ) = t R(M − Aa p0 ), t Rp0 ),

(2.296)

and thus we have |p0 | = |p0 |,

M  .p0 = M.p0 .

(2.297)

Hence the coadjoint orbit of a point p = (M, p0 ), if p0 = 0 is of codimension 2 (if |p0 | = 1 and M.p0 = 0, then the coadjoint orbit of p is identified with T S2 ). If p0 = 0, then p0 = 0, |M  | = |M|, the coadjoint orbit of p, is 2-dimensional.

2.7.5 Velocity Field of a Rigid Body The conservation of distance between two material points during evolution implies (xt − yt , xt − yt ) = (Rt (x − y), Rt (x − y)) = (x − y, x − y). By derivation, using (2.269), we obtain 3 ˙ (Az, z) + (z, Az) = 0 with A = R−1 t Rt , ∀z = x − y ∈ R ,

(2.298)

˙ t R−1 or also A = R t . The displacement of the rigid body by isometries implies the equiprojection of the velocity field: A is a skew-symmetric operator in R 3 , hence

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such that Ax = μ × x, ∀x ∈ R 3 , with μ ∈ R 3 , called the instantaneous angular velocity. Note that here it is a question of vector fields on R 3 (or on E3 ), but not on E3 . Remark 25 On the derivative maps (lg )∗ and (rg )∗ . It is essential to observe the difference between multiplication on the left and that on the right of the group D(3) on its tangent space: multiplication on the left by g = (R, a) in D(3) on A = (μ, α) ∈ Te D(3) is such that (lR )∗ A ∈ Tg D(3), and thus (lg )∗ = (lR )∗ (with R identified with (R, 0)). Recall that the action of A = (μ, α) (respectively (lg )∗ A) in R 3 is given by the affine function x → μ × x + α (respectively R(μ × x + α)), whereas multiplication on the right by g = (R, a) in D(3), (rg )∗ A ∈ Tg D(3), is given in R 3 by the affine function x → (μ × (Rx + a) + α). ¶ A main property of a Lie group is that its tangent space may be identified with a product that is T G ≡ G × g, with g its Lie algebra, thanks to its Maurer–Cartan forms, either left ωl or right ωr , with values in Te G, defined for all g ∈ G by ωgl (v) = (lg −1 )∗ v,

∀v ∈ Tg G,

lg −1 s = g −1 s, ∀s ∈ G,

ωgr (v) = (rg −1 )∗ v,

∀v ∈ Tg G,

rg −1 s = sg −1 , ∀s ∈ G.

(2.299)

Thus ωgl and ωgr are isomorphisms from Tg G onto Te G, with ωe = I (the identity), and ωgl is invariant on the left, that is, lg∗ (ω) = ω, ∀g ∈ G, and ωgr is invariant on the right. These two differential forms are related by ωsr = Ads .ωsl = ωsl Ads ,

(2.300)

since ls −1 Int s = (Int s) ls −1 = r(s −1 ),

∀s ∈ D(3).

(2.301)

We now use these forms in the framework of the displacement group. From vˆ = g˙ ∈ Tg D(3), we define the elements v S and v E ∈ Te D(3) = d3 by def

˙ = (lg −1 )∗ g, ˙ v S = ωgl (g)

def

and v E = ωgr (g) ˙ = (rg −1 )∗ g; ˙

(2.302)

v S and v E are representatives of the velocity of the rigid body, with Lagrange (respectively Euler) coordinates, said to be relative to the body (respectively to the space); they may be identified with affine transformations whose linear part is a skew-symmetric matrix, called wrenches; v S and v E are also identified with the pairs of vectors in R 3 v S = (μS , voS ),

v E = (μE , voE ),

(2.303)

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255

where μS and μE are respectively called the angular velocity relative to the body ˙ a), and he angular velocity to the space, and are expressed, for g˙ = (R, ˙ by ˙ R−1 a) v S = (R−1 R, ˙ = (AμS , voS ), ˙ −1 , −RR ˙ −1 a + a) ˙ = (AμE , voE ) = (AμE , −AμE a + a). ˙ v E = (RR

(2.304)

We can also identify v S (respectively v E ) with a vector field on D(3) invariant on the left (respectively on the right) by vsS = (ls )∗ v S , s ∈ D(3) (respectively by vsE = (rs )∗ v E ). The relation between the angular velocities is directly obtained from the time derivative of Rt : ˙ = AμE R = RAμS , R

˙ −1 , AμS = R−1 R, ˙ thus AμE = RR

(2.305)

giving AμE = RAμS R−1 = ARμS , and thus μE = RμS . Velocities of the points in the rigid body in R 3 and E3 for the frame u˜ 0 . The velocity of a material point of a body at xt = gt .x = Rt x + at ∈ R 3 in the frame u˜ 0 is given in R 3 by ˙ t .x + a˙ t =AμE Rt .x + a˙ t = v E .xt , x˙ t = g˙t .x = R = Rt AμS .x + a˙ t = Rt (v S .x),

(2.306)

where v E , v S , are identified with the affine functions v E .y = (AμE )y + voE , v S .x = (AμS )x + voS ,

with voE = a˙ − (AμE )a, with voS = R−1 ˙ t a,

(2.307)

with time-dependent vectors μS , voS , μE , voE . The velocity vMt = M˙ t of a material point Mt of a rigid body in E3 is obtained by taking the derivative of the expression (2.270): vMt = u0 (g˙t .x) = u0 (v E .xt ) = ut (v S .x). ¶

(2.308)

Relations between the different expressions for the velocity. (i) Relations between the velocities v S and v E in d3 . The expression of v S as a function of v E is given by v S = Adg −1 v E = Ad(R−1 ,−R−1 a) (μE , voE ),

(2.309)

and thus by (2.304), v S = (μS , voS ) = Rt−1 ◦ v E ◦ gt = (R−1 μE , R−1 (voE + μE × a)),

(2.310)

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2 Classical Mechanics

whence the matrix relations between the two velocities are  S   E   E  S μ I 0 μ I 0 μ μ −1 =R , = . R S E E vo vo vo voS −Aa I Aa I (ii) Darboux differential. Relations between the estimates of the frame velocity in T (E3 ) through v S , v E in d3 . Let φ0 be the map (see (2.267)) φ0 : u˜ ∈ E3 → g = (u˜ 0 )−1 u˜ ∈ D(3),

(2.311)

whose inverse φ0−1 is the map φ0−1 : g ∈ D(3) → u˜ = u˜ 0 ◦ g ∈ E3 .

(2.312)

Then the derivative T (φ0 ) transforms the tangent space T (E3 ) into T (D(3)), so that for all vˆ = u˙˜ ∈ Tu˜ (E3 ), with the notation φ0∗ = (φ0 )∗ , we have φ0∗ (v) ˆ = g˙ = v˜ ∈ Tg D(3).

(2.313)

This transforms the velocity of a moving frame u˜ t into the velocity of displacement with respect to the fixed frame u˜ 0 , with u˜ t = u˜ 0 ◦ gt . Then we obtain the velocities v S , v E ∈ d3 identified with R 3 × R 3 from the velocity vˆ ∈ Tu˜ (E3 ) by ˆ v S = ωl (φ0∗ (v)),

v E = ωr (φ0∗ (v)). ˆ

(2.314)

Let πˆ l and πˆ r be the left and right Darboux differentials of φ0 , defined by πˆ l = φ0∗ (ωl ),

πˆ r = φ0∗ (ωr ),

(2.315)

which are differential forms on E3 with values in d3 . They are the pullback forms of the (left or right) Maurer–Cartan differential forms ωl,r on D(3) by φ0 . Then (2.314) is given by v S = πˆ l (v) ˆ = ωl (φ0∗ (v)), ˆ

v E = πˆ r (v) ˆ = ωr (φ0∗ (v)). ˆ

(2.316)

Thus the left (respectively right) Darboux differential transforms the velocity of the frame vˆ ∈ Tu˜ (E3 ) into the velocity v S (respectively v E ) in d3 . Remark 26 Let P = E3 × D(3) be the trivial principal fibre bundle over E3 with the group D(3). Then from this bundle we can define the trivial fibre bundle Q = E3 × E3 over M = E3 , which is the set of pair of Euclidean

2.7 Rigid Bodies

257

frames (u˜ S , u) ˜ with projection πS , so that πS (u˜ S , u) ˜ = u˜ S ,49 with the group of displacements G = D(3) acting on the right on u˜ ∈ E3 . From (u, ˜ a) ˜ ∈ P , we obtain (u, ˜ u˜ a) ˜ ∈ Q. We can identify every u˜ ∈ Qu˜ S with a map u˜ from R 3 × R 3 onto Tu˜ S (E3 ). Then we have two natural sections of Q, σS (u˜ S ) = (u˜ S , u˜ S ),

σ 0 (u˜ S ) = (u˜ S , u˜ 0 ),

and we can identify u˜ S (respectively u˜ 0 ) with the map ˆ u˜ S (v S ) = v,

respectively u˜ 0 (v E ) = v. ˆ

(2.317)

The inverses of these maps are identified with the differential forms ωu˜ S , ωu˜ on the Euclidean frame E3 with values in R 3 × R 3 , so that for all vˆ ∈ Tu˜ S (E3 ), we have 0

u˜

ωu˜ SS (v) ˆ = (lu˜ −1 )∗ vˆ = v S , S

ωuu˜˜ S (v) ˆ = (l(u˜ 0 )−1 )∗ vˆ = v E ; 0

(2.318)

˜ with lu˜ −1 (respectively l(u˜ 0 )−1 ) the map from E3 into D(3), so that lu˜ −1 v˜ = u˜ −1 S v, S

S

˜ (vjS ) (respectively (vjE )) are the components of vˆ respectively l(u˜ 0 )−1 v˜ = (u˜ 0 )−1 v; for the frame u˜ S (respectively u˜ 0 ) linked to the rigid body (respectively fixed with the space). ¶

Kinetic Energy of a Rigid Body Let M (respectively Mt ) be a connected bounded domain occupied by a rigid body in E3 at time 0 (respectively t). Let ρ be the specific mass of the body such that supp ρ = M. Thisis a positive integrable function, whose smoothness does not matter. Then m =  ρ(x) dx is the total mass. Let u˜ 0 be a particular Euclidean frame (taken at initial time). The kinetic energy of the rigid body at time t is defined with xt = gt x by E

1 = 2

kin def



1 |x˙t | ρ(x)dx = 2 0

 |g˙t x|2 ρ(x)dx,

2

(2.319)

0

with 0 = (u˜ 0 )−1 M = u˜ −1 t (Mt ). Thus with (2.306), we have E kin =

49 The

1 2



 A 0

μE

2 1 Rt .x + a˙ t  ρ(x)dx = 2

frame u˜ S being linked to the rigid body.

 0

  Rt A S .x + a˙ t 2 ρ(x)dx. μ

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2 Classical Mechanics

Then the kinetic energy is given by E kin (v) ˆ =

1 2



  2   1   E  S 2 v .gt x  ρ(x)dx = v .x  ρ(x)dx. 2 0 0

(2.320)

2.7.6 Inertial Riemannian Metrics The kinetic energy of a rigid body naturally leads one to define a Riemannian metric on the space of Euclidean frames E3 , so that we will define from the identity two invariant Riemannian metrics on the displacement group D(3), one invariant on the left, the other on the right. With a Euclidean frame u, ˜ M is represented in R 3 by the domain  such that M = u(), ˜ and the specific mass is represented by ρ ◦ u, ˜ still  denoted by ρ. Let ν(dx) be the measure defined by ν(dx) = m1 ρ(x) dx, so that  ν(dx) = 1. Metric on d3 We first specify the metric at the unit e of D(3) for a Euclidean frame u. ˜ Let z = (μ, α) ∈ d3 ; we have z.x = μ × x + α, and  = u˜ −1 M, and hence  def |μ × x + α|2 ν(dx). (2.321) gue˜ (μ, α) = 

Let G ∈ E3 be the center of mass, and let xG ∈ R 3 be the position of the center of mass of the rigid body defined by def





xG =

(x − xG ) ν(dx) = 0.

x ν(dx), thus 

(2.322)



Then the metric on D(3) at the unit e is given by gue˜ (μ, α)

 |μ × (x − xG ) + μ × xG + α|2 ν(dx)

= 



(2.323) |μ × (x − xG )| ν(dx) + |μ × xG + α| . 2

=

2



The inertia tensor of the body with respect to the center of mass is the symmetric positive definite matrix I G such that 



  μ × x  2 ν(dx  ),

|μ × (x − xG )|2 ν(dx) =

(I G μ, μ) = 

G

(2.324)

2.7 Rigid Bodies

259

with x  = x − xG = τ−xG (x), and G = τ−xG  the domain with center G; τxG , the translation by the vector xG of the frame u, ˜ gives the frame u˜ G such that u˜ = u˜ G ◦ τxG . Then  2  2 |μ × x  |2 = |μ|2 |Pμ⊥ x  |2 = |μ|2 x   − μ.x   ; the matrix I G is expressed by def



(I G )ij =

(x 2 δ ij − x i x j ) ν(dx).

(2.325)

G

Then let αG = α + μ × xG .

(2.326)

geu˜ G (μ, αG ) = (I G μ).μ + |αG |2 .

(2.327)

Then the metric is

This metric splits into two independent metrics, one corresponding to rotations, the other to translations. Then we can diagonalize the matrix I G , thanks to a rotation RI . Let −1 −1  xˆ = R−1 I x = RI τ−xG (x) = gI x, −1 −1 I = R−1 I G = RI τ−xG  = gI 

= gI−1 u˜ −1 (M) = (u˜ I )−1 (M),  ˆ = 0, ∀i = j with whence  = gI (I ) and u˜ I = u˜ gI ; we have I (xˆ i xˆ j ) ν(d x) the new variables. Thus    2    −1  G  2   μ × x ν(dx ) = (I μ, μ) = ˆ (R μ) × xˆ  ν(d x). G

Let μ = R−1 μ and Ii = (I G μ, μ) =



I

 I

 i 2 xˆ  ν(d x). ˆ Then

 2  2  2  2 (1 − μi  )Ii = μ1  (I2 + I3 ) + μ2  (I3 + I1 ) + μ3  (I1 + I2 ).

Principal axis of the inertia tensor. The eigenvectors of the “inertia tensor” I G are called the axes of inertia of the rigid body, and the corresponding Euclidean frame u˜ 0 is also denoted by uIG . The eigenvalues (I1 , I2 , I3 ) of I G are such that I1 = I2 + I3 , I2 = I3 + I1 , I3 = I1 + I2 .

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2 Classical Mechanics

The choice as reference Euclidean frame of uIG is built with the eigenvectors ej of I, and the center of mass is natural. From a Euclidean frame u, ˜ we will denote by gI the displacement that diagonalizes the metric gue˜ with the center of mass, thus so that uIG = u.g ˜ I . The metric of d3 at e for this frame is then expressed as a bilinear form by uI

ge G ((μ , α  ), (μˆ  , αˆ  )) =



Ii (μ )i (μˆ  )i + α  .αˆ  .

(2.328)

Finally, we have the relations 

−1  μ = R−1 I μ, α = RI (μ × xG + α),

thus

μ α



= (Ad gI−1 )

 μ , α (2.329)

that is, 

μ α



=

I −AR−1 xG I

0 I



   μ I 0 μ −1 RI = RI . α −AxG I α −1

Thus we have the relation between the metrics uI

gue˜ (μ, α) = ge G (μ , α  ),

(2.330)

and then uI

ge G = gue˜ .Ad gI

with uIG = u.g ˜ I.

(2.331)

Let v E = (μE , v0E ) = (μ, α). Then (μ , α  ) = v I = (μI , v0I ) = Ad gI−1 v E = Ad gI−1 Ad g(v S ) = Ad gr (v S ), with gr = gI−1 g, and thus g = gI gr . The relation (2.330) is uI

gue˜ (v E ) = ge G (v I ).

(2.332)

The question is to have the choice of taking uIG = u˜ 0 , at the initial time, so that ˙ a) ˙ I , x˙G ) and gr = (I, 0), RI = R, xG = a; then v˜ = g˙ = (R, ˙ = (R 0 αG = v E xG = gg ˙ −1 xG = gx ˙ G = vG .

This choice is not always possible, for instance in the case of a rigid body with a fixed point that is not the center of mass, and in the case of two rigid bodies with a link.

2.7 Rigid Bodies

261

The metric for the velocity v E with xG = a is given by 2    gue˜ (v E ) = (I G R−1 μE , R−1 μE ) + μE × a + v0E  = (RI G R−1 μE , μE ) + (−Aa μE + v0E , −Aa μE + v0E )

(2.333)

= (I˜ G μE , μE ) − (2Aa μE , v0E ) + (v0E , v0E ), with I˜ G = RI G R−1 − A2a .

(2.334)

The metric operator Gue˜ from d3 onto d3∗ , so that gue˜ (v E ) = < Gue˜ (v E ), v E >, is given by 

Gue˜

=

I˜ G Aa −Aa I

(2.335)

,

with I˜ G given by (2.334). Thus we can define the momenta (up to the factor of mass) by uI

pI = (MI , poI ) = Ge G v I ,

pE = (ME , poE ) = Gue˜ v E ,

(2.336)

with  p = E

I˜ G μE + Aa voE −Aa μE + voE

,

I



p =

I G μI v0I



 =

I G R−1 I μ −1 RI αG

.

(2.337)

Remark 27 The change of the inertia tensor I G by translation Ta is given by (I˜ G μ, μ) = (I G μ, μ) + |μ × a|2 ,

(2.338)

with μ ∈ S 2 , and |μ × a|2 = |a|2 − (μ.a)2 the square of the distance between the axes (μ, O) and (μ, G). Thus we have as in (2.334) (with R = I ) the relation (this is the Steiner theorem) I˜ G = I G − (Aa )2 ¶. On the metric on o(3). From (2.321), we could define the metric for the rotation group alone, on o(3), by the trace ˆ ue˜ Aμ ) gue˜ (μ) = tr (t Aμ G ˆ =G ˆ ue˜ given by G ˆ ij = with the matrix G



x

i x j ν(dx).

(2.339) Then since t Aμ = −Aμ ,

ˆ =| μ |2 tr (Pμ⊥ G), ˆ gue˜ (μ) = −tr (A2μ G)

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2 Classical Mechanics

with Pμ⊥ = I − Pμ the projection on the orthogonal space of μ. Thus ˆ ˆ > = < μ, Iμ >, gue˜ (μ) =| μ |2 (tr G)− < μ, Gμ ˆ − G, ˆ see (2.325), and the identity matrix I . with the matrix I = I (tr G) Triangular inequalities. From (2.325), we have the inequalities  < I G μ, μ > ≤ μ2

x 2 ν(dx). G

 The trace of I G is tr I G = 2 G x 2 ν(dx), and thus < I G μ, μ > ≤ 12 tr I G , which implies the following inequalities between the eigenvalues of I G : Ij ≤

1  (I + I2 + I3 ), 2 1

j = 1, 2, 3,

and hence the triangular inequalities I1 ≤ I2 + I3 , I2 ≤ I3 + I1 , I3 ≤ I1 + I2 . In the case of a rigid body with a symmetry (see below), with I1 = I2 and I1 < I3 , we have the inequality I3 ≤ 2I1 . ¶ Some special cases. When the body has a symmetry of order n with respect to an axis that we can take as e3 , then the inertia tensor I G must commute with the matrix Rθe3 with θ = 2π n . Thus with α = cos θ, β = sin θ, we have ⎞ α −β 0 = ⎝β α 0⎠. 0 0 1 ⎛

Rθe3

For a binary symmetry (the case of a propeller), θ = π, and thus β = 0, α = −1. The commutativity of I G with Rπe3 implies that I G is a matrix with the shape ⎞ a11 a12 0 = ⎝ a12 a22 0 ⎠ . 0 0 I3 ⎛

IG

A higher order of symmetry (p ≥ 3) implies that I G is diagonal and such that ⎛

IG

⎞ I1 0 0 = ⎝ 0 I1 0 ⎠ . 0 0 I3

2.7 Rigid Bodies

263

The inertia tensor of a homogeneous ball of unit radius (with the center of mass at the origin) is expressed by I G = 25 I . Recall that the metric, invariant on the left and on the right, is expressed by (see [Kob-Nom, ch. IV.1, p. 155]) 1 (Aμ , α), (Aμ , α  ) → − tr (Aμ .Aμ ) + α.α  = μ.μ + α.α  . 2

¶

(2.340)

Metric on d3∗ I

The dual space of d3 is equipped with a metric (ˆguG )−1 given on p = (M, p0 ), with the frame uIG , by I

(ˆguG )−1 (M, p0 ) = ((I G )−1 M).M + |p0 |2 .

(2.341)

ˆ u˜ on d ∗ : In the frame u, ˜ see (2.335), with a = xG , we obtain the metric tensor G 3 ˆ u˜ )−1 = (G



−(I G )−1 Aa (I G )−1 Aa (I G )−1 I − Aa (I G )−1 Aa

.

(2.342)

Inertial Riemannian Metrics on D(3) and E3 From the metric (2.321) on d3 , we can define two Riemannian metrics on D(3), one gl linked to the rigid body and the Lagrange coordinates and left invariant, the other gr linked to the space and the Euler coordinates and right invariant. For an evolution (st ) in D(3), with s˙t = v, ˜ we define the metric gl (respectively gr ): 0

gls (˙s ) = gue (v S ) =



   S 2 v .x  ν(dx),

(2.343)

    E 2 v .y  ν(dy).

(2.344)

0

respectively grs (˙s ) = gue˜ (v E ) =



Then a left (respectively right) invariant Riemannian metric gl (respectively gr ) on D(3) is defined (as a quadratic form), thanks to the left (respectively right) Maurer– Cartan form ωl (respectively ωl ), by

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2 Classical Mechanics

 gls (z) =

0

 grs (z)

 2  l  ωs (z).x  ν(dx), (2.345)

  r ω (z).y 2 ν(dy),

=

∀s ∈ D(3), z ∈ Ts D(3),

s



with 0 = (u0 )−1 (M). We recall that ωsl (˙s ) = v S (respectively ωsr (˙s ) = v E ). Relations with the Maurer–Cartan forms. 0

(i) The Riemannian metric gl on D(3) and the metric gue on d3 are related via the left Maurer–Cartan form 0

gls = gue ◦ ωsl ,

0

respectively gls = gue ◦ (ωsl ⊗ ωsl ).

(2.346)

The expression given by (2.345) is positive and quadratic with respect to z. The left invariance of gl is a consequence of the left invariance of the Maurer–Cartan form: (ls1 )∗ gl = gl , since ((ls1 )∗ gl )s = gls1 s (ls1 )∗ = gue ◦ ωsl 1 s = gls1 s . 0

(ii) The Riemannian metric gr on D(3) and the metric gue˜ on d3 are related via the right Maurer–Cartan form by grs = gue˜ ◦ ωsr ,

respectively grs = gue˜ ◦ (ωsr ⊗ ωsr ).

(2.347)

We also have the right invariance of gr : (rs1 )∗ gr = gr . Let ζ (g) = g −1 . Then this map in D(3) is such that ζ 2 = I , and ζ φ0 = ψ0 ,

ζ ∗ ωl = −ωr ,

ζ ∗ ωr = −ωl ,

ζ ∗ gl = gr ,

ζ ∗ gr = gl . (2.348)

Proof We have ζ φ0 (u) ˜ = ζ (u˜ −1 ˜ = u˜ −1 u˜ 0 = ψ0 (u). ˜ Then since lg ζ = ζ rg −1 0 u) −1 −1 and ζ∗ (v) = −g vg , ∀v ∈ Tg D(3), it follows that ζ∗ (v) = −v, ∀v ∈ d3 . Therefore, (ζ ∗ ωl )g = ωgl −1 ζ∗ = (lg )∗ ζ∗ = (lg ζ )∗ = (ζ rg −1 )∗ = ζ∗ (rg −1 )∗ = ζ∗ ωgr = −ωgr . Then since gr and gl are quadratic, we deduce that ζ ∗ gl = gr , which we can prove directly. Thus ζ is an isometry with respect to the metrics gr , gl . ¶ We now define a Riemannian metric on the space of Euclidean frames. Definition 2 A Riemannian metric gˆ on E3 is defined (as a quadratic form) by gˆ u˜ (v) ˆ = gue (v S ) = gue˜ (v E ) 0

with vˆ = u0 .v˜ = (φ0−1 )∗ v˜ ∈ Tu˜ E3 .

(2.349)

2.7 Rigid Bodies

265

The Riemannian metric gˆ on E3 as a quadratic form (respectively as a bilinear form) is given via the right Darboux differential, see (2.315), by gˆ u˜ = gue˜ ◦ πur˜ ,

respectively gˆ u˜ = gue˜ ◦ (πur˜ ⊗ πur˜ ).

(2.350)

We can review the metric on E3 in the following equivalent way. With the map φ0 from E3 onto D(3) such that φ0 (u) ˜ = (u0 )−1 u˜ = g ∈ D(3) (see (2.311)), let ψ0 be the map defined by ψ0 (u) ˜ = u˜ −1 u0 = g −1 .

(2.351)

˜ = u.s, ˜ u˜ ∈ E3 , s ∈ D(3). Then the maps φ0 and ψ0 satisfy Let r˜s (u) φ0 ◦ r˜s = rs ◦ φ0 ,

ψ0 ◦ r˜s = ls −1 ◦ ψ0 ,

(2.352)

since ˜ = gs = rs (φ0 (u)), ˜ φ0 (˜rs (u))

ψ0 (˜rs (u)) ˜ = s −1 g −1 = ls −1 ψ0 (u). ˜

Proposition 4 The Riemannian metric gˆ on E3 is such that gˆ = φ0∗ (gr ) = ψ0∗ (gl ).

(2.353)

Moreover, this metric satisfies (with r˜s u˜ = us) ˜ r˜s∗ gˆ = gˆ .

(2.354)

Proof With φ0 (u) ˜ = g, ψ0 (u) ˜ = g −1 , we have (φ0∗ gr )u˜ = gue˜ .ωgr .(φ0 )∗ = gue˜ (rg −1 φ0 )∗ , (ψ0∗ gl )u˜ = (gue .ωgl −1 (ψ0 )∗ = gue (lg ψ0 )∗ . 0

0

˜ = ((u0 )−1 u)g ˜ −1 = e and lg ψ0 (u) ˜ = g.(u˜ −1 u0G ) = e, so that We have rg −1 φ0 (u) we obtain as in (2.349), (φ0∗ gr )u˜ (v) ˆ = gue˜ (v E ) = (ψ0∗ gl )u˜ (v) ˆ = gue (v S ). 0

˜ = gs = rs φ0 (u), ˜ and Then we prove (2.354). We have φ0 r˜s = rs φ0 , since φ0 r˜s (u) thus r˜s∗ (ˆg) = r˜s∗ (φ0∗ gr ) = (φ0 r˜s )∗ gr = (˜rs φ0 )∗ gr = φ0∗ r˜s∗ (gr ) = φ0∗ (gr ) = gˆ .

¶

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2 Classical Mechanics

Remark 28 On the Riemannian metric operators Gr , Gl on D(3). The map Grs from the tangent space Ts D(3) onto Ts∗ D(3) that is the metric operator defined by < Grs z, z˜ > = grs (z, z˜ ), ∀z, z˜ ∈ Ts D(3)

(2.355)

is deduced from Gre , thanks to the right invariance of the metric gr with respect to the Maurer–Cartan form ωr : Grg = t ωgr ◦ Gre ◦ ωgr .

(2.356)

This implies the following relation for the tensor I G , with s1 = (R1 , 0): G G R−1 1 Is1 R1 = Ie ,

∀s1 ∈ D(3).

(2.357)

By exchanging ωr for ωl , we obtain similar relations for the metric operator Gls .

¶

The (right) Darboux differential π r = ωr ◦ (φI )∗ , so that π r (v) ˆ = v E , is an isometry from Tu˜ E3 into Te D(3) = d3 for the corresponding metrics. Momentum and Angular Momentum With the metric operator Grg from Tg D(3) onto Tg∗ D(3) associated with grg , we define the momentum p˜ and the angular momentum M of a rigid body from its velocity v˜ = g˙ ∈ Tg D(3) by p˜ = (M, po ) = mGrg (v). ˜

(2.358)

ˆ be the metric operator from T (E3 ) onto T ∗ (E3 ) associated with gˆ . With Let G vˆ = u˙˜ ∈ Tu˜ (E3 ), the momentum pˆ is defined by ˆ u˜ v. ˆ pˆ = mG ˆ u˜ vˆ is With the velocity v˜ = (φI )∗ (v) ˆ ∈ Tg D(3), the momentum pˆ = mG −1 ∗ ∗ S such that pˆ = (φI ) (p), ˜ or p˜ = (φI ) (p). ˆ The momenta p and pE (at the origin, respectively said to be relative to the body and to the space) and the corresponding angular momenta are defined by (2.337). The equivalence between the given definitions is simply a consequence of (2.309). We have 2E kin (v) = < p, ˆ vˆ > = < p, ˜ v˜ > = < pE , v E > = < pS , v S > .

(2.359)

From (2.337) and (2.359), we deduce the relation uI

pS = Gue˜ Adg −1 (Ge G )−1 pE = t Adg pE .

(2.360)

2.7 Rigid Bodies

267

With (2.337) and the mass m, the momentum pE ∈ T ∗ D(3) is expressed by I

M E = m((I uG μE + xG × voE ),

poE = m(voE − xG × μE ),

(2.361)

and (see (2.326)) poE = mvG . The momentum pS = (M S , poS ) is given by (2.337), (2.335), M S = m(I˜ − (AxG )2 μS + xG × voS ),

poS = m(voS − xG × μS ),

(2.362)

with v S = (μS , α S ), I = I˜ G , and xG = a. The inverse relation is given with (2.342) by 

μS αS



ˆ le )−1 = (G



MS poS



 =

−I −1 Aa I −1 Aa I −1 I − Aa I −1 Aa



MS poS

.

(2.363)

2.7.7 Modelling of Forces. Wrenches From a given “exterior” force field acting on a rigid body, we will define a differential form on D(3) and two wrenches (i.e., elements of the dual d3∗ of its − → Lie algebra), which represent the force field. Let f be a given field of volumetric forces (“applied to the body”), which is a map from the domain M ⊂ E3 of the rigid body into T E3 . The choice of a reference (fixed) Euclidean frame u˜ 0 allows one to define the force field f in R 3 (by transport by the inverse of u˜ 0 ) by − → f ◦ u˜ 0 = u0 ◦ f. Let 0 = u˜ −1 (M), that is, 0 = u˜ −1 ˜ 0 )−1 M, as in the previous timet (Mt ) = (u → − dependent notation (see (2.319). The power P of the force field f in E3 for the velocity field v˜ ∈ Tu˜ E3 is expressed by  P = < Fu˜ , v˜ > =

− → f (u(x)). ˜ v(x) ˜ dx. 0

→ − → − Then f (u(x)) ˜ = f (u˜ 0 (g(x))) = u0 (f (g(x))) and − → ˙˜ ˙ = f (g(x)).g(x). ˙ f (u(x)). ˜ u(x) = u0 f (g(x)).u0 (g(x)) Expression of the wrench with respect to D(3). We have 



P=

˙ + a) f (Rx + a).(Rx ˙ dx,

f (g(x)).g(x) ˙ dx = 0

0

(2.364)

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2 Classical Mechanics

which defines the force field F on D(3) by  Fg = (Mg , Rg ),



(Mg )ij =

fi (g(x))x j dx,

Rg =

0

f (g(x))dx. 0

(2.365)

Then we have ˙ + Rg .a. P = < Fg , g˙ > = tr (Mg R) ˙

¶

(2.366)

Expression of the wrench with respect to d3 . The power P of this force field and for the velocity field g˙ ∈ Tg D(3) is, with the variable y = g(x) ∈  = g(0 ), 

f (y)g˙ ◦ g −1 (y) dy.

P=

(2.367)



Then the wrench F E ∈ Te∗ D(3) is defined with v E = ωgr g˙ = g˙ ◦ g −1 (y) by  P = < F E , vE > = f (y).(v E .y) dy. (2.368) 

The wrench

∈

FS

Te∗ D(3)

is defined with f S = R−1 f ◦ g and v S = R−1 g˙ by  f S (x).(v S .x) dx. (2.369) P = < F S , vS > = 0

The wrench Fu˜ ∈ Tu˜∗ (E3 ) is defined in a similar way, so that the power P is expressed with Fg ∈ Tg∗ (D(3)), F E ∈ d3∗ by P = < Fu˜ , u˙˜ > = < Fg , g˙ > = < F E , v E >,

(2.370)

and we have the relations or Fg = ((φ0−1 )∗ F  )g ,

F  = φ0∗ (F ),

F E = Geu˜ ωgr (Gug˜ )−1 F. 0

(2.371)

With v E = (μE , voE ), we then have  P= 

f E (y).(AμE y + v0E ) dy = MFE μE + RFE v0E .

(2.372)

Thus F E = (MFE , RFE ) ∈ R 3 × R 3 with  MFE =

 y × f E (y) dy, 

RFE =

f E (y) dy. 

We call MFE the moment of the wrench F E , and RFE its resultant.

(2.373)

2.7 Rigid Bodies

269

Application: Associated wrench with a uniform gravitational field directed along the (fixed) direction e30 . The density of the gravitational force is here defined (in the natural frame of R 3 ) by f (x) = (0, 0, −gρ(x)) = −gρ(x)3 (with g gravity constant, positive, with the usual choice of opposite directions for f and 3 ). The wrench of the gravitational forces (relative to the space) is then defined by MFE = −gm xG × 3 ,

RFE = −gm 3 .

(2.374)

S = g −1 (x ) = R−1 (x − a), the wrench F S is With e3 = R3 , and with xG G G S × e3 , MFS = −gmxG

RFS = −gm e3 .

(2.375)

Force field depending on the velocity. If there is a density of volumetric forces, then the power of the force is given by  P = < F(g,g) ˙ , g˙ > =

f E (y, v E .y).(v E .y) dy. 

2.7.8 Covariant Derivative of the Velocity Since the (left or right) Maurer–Cartan form is an isometry, the covariant derivative of the velocity field v = vgt = s(gt ) on D(3) linked with the Riemannian metric g˜ l is transported to the unit of the group via ωr or ωl (see [Kob-Nom, ch. IV.2, Prop. 2.5, p. 161] and equation (2.542) in the appendix). The covariant derivative of the velocity field vgt on D(3) linked to the Riemannian metric g˜ r is transported to the unit element of the group via the (right) Maurer–Cartan form by ωgr t (∇vr s) =

dv E − B(v E , v E ), dt

(2.376)

with B(v E , v E ) ∈ d3 given by (see the appendix, (2.538) and (2.539)) def

∗ E B(v E , v E ) = (adG,v E )v E = G−1 e adv E Ge v .

(2.377)

Then the covariant derivative of the velocity field sˆ (u˜ t ) = vˆu˜ t = vˆ linked to the Riemannian metric gˆ on E3 is transported to the unit element of the group via the (right) Darboux differential π r by πur˜ t (∇vˆ sˆ ) = ωgr t (∇vr s).

(2.378)

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2 Classical Mechanics

(With ωgl t v = v S we have ωgl t (∇vl s) = ∇vl S (ωgl t s) =

dv S − B(v S , v S ).) dt

(2.379)

We give the components of the covariant derivative of the velocity field at the unit of SO3 or at the reference frame u˜ 0 . Let (Xj ) be the vectors of u˜ 0 . Since the frame is Euclidean but not orthonormal for the metric of the rigid body, we have g(Xi , Xj ) = Ii δi,j . Then we can use the relation (2.530), so that we obtain 2g(∇Xi Xj , Xk ) = g([Xi , Xj ], Xk ) + g([Xk , Xi ], Xj ) + g([Xk , Xj ], Xi ). (2.380) k Since ∇Xi Xj = i,j Xk , the Riemann–Christoffel symbols are given by k Ik = i,j,k [Ik + Ij − Ii ]. 2 i,j

Thus k i,j = i,j,k

1 1 1 [Ik + Ij − Ii ] = i,j,k (1 + [Ij − Ii ]). 2Ik 2 Ik

(2.381)

We can also obtain this result thanks to the formula (∇X Y )e =

1 ([X, Y ] − B(X, Y ) − B(Y, X)), 2

with B(Xi , Xj ) given by g(Xk , B(Xi , Xj )) = g([Xj , Xk ], Xi ) = j,k,i Ii , so that B(Xi , Xj ) = i,j,k Ii I1k Xk . Then for a rigid body with a fixed point, the Euler  j equation for the angular velocity μ = μ Xj ∈ T SO(3), which is the geodesic equation on SO(3), is (if (k, i, j ) is a cyclic permutation of (1, 2, 3)) 1 dμk + [Ii − Ij ])μi μj = 0. dt Ik Remark 29 Another method to calculate the covariant derivative at the unit of D(3) is to use the general formula relative to the adjoint. In the Euclidean frame u˜ 0 = uIG , 0 with v = (μ, vG ) and I˜ = I u˜ , we have  adG,v v = −

I˜ −1 0 0 I



A μ A vG 0 Aμ



I˜ 0 0I



μ vG

,

(2.382)

and thus  B(v, v) = (adG,v )v = −

˜ I˜ −1 (μ × Iμ) . μ × vG

(2.383)

2.7 Rigid Bodies

271

Using the decomposition of I˜ with the inertial principal axes, we obtain ˜ = (μ2 μ3 I3 − I2 , μ3 μ1 I1 − I3 , μ1 μ2 I2 − I1 ). I˜ −1 (μ × Iμ) I1 I2 I3 ˜ are given by ak = Thus the components of I˜ −1 (μ × Iμ)

1 Ik kij μi μj Ij .

(2.384) ¶

2.7.9 Euler Equation for a Rigid Body Let F  be the differential form on E3 that represents the force field on the ˆ u˜ vˆ in system. Then in the space of Euclidean frames, the momentum pˆ = mG Tvˆ∗ (E3 ), vˆ ∈ Tu˜ E3 due to this force must satisfy the fundamental equation of mechanics (with the covariant derivative) ∇vˆ pˆ = Fu˜ .

(2.385)

We recall that from this equation, the power of the force implies the change of kinetic energy: P = < F  , vˆ > = Lvˆ E kin (v) ˆ =

m g(v, ˆ v) ˆ = mg(∇vˆ v, ˆ v) ˆ 2

= m < G(∇vˆ ), vˆ > = < ∇vˆ p, ˆ vˆ > . This fundamental equation is made explicit using a specific Euclidean frame u˜ 0 , and the associated map φ0 , so that φ0 (u) ˜ = (u˜ 0 )−1 u˜ = g ∈ D(3). Since φ0 is an isometry from E3 onto D(3) (equipped with the corresponding metric), the covariant derivative is naturally transformed into the corresponding covariant derivative in Tg (D(3)). The Euler equation on the group D(3). Let F = (φ0−1 )∗ (F  ) be the differential form on D(3), g ∈ D(3), defined by (2.371) and (2.365) from the field of volumetric forces f . The fundamental equation of mechanics relative to the momentum p˜ = mGv, ˜ with x(t) = gt , v˜ = g˙ = φ0∗ (v) ˆ ∈ Tg D(3), is ∇v˜ p˜ = Fg ,

(2.386)

and the fundamental equation on the velocity is ∇v˜ v˜ = m−1 G−1 g Fg .

(2.387)

This equation is also made explicit by transport to the unit of the group D(3). The relations between velocities and momenta are indicated in the following diagram:

272

2 Classical Mechanics t (r

t (l

g∗ )

g∗ )

pE ∈ Te∗ D(3) ←− p ∈ Tg∗ D(3) −→ pS ∈ Te∗ D(3) ↑ mGeu˜

↑ mGug˜

0

↑ mGue˜

ωgr

ωgl

v E ∈ Te D(3) ←− g˙ ∈ Tg D(3) −→ v S ∈ Te D(3), Euler equation on d3 and d3∗ . Then we write (2.387) in the Lie algebra d3 in the following form (with F = F E or F S ): dv − B(v, v) = m−1 G−1 e F. dt

(2.388)

Let us apply Ge to (2.388). Recall that GB(u, u) = {u, Gu} (see (2.539) below). We obtain the Euler equation in d3∗ , dp − {v, p} = F, dt

(2.389)

with p = mGv, {v, p} = adv∗ p given by (2.295). With the right transport and the wrench (2.371), the Euler the equation for pE = (M E , p0E ) is ˜ = Gu˜ (ωr ∇v(t ˜ ) v(t)) 0

dpE  E E  − v , p = F E. dt

(2.390)

With the left transport, the Euler equation for pS is Gu˜ (ωl ∇v(t ˜ = ˜ ) v(t))

dpS  S S  − v , p = F S. dt

A matrix version of equation (2.389), with p = (M, po ) and v = (μ, vo ), is 

dM/dt dpo /dt



 +

A μ A vo 0 Aμ



M po



 =

MF RF

,

(2.391)

from which we deduce the equation of motion for the rigid body, with (μ, vo ) given from (M, po ) by (2.363) for the left transport. Proposition 5 The evolution equation of the rigid body relative to its momentum (M, p0 ) ∈ d3∗ , with given forces by the wrench (MF , RF ), is the Euler equation 

(i) dM dt + Aμ M + Avo p0 = MF , 0 (ii) dp dt + Aμ p0 = RF ,

in the case of gravitational forces, the wrench is given by (2.375).

(2.392)

2.7 Rigid Bodies

273

Remark 30 The force field f is generally known (or assumed to be known) in E3 , thus in R 3 , but F E depends a priori on the position of the body, as we can see for gravitational forces: RFE = −gme3 is known, but that is not the case for E (t) × e 0 , which depends on the evolution of the center of mass of MFE = −gm xG 3 the rigid body. ¶ Free rigid body (without exterior forces). From the Euler equation (2.390), if there is no exterior force, that is, F E = 0 (and F S = 0), then ∇v˜ v˜ = 0: the free evolution of the rigid body is along the geodesics. We will prove the following properties: 

   v S , pS = v E , pE = 0,

dpE dpS = = 0, dt dt

dv S dv E = = 0. dt dt (2.393) This will imply that p0E = mvG (see (2.361)) is constant, that the motion of the center of mass is a straight line, and that the axis of the rotation is along this straight line. Proof of (2.393) With (2.360), we have pS = t Ads (pE ) using the variable s = g. We first prove that (i)

d Ads = (ad v E ) ◦ Ads , dt

    (ii) v S , pS = t Ads ( v E , pE ).

(2.394)

From (2.286), we have with s = (R, a), d Ad(R,a) = dt





 0 AμE 0 0 I 0 R. R+ AμE R = Aa˙ + Aa AμE AμE Aa˙ 0 Aa I

We have also ad v ◦ AdR,a = E

AμE 0 Av E AμE 0



I 0 Aa I



R=

Av E 0

0 AμE + AμE Aa AμE

R.

Moreover, v0E = a˙ − AμE a = a˙ − μE × a. Thus Av E + AμE Aa = Aa˙ − AμE ×a + AμE Aa = Aa˙ + Aa AμE , 0

thanks to the Jacobi identity. This is (2.394)(i). Then (2.394)(ii) is a consequence of the relation pS = Ads∗ (pE ) and < [v S , w], pS > = < [Ads −1 )v E , w], Ads∗ (pE ) > = < [v E , Ads w], pE >

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2 Classical Mechanics

for all w ∈ d3 , and thus   < w,t (ad v S )pS > = < w, v S , pS >   = < Ads w, (ad ∗ v E )pE > = < Ads w, v E , pE >, which is (2.394)(ii). Then from (2.394), we have by transposition     d Ads∗ pE = Ads∗ (ad ∗ v E )pE = Ads∗ v E , pE = v S , pS . dt And with the relation pS = Ads∗ (pE ) and the Euler equation of the geodesic, we have   d dpS  S S  dpE − v , p = Ads∗ − Ads∗ v E , pE + ( Ads∗ )pE = 0, dt dt dt we obtain dpS  S S  d dpE  E E  − v , p = Ads∗ − v , p + ( Ads∗ )pE dt dt dt   d = ( Ads∗ )pE = v S , pS = 0. dt Thus we obtain (2.393). This implies that pS = (μS , p0S ), pE = (μE , p0E ) and v S , v E are constant, so that vG is constant, and since p0S = t Rp0E = mt RvG , the axis of the rotation R is along vG , and then p0S = p0E . ¶ Remark 31 If p0S = 0, then vG = 0, and the center of gravity is a fixed point. In this case,50 with μj = I1j MjE , the free Euler equation of the momentum is dMiE  E + ij k μE j Mk = 0. ¶ dt We can find this geodesic equation in the following manner. Let u = (ej ) ∗ be a local  frame in T D(3),  and (sj ) a dual local frame for T D(3) such that v = μj ej , and M = Mj sj . Let A be a unit geodesic field on D(3), and let ωˆ be the Hilbert form on D(3) (see (2.230)), so that ωˆ = GA, ω(A) ˆ = 1. We define the connection form (ωk,i ) on D(3) by (ωk,i )v = kij μj ⊗ ωˆ v . Thus we have (ωk,i )v (A) = kij μj . Then the Euler equation is given by ∇A M =

50 And

 dMi  + Mk kij μj )si = 0. ( dt

more generally when there is a fixed point.

2.7 Rigid Bodies

275

Remark 32 The connection form ω on the principal bundle P = E3 × D(3). Let π, respectively πD(3) , be the natural projection from P onto E3 , respectively D(3). Let ω˜ be the connection  form on E3 , valued in d3 so that the  covariant derivative of a section s = μ s of T (E ) is given by ∇s = dμj sj + j j 3  μk ω˜ k,j sj . Let ωl be the left Maurer–Cartan form on D(3). Then the connection form ω on P , valued in d3 , is given by ∗ l −1 ωu, ˜ u˜ π∗ , ˜ a˜ = (πD(3) ω )u, ˜ a˜ + Ad (a˜ ) ω

∀(u, ˜ a) ˜ ∈ P.

(2.395)

Proof Let A∗ be the vector field on P induced by A ∈ d3 . Then in order that ω be a connection form, we have to prove the following properties (see [Kob-Nom, ch. II.1, p. 64]): ω(A∗ ) = A,

rb∗˜ ω = Ad (b˜ −1 ) ω,

(2.396)

˜ thus (r ∗ ω)(u, with rb˜ the right multiplication by b; ˜ a) ˜ = ω(u, ˜ (rb˜ )∗ . ˜ a˜ b) ˜ b

(A∗ )

ωal˜ (πD(3) )∗ A∗

= 0, and = ωal˜ Aa˜ = A, and then ω(A∗ ) = A. (i) We have π∗ ˜ a) ˜ = a˜ b˜ = rb˜ a, ˜ and thus (ii) We have also πD(3) rb˜ (u, ∗ ∗ l −1 l (rb∗˜ πD(3) ωl )u, ˜ a˜ = rb˜ ωa˜ = Ad (a˜ )ωa˜ .

Then we obtain (2.396) from ˜ −1 )ω˜ u˜ π∗ (r ˜ )∗ = Ad (b˜ −1 )[Ad (a˜ −1 )ω˜ u˜ π∗ ]. rb∗˜ Ad (a˜ −1 )ω˜ u˜ π∗ = Ad ((a˜ b) b Now let τ be a curve in E3 , with τ (t) = u(t), ˜ t ∈ R, and let τˆ (t) = (u(t), ˜ a(t)), ˜ τˆ be a lift of τ in P . We want to determine a˜ so that τˆ = τ∗ is a horizontal lift of τ . ˙˜ a). ˙˜ Then we must have ω (X) = 0. With the connection form (2.395), Let X = (u, we have ˙˜ + Ad (a˜ −1 )ω˜ u˜ (u) ˙˜ = 0. ω (X) = ωal˜ (a)

(2.397)

˙˜ = a˙˜ a˜ −1 ∈ d3 . Then for a given Thus a˜ must satisfy ω˜ u˜ (u) ˜˙ = −a˙˜ a˜ −1 = −ωar˜ (a) (smooth) curve τ in E3 , we obtain (thanks to the lemma of [Kob-Nom, ch. II.3, p. 69]) a unique curve (a˜ t ) in d3 such that a˜ 0 = e and (a˙˜ a˜ −1)t = Yt , with Yt = ˙˜ ω˜ u˜ (u(t)). With the reference frame u˜ 0 , let g = (u˜ 0 )−1 u˜ = φ0 (u) ˜ ∈ D(3), as in (2.311). Then u˙˜ = (φ0−1 )∗ v, ˜ with v˜ = g, ˙ so that ˙˜ ((φ0−1 )∗ ω) ˜ g v˜ = ω˜ u˜ (φ0−1 )∗ v˜ = −ωar˜ (a).

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2 Classical Mechanics

We have v E = ωr (v), ˜ v˜ = (rg˜ )∗ v E . Thus ˙˜ ω˜ u˜ (φ0−1 rg˜ )∗ (v E ) = −ωar˜ (a). Then ω˜ u˜ (φ0−1 rg˜ )∗ = (φ0−1 rg˜ )∗ ω˜ at e is the connection form ω˜ e on d3 , which is  k given by the Riemann–Christoffel symbols (2.381) (ω˜ e )i,j = i,j ek , so that the E h −1 ˙ relation between the velocity v and v = a˜ a˜ in d3 is ˙˜ ω˜ e (v E ) = −v h = −ωar˜ (a).

(2.398)

If v E is a solution of the Euler equation, this implies that the curve τ is a geodesic, and thus v E = v h , so that v E satisfies the equation ω˜ e (v E ) + v E = 0. ¶ Remark 33 On units. The components of a wrench associated with a force given by its moment and its resultant do not have the same dimension from the units point of view; likewise for components of velocities and momenta of D(3). With three dimensions, with v = (μ, vo ), v = v E = (μE , voE ), or v = v S = (μS , voS ), p = (M, po ), and F = (MF , RF ), the units are −1 ; [μ] = T −1 , [vo ] = LT −1 ; [M] = ML2 T −1 , [po ] = MLT   [MF ] = ML2 T −2 , [RF ] = MLT −2 , [E] = ML2 T −2 , M 2 = ML4 T −2 ,

the units of the inertia tensor being [mI] = ML2 , [I] = L2 .

¶

2.7.10 Some Angular Parametrizations of SO(3) In order to write explicitly the components of the Euler equations, a parametric representation of the manifold SO(3) of rotations in R 3 is useful. Of course various representations are possible.

Parametrization with Rotational Axis A first example is given, using the rotation of the μ-axis through the angle θ with (θ, μ) ∈ (2πT ) × S 2 → Rθμ ∈ SO(3). Using spherical coordinates of the unit sphere S 2 leads to a parametric representation of SO(3). But the relation Rθμ = R2π−θ points out that this map is not bijective, since every R ∈ SO(3) −μ that is different from the unit I is the image of two elements. We obtain a (partial) parametric representation by restricting the map’s domain to the set ]0, π[×S 2 . But

2.7 Rigid Bodies

277

the unit of the group is not obtained by this parametric representation.51 Notice that the sphere S 2 is identified with the quotient SO(3)/SO(2) if we identify the torus 2πT with SO(2), so that we could use the fact that S 2 is a symmetric space (see [Kob-Nom, Vol. 2, ch. XI.10, p. 264]).

Parametric Representation by the Euler Angles Let u0 = (e10 , e20 , e30 ) (the reference frame) and u = (e1 , e2 , e3 ) be two Euclidean frames with the same origin O of E3 . There exists one and only one rotation R that transforms the frame u0 into u. The transformation of the vectors ej0 by a rotation R is written by52 Rej0 = ej ,

j = 1, 2, 3.

(2.399)

3 We identify a frame u = (e1 , ie2 , e3 ) with the corresponding 0linear map u from R into E3 , so that u(ξ ) = ei ξ . Then the transformation of u into u corresponding to (2.399) is given by u = u0 t R. A parametric representation of the rotation R is obtained with the Euler angles (θ, φ, ψ), so that R reads as the following product: φ e3

ψ e3

R = R 0 Rθe0 R 0 . 1

(2.400)

With φ = θ = ψ = 0, we have R = e (also denoted by I ), the identity of the group SO(3). Using the following property Rθgμ = gRθμ g −1 ,

∀g ∈ SO(3), μ ∈ S 2 ,

(2.401)

which is valid for every angle θ , we define the unit vector eN called the nodal vector orthogonal to e30 and e3 by eN = λe30 × e3 ,

with λ =

1 > 0. sin θ

51 It may be interesting (notably in quantum physics) to use a parametric representation of the universal covering of SO(3) (of order 2), which is given by the unitary group SU (2) and which is identified with the sphere S 3 or with the group of quaternions of norm 1. 52 For instance, we write  cos φ sin φ −φ . (e10 , e20 )R 0 = (e10 , e20 ) e3 − sin φ cos φ

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2 Classical Mechanics

Following this definition of the vector eN , we assume that e30 and e3 are not collinear, and thus θ = 0, π, and hence 0 < θ < π. Then we can define the Euler angles by φ e3

R 0 e10 = eN ,

RθeN e30 = e3 ,

Rψ e3 eN = e1 ,

(2.402)

and thus θ = angle (e30 , e3 ), φ = angle (e10 , eN ), ψ = angle (eN , e1 ).

(2.403)

Then R = R(θ,φ,ψ) is also expressed by φ e3

θ R = Rψ e3 ReN R 0 ,

ψ e3

ψ+φ , e30

φ e3

and thus R = RθeN (R 0 R 0 ) = RθeN R

(2.404)

which is easily verified, for instance, by showing that (2.399) is carried out by (2.404). With (2.400), we also have R = RR(θ,φ,ψ) R−1 = Rφe3 Rθe1 Rψ e3 .

(2.405)

ˆ gives a parametrization in SO(3), The following mapping R ˆ R (θ, φ, ψ) ∈ U˜ = ]0, π[ × 2πT × 2πT −→ R(θ,φ,ψ) ∈ SO(3),

(2.406)

ˆ is a diffeomorphism from U = ]0, π[ × ]0, 2π[ × ]0, 2π[ onto RU ˆ . We and R 0 ˆ verify that R is injective on U by application on the frame u : the obtained frame ˆ does not contain the unit element I defines Euler angles in a unique way. But RU 0 53 of the rotation group of axis e3 . ˆ The derivative R ˆ ∗ of the map Rˆ An important property of the mapping R. ∂ ∂ ∂ transforms the frame of tangent vectors ( ∂θ , ∂φ , ∂ψ ) (at (θ, φ, ψ)) into the frame of TR SO(3): vR = (AeN R, Ae0 R, Ae3 R) = (AeN , Ae0 , Ae3 )R. 3

3

We transform vR into the frame v E = vR R−1 = (AeN , Ae0 , Ae3 ) of TI SO(3), 3 thus of the Lie algebra o(3) of SO(3), by multiplication on the right by R−1 , and we identify v E with the frame v I = (eN , e30 , e3 ) of E3 that is nonorthogonal (thus non-Euclidean),

53 The fact that the Euler parametrization does not give a diffeomorphism in a neighborhood of the unit may be viewed by taking the derivative of (2.400) with respect to the angles θ, φ, ψ at 0, which gives the nonindependent vector fields Ae0 , Ae0 , Ae0 . 3

1

3

2.7 Rigid Bodies

ˆ∗ R



∂ ∂θ

279

= AeN R,

Rˆ ∗



∂ ∂φ

= Ae0 R, 3

ˆ∗ R



∂ ∂ψ

= Ae3 R,

(2.407)

which simply corresponds to the relations ∂ ˆ ˆ∗ R=R ∂θ



∂ ∂θ

,

∂ ˆ ˆ∗ R=R ∂φ



∂ ∂φ

,

∂ ˆ ˆ∗ R=R ∂ψ



∂ ∂ψ



ˆ∗ Thus we can identify the frame v I = (eN , e30 , e3 ) with the frame R

.

(2.408)

∂ ∂ ∂ ∂θ , ∂φ , ∂ψ

 .

Change of frames between v I = (eN , e30 , e3 ), u0 , and u. (i) Exchange the frame v I for u0 = (e10 , e20 , e30 ). The spherical coordinates of e3 in the frame u0 are expressed as functions of the Euler angles by (θ, φ − π2 ), and ψ is the angle of rotation with respect to e3 , which gives the following expressions for e3 and eN : e3 = sin θ sin φ e10 − sin θ cos φ e20 + cos θ e30 , eN = cos φ e10 + sin φ e20 .

(2.409)

(ii) Exchange the frame v I for u = (e1 , e2 , e3 ). −ψ

With e30 = R−θ eN e3 and eN = Re3 e1 (see (2.402)), we have e30 = cos θ e3 + sin θ (cos ψ e2 + sin ψ e1 ), eN = cos ψ e1 − sin ψ e2 .

(2.410)

We can verify that the frames v I and u have the same orientation using the exterior product. Thus we have the relations v I = u0 M0I = uMSI ,

(2.411)

with the transition matrices M0I , MSI = RM0I (which are not in SO(3)), ⎛

M0I

⎞ cos φ 0 sin θ sin φ = ⎝ sin φ 0 − sin θ cos φ ⎠ , 0 1 cos θ

⎛

MSI

⎞ cos ψ sin θ sin ψ 0 = ⎝ − sin ψ sin θ cos ψ 0 ⎠ . 0 cos θ 1

Expression for the angular velocity vectors μE and μS with Euler angles. → The components of the instantaneous angular velocity vector − μ are given by ∂ ∂ ∂ 54 ˙ ψ) ˙ in the frame ( , , ). Then (θ˙ , φ, ∂θ ∂φ ∂ψ

54 If

→ we identify u0 with the canonical frame of R 3 , we can identify − μ with μE .

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2 Classical Mechanics

− → μ = u0 (μE ) = u(μS ) = θ˙ eN + φ˙ e30 + ψ˙ e3 .

(2.412)

˙ = (θ˙ ∂ + φ˙ ∂ + ψ˙ ∂ ) ∈ T U ; then we have Let ∂θ ∂φ ∂ψ ˙ =R ˆ ∗ ˙ 0 + ψA ˙ e3 )R = A− ˙ = (θ˙ AeN + φA → R μ R. e 3

(2.413)

Substituting (2.409) then (2.410) into the expression (2.412) for μE , we obtain ˙ with the corresponding column vector) (identifying − → ˙ = u0 M0I ( ) ˙ = uMSI ( ), ˙ μ = v I ( )

˙ = (θ˙ , φ, ˙ ψ), ˙ with t

(2.414)

which gives the instantaneous angular velocity vector μS and μE in matrix form ˙ μS = MSI ( ),

˙ μE = M0I ( ).

(2.415)

l = R ˆ ∗ (ωl ), which is the Darboux differentials. The left Darboux differential π ˆ (see (2.406)), is pullback of the left Maurer–Cartan form ω = ωl of SO(3) by R such that l ˙ ˙ = AμS . ˆ ∗ (ωl ), ˙ > = < ωl , Rˆ ∗ ( ) ˙ > = R−1 Rˆ ∗ ( ) ˙ = R−1 R , >=< R < π

We have also l = AR−1 (eN ) dθ + AR−1 (e0 ) dφ + Ae0 dψ, π 3 3 l = R−1 (e )dθ + R−1 (e 0 )dφ + e 0 dψ = u0 M d . identified with π¯ N SI 3 3 The right Darboux differential π r = Rˆ ∗ (ωr ) is given by r ˙ ˆ ∗ ( ) ˙ > = AμE , < π , > = < ωr , R

and thus π r = AeN dθ + Ae0 dφ + Ae3 dψ, which is identified with π¯ r = eN dθ + 3

e30 dφ + e3 dψ. Inertia metric on SO(3) with the Euler angles. Taking the frame u with  the axes of inertia of the rigid body, the inertia metric is given by gIe (μS ) = Ij (μSj )2 . But we have to use the relations (2.415) with respect to the Euler angles on U˜ (see (2.406)), with = (θ, φ, ψ). The result is not very nice, because the frame v I is not orthogonal. Then the inertia metric gI is given by ˆ ∗ gI ) = I1 [cos ψ dθ + sin θ sin ψ dφ]2 (R + I2 [− sin ψ dθ + sin θ cos ψ dφ]2 + I3 [dψ + cos θ dφ]2 .

(2.416)

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281

We can obtain this formula thanks to the left Darboux differential by ˆ ∗ (gue 0 ◦ ω)( ) ˆ ∗ ( ) ˙ =R ˙ = gue 0 (ωR ˙ = gue 0 (π l ( )) ˙ = gue 0 (μS ). Rˆ ∗ gI ( ) (2.417) For a rigid body (and for rotations alone), the metric expressed in the Euler angles is the pullback metric by R of the inertia metric on SO(3) (due to the inertia metric of D(3)). The tensor of the inertia metric I , with the Euler angles, is such that ˙ I ˙ > = < MSI , ˙ IMSI ˙ > = < μS , IμS > = gue 0 (μS ). < , Then it is expressed by I = t MSI IMSI , with I = I G (see (2.324)), which gives the matrix representation of I in the frame (dθ, dφ, dψ), ⎞ (I1 −I2 ) sin θ sin 2ψ 0 I1 cos2 ψ + I2 sin2 ψ 2 ⎝ (I1 −I2 ) sin θ sin 2ψ (I1 sin2 ψ + I2 cos2 ψ) sin2 θ + I3 cos2 θ I3 cos θ ⎠ , 2 0 I3 cos θ I3 ⎛

or also with I1,2 = 12 (I1 + I2 ), I1,2 = 12 (I1 − I2 ), ⎛

⎞ I1,2 + I1,2 cos 2ψ I1,2 sin θ sin 2ψ 0 ⎝ I1,2 sin θ sin 2ψ (I1,2 − I1,2 cos 2ψ) sin2 θ + I3 cos2 θ I3 cos θ ⎠ . I3 0 I3 cos θ In the case of an axially symmetric rigid body, with I1 = I2 = I1,2 , the inertia metric is simply expressed by ˆ ∗ gI ) = I1,2 [(dθ )2 + sin2 θ (dφ)2 ] + I3 [dψ + cos θ dφ]2 . (R Expression of forces with Euler variables. Using (2.415), we have with the Euler angles, ˙ ˙ > = MFS .MSI ( ), < MFS , μS > = < MFS , MSI and with v◦S = R−1 a, ˙ < RFS , v◦S > = < RRFS , a˙ > . This allows us to write the power P using (2.372). ¶

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Another Angular Parametrization of SO(3) A rotation R is defined from the frame (e10 , e20 , e30 ) of E3 and the angles ϕ1 , ϕ2 , ϕ3 by55 R = R1 R2 R3 ,

ϕ ei

Ri = R 0i .

with

(2.418)

At ϕi = 0, we have Ri = I , and thus R = I if ϕi = 0, ∀i. We again write ei = Rei0 . Then RR 0i R−1 = R ϕ ei

ϕi

Rei0

(2.419)

ϕ

= Reii , and thus R = RRR−1 = Rϕe11 Rϕe22 Rϕe33 .

(2.420)

Ri |ϕi =0 = Ae0 , and On taking the derivative, we obtain ∂∂ϕ i i

∂R = R1 Ae0 R2 R3 = R1 Ae0 R−1 1 R = AR1 e20 R, 2 2 ∂ϕ2 and thus ∂R = Ae0 R, 1 ∂ϕ1

∂R = AR1 e0 R, 2 ∂ϕ2

∂R = RAe0 = ARe0 R. 3 3 ∂ϕ3

Then we verify that ϕ e1

ϕ e2

Re30 = R 01 R 02 e30 = sin ϕ2 e10 − sin ϕ1 cos ϕ2 e20 + cos ϕ1 cos ϕ2 e30 , (2.421)

R1 e20 = cos ϕ1 e20 + sin ϕ1 e30 . The vectors e10 , R1 e20 , Re30 are independent if the determinant (e10 , R1 e20 , Re30 ) is nonzero. Now (e10 , R1 e20 , Re30 ) = cos 2ϕ1 cos ϕ2 = 0 if ϕ1 = ± π4 , ϕ2 = ± π2 . Let O = {(ϕ1 , ϕ2 , ϕ3 ) ∈] −

π π π π , + [×] − , + [×] − π, π[}. 4 4 2 2

The map Rˆ : (ϕ1 , ϕ2 , ϕ3 ) → R = R 01 R 02 R ϕ e1

ϕ e2

ϕ3 e30

is a diffeomorphism from O onto

ˆ ⊂ SO(3). Then we can identify the frame RO

control theory, see for instance [Isi, ch. 5.5, p. 264], we write ϕ1 = ψ, ϕ2 = −θ, ϕ3 = φ (ψ, θ, φ are respectively called roll, pitch, yaw).

55 In

2.7 Rigid Bodies

283

−1 −1 v I = (e10 , R1 e20 , Re30 ) = (R−1 e1 , R1 R−1 e2 , e3 ) = (R−1 e3 Re2 e1 , Re3 e2 , e3 )

 ˆ ∗ ∂ , ∂ , ∂ ; v I is a (using RR1 R−1 = Re1 , see (2.401)), with the frame R ∂ϕ1 ∂ϕ2 ∂ϕ3 frame of TI SO(3), thus of the Lie algebra o(3), or of E3 , but this is not a Euclidean frame, since e10 and Re30 are not orthogonal. Then we have relations between the frames u0 = (e10 , e20 , e30 ), u = (e1 , e2 , e3 ), and v I : v I = u0 M0I = uMSI ,

(2.422)

with the matrices ⎛

M0I

⎛ ⎞ ⎞ 1 0 sin ϕ2 cos ϕ2 cos ϕ3 sin ϕ3 0 = ⎝ 0 cos ϕ1 − sin ϕ1 cos ϕ2 ⎠ MSI = ⎝ − cos ϕ2 sin ϕ3 cos ϕ3 0 ⎠ . sin ϕ2 0 sin ϕ1 cos ϕ1 cos ϕ2 0 1

New expression of the angular velocity vectors μE , μS . Then the components of → the instantaneous angular velocity vectors − μ are expressed by (ϕ˙ 1 , ϕ˙2 , ϕ˙3 ) in the I frame v . Then we have − → μ = u0 (μE ) = u(μS ) = ϕ˙1 e10 + ϕ˙2 R1 e20 + ϕ˙ 3 Re30 ∈ E3 . Let ϕ˙ =



(2.423)

ϕ˙ i ∂ϕ∂ i . Indeed, we have

ˆ ∗ (ϕ) ˙ = R˙ = R



ϕ˙ i

∂R → = (ϕ˙1 Ae0 + ϕ˙ 2 AR1 e0 + ϕ˙ 3 ARe0 )R = A− μ R. 1 2 3 ∂ϕi

(2.424)

On identifying ϕ˙ with − → μ = v I .ϕ˙ = u0 M0I ϕ˙ = uMSI ϕ, ˙ with t ϕ˙ = (ϕ˙1 , ϕ˙2 , ϕ˙3 ),

(2.425)

we obtain the instantaneous angular velocity vectors μS and μE , with matrix notation μS = MSI (ϕ), ˙

μE = M0I (ϕ). ˙

(2.426)

ˆ ∗ (ωl ) of the map R ˆ gives Darboux differentials. The Darboux differential π l = R (as for the Euler angles) ˆ ∗ (ϕ) ˙ = AμS . ˙ > = R−1 R ˙ = R−1 R < πϕl , ϕ˙ > = < Rˆ ∗ (ω), ϕ˙ > = < ω, Rˆ ∗ (ϕ) We also have πϕl = AR−1 e0 dϕ1 + AR−1 e0 dϕ2 + Ae0 dϕ3 , 1

3

2

3

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identified with 0 0 0 π¯ ϕl = R−1 e10 dϕ1 + R−1 3 e2 dϕ2 + e3 dϕ3 = u MSI dϕ.

The right Darboux differential π r = Rˆ ∗ (ωr ) is given by ˙ > = AμE , < πϕr , ϕ˙ > = < ωr , Rˆ ∗ (ϕ) and thus πϕr = Ae0 dϕ1 + AR1 e0 dϕ2 + ARe0 dϕ3 , 1

2

3

which is identified with π¯ ϕr = e10 dϕ1 + R1 e20 dϕ2 + Re30 dϕ3 = v I dϕ = u0 M0I dϕ. Expression of the inertia metric for rotations with this parametrization. With the frame u taken with the axes of inertia of the rigid body, the inertia metric expressed 0 by gue , as a function of ϕ, is given from (2.426) by ˆ ∗ gI )ϕ = I1 [cos ϕ2 cos ϕ3 dϕ1 + sin ϕ3 dϕ2 ]2 (R + I2 [− cos ϕ2 sin ϕ3 dϕ1 + cos ϕ3 dϕ2 ]2 + I3 [dϕ3 + sin ϕ2 dϕ1 ]2 .

(2.427)

The metric expressed with these angles is again the pullback metric by R of the inertia metric of SO(3) due to the inertia metric of D(3), and we have, as in (2.417), with the Darboux differential, (Rˆ ∗ gI )(ϕ) ˙ = gue (μS ), 0

ˆ ∗ (ωl ) = gue ◦ π l . and Rˆ ∗ gI = gue ◦ R 0

0

The inertia tensor I = (g(ϕ)i,j ) associated with the frame u is then expressed with the angles ϕi by Iϕ = t MSI IMSI , or in matrix representation, with I1,2 = 1 2 (I1 − I2 ), by ⎛

⎞ (I1 cos2 ϕ3 + I2 sin2 ϕ3 ) cos2 ϕ2 + I3 sin2 ϕ2 I1,2 cos ϕ2 sin 2ϕ3 I3 sin ϕ2 ⎝ ⎠. I1,2 cos ϕ2 sin 2ϕ3 I1 sin2 ϕ3 + I2 cos2 ϕ3 0 0 I3 I3 sin ϕ2 In the case of an axially symmetric rigid body, with I1 = I2 = I1,2 , the inertia metric is (Rˆ ∗ gI )ϕ = I1,2 [cos2 ϕ2 (dϕ1 )2 + (dϕ2 )2 ] + I3 [dϕ3 + sin ϕ2 dϕ1 ]2 .

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285

Parametrization with the Exponential on S 3 The parametrization of SO(3) by spherical coordinates of the sphere56 S 3 or the ball Bπ = {μ ∈ R 3 , | μ |≤ π}, using the exponential mapping as in (2.279), leads to an orthonormal frame of o(3). Let ν = νφ,α ∈ S 3 , α = αθ ϕ ∈ S 2 be φ the spherical Gauss map from SO(3) on S 3 that is given, for the rotation Rα (with 2 α ∈ S identified with the quotient space SO(3)/SO(2)), by νφ,α = (cos φ, sin φ α),

α = (cos θ, sin θ cos ϕ, sin θ sin ϕ),

(2.428)

with 0 ≤ φ ≤ π, 0 ≤ θ ≤ π, 0 ≤ ϕ < 2π. By derivation of α with respect to θ , then ϕ, we obtain ∂α = (− sin θ, cos θ cos ϕ, cos θ sin ϕ) ∂θ ∂α eϕ = = (0, − sin θ sin ϕ, sin θ cos ϕ) = sin θ e˜ϕ . ∂ϕ

eθ =

(2.429)

Then (α, eθ , e˜ϕ ) is an orthonormal direct frame in R 3 (for the usual metric), and we have α × eθ = e˜ϕ ,

eθ × e˜ϕ = α,

e˜ϕ × α = eθ ,

(2.430) φ

corresponding to the Lie algebra o(3). The partial derivatives of Rα = exp Aφα with respect to the variables φ, θ, ϕ (using (2.281) with μ = φα ∈ R 3 ) give the following elements of the Lie algebra o(3): φ

Ae N = φ

∂Rα −φ Rα , ∂φ

φ

Ae N = θ

∂Rα −φ Rα , ∂θ

φ

AeϕN =

∂Rα −φ Rα . ∂ϕ

We observe that Rϕe3 (e2 ) = e˜ϕ ,

Reθ˜ϕ (e3 ) = α.

Thus we have the relations ϕ θ −ϕ φ ϕ −θ −ϕ Rφα = Rθe˜ϕ Rφe3 R−θ e˜ϕ = Re3 Re2 Re3 Re3 Re3 Re2 Re3 ,

and then with ζ = Rθe2 (e3 ), we obtain Rφα = Rϕe3 Rζ R−ϕ e3 . φ

implicitly use the fact that S 3 is a 2-sheeted universal cover of SO(3), since ν and −ν give the same rotation, and S 3 is identified with the quotient space SO(4)/SO(3); see [Kob-Nom, T2, Ch XI.10]. Note that the map φ → νφ,α gives geodesics of the sphere S 3 . 56 We

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Taking the derivative, we have φ

φ

∂Rα = Ae3 Rφα − Rφα Ae3 , ∂ϕ

φ

∂Rα = Ae˜ϕ Rφα − Rφα Ae˜ϕ . ∂θ (2.431) φ Since e3 = R−θ α = (cos θ )α−(sin θ )e , R e ˜ = (cos φ) e ˜ −(sin φ)e , we obtain θ ϕ ϕ θ α e˜ϕ ∂Rα = Aα Rφα , ∂φ

∂ ∂ ∂ , ∂θ , ∂ϕ ), the frame v I = (eφN , eθN , eϕN ) in o(3), which we can identify with R∗ ( ∂φ by

eφN = α, eθN = −Rφα e˜ϕ + e˜ϕ = sin φ eθ + (1 − cos φ)e˜ϕ ,

(2.432)

eϕN = −Rφα e3 + e3 = sin θ [(−1 + cos φ) eθ + sin φ e˜ϕ ]. We verify that these vectors are orthogonal, and that their norms are57 | eφN |= 1,

| eθN |= 2 sin

φ , 2

| eϕN |= 2 sin θ sin

φ ; 2

I of the Lie algebra o(3) (for the natural thus we obtain an orthonormal frame vN I = (e N , eN , e N ): metric) with the corresponding normalized unit vectors vN ϕ φ θ



I N N vN = (eN φ , e θ , eϕ ) = α, eθ , e˜ϕ



⎞ 1 0 0 ⎝ 0 cos φ − sin φ ⎠ . 2 2 0 sin φ2 cos φ2 ⎛

(2.433)

φ/2

I = (α, e , e˜ )R Thus vN θ ϕ α . Then the (right) instantaneous angular velocity vector E I ˙ ˙ ˙ θ˙ , ϕ), μ = v ( ), with = (φ, ˙ is given by

⎛ ⎞ ⎛ ⎞ φ˙ θ˙˜   ⎜ ⎟ μE = (eφN , eθN , eϕN ) ⎝ θ˙ ⎠ = α, eθ , e˜ϕ Rφ/2 (2 sin φ2 ) θ˙ ⎠ . α ⎝ ϕ˙ (2 sin φ2 sin θ ) ϕ˙ Then the right Darboux differential π r of the exponential, or of the mapping = φ r = A dφ + A dθ + A dϕ, identified (φ, θ, ϕ) $→ Rα ∈ SO(3), is given by π α eϕN eN θ

r = α dφ + e N dθ + e N dϕ = v I d . with π¯ ϕ θ The (left) instantaneous angular velocity vector μS is such that −φ/2 . μS = R−φ α μE = (α, eθ , e˜ϕ )Rα

57 Recall

that with 0 < φ < π, 0 < θ < π, we have sin

φ 2

> 0, sin θ > 0.

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287

The inner product in o(3) is given by μ.μ = − 12 tr (Aμ Aμ ) (which corresponds, up to a factor, to the Killing–Cartan form of o(3)). The invariant Riemannian metric on SO(3) is given by φ g(μ) = φ˙ 2 |α|2 + θ˙ 2 |eθN |2 + ϕ˙ 2 |eϕN |2 = φ˙ 2 + 4 sin2 [θ˙ 2 + ϕ˙ 2 sin2 θ ], 2 in contrast to the Riemannian metric on S3 (induced by the metric of R 4 ), g = dφ 2 + sin2 φ[dθ 2 + sin2 θ dϕ 2 ]. Then choosing (α, eθ , e˜ϕ ) as inertia frame, we obtain the (left) inertial Riemannian metric gI on SO(3)\{e} (on the chart due to angles = (φ, θ, ϕ), or the pullback  φ of the Riemannian metric gI (μ) = Ij (μj )2 by the map → Rα ), gI (μ) = I1 φ˙ 2 + 4 sin2

φ [I2 θ˙ 2 + I3 ϕ˙ 2 sin2 θ ]. 2

(2.434)

Thus we see that this parametrization of SO(3) gives, for the inertial Riemannian metric, simpler formulas than those given previously, since the frame v I is orthogonal.

2.7.11 Potential Energy for Forces with Potential For an “exterior” force field f given in E3 or in R 3 that is the derivative of a potential, it is a priori not obvious that the corresponding force field F on T D(3) (or the wrench F E or F S ) can also be the derivative of a potential. At first we verify that in the case of uniform gravitational forces, the field F is effectively the derivative of a potential. It is sufficient to show that the moment MF of F is the derivative of a potential. For g = (R, 0), taking the function 0 U (g) = gm(xG .e30 ) = gm(RxG ).e30

on D(3), we have for every vector field X on SO(3), identified with a vector μ of R 3 by XR = Aμ R, and with e30 identified with the canonical vector 3 of R 3 , 0 ).e30 ) = gm (Aμ xG ).e30 < dU, X > = LX (U ) = gmLX ((RxG

= gm (μ × xG ).3 = gmμ.(xG × 3 ) = < −MF , X >,

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which implies F = −dU . For an axially symmetric rigid body, the center of mass −→ G is on the axis (e3 , O), OG = u0 (xG ) = le3 (with constant l), and MF = −gml e3 × e30 = −gml sin θ eN ,

and U = gmle3 .e30 = gml cos θ,

which gives MF = −( ∂U ¶ ∂θ )dθ . In a more general setting, we have the following. Proposition 6 If the “exterior” force field f in R 3 is the derivative of a potential, then the corresponding force field F on T D(3) also is (locally) the derivative of a potential. Proof We use the parametrization of SO(3) with (ϕ1 , ϕ2 , ϕ3 ) ∈ O. Let u˜ ν = ( ∂ϕ∂ 1 , ∂ϕ∂ 2 , ∂ϕ∂ 3 ) be the corresponding frame of TI SO(3). Then the expression (2.424) for the velocity is ˙ = Rˆ ∗ ( R



ϕ˙i

 ∂ ˆ ∗ (u˜ ν (ϕ)) )=R ˙ = ϕ˙ i AμE R, j ∂ϕi

with (μE j ) = (1 , R1 2 , R3 ).

The power of forces is obtained from (2.366) with the velocity vector v E . Recall that v E = (μE , voE ), with voE = a˙ − μE × a, and that P is expressed by (2.372), or also by P = M˜ FE .μE + RFE .a, ˙ with M˜ FE = MFE − a × RFE =

 Ry × f (gy) dy.

(2.435)



Hence P = < Fg , g˙ > = αj = M˜ FE .μE j,



(αj ϕ˙j + βj a˙ j ), with

βj = (RFE )j .

(2.436)

Then the force field F is locally the derivative of a potential if there exists a function ∂U ∂U U (ϕ, a) such that αj = ∂ϕ , βj = ∂a . A necessary condition is that the Schwarz j j conditions be satisfied: def

(i) Iij =

∂βj ∂αj ∂αi ∂αj def ∂βi def ∂βi − = 0, (ii) Jij = − = 0, (iii) Kij = − = 0. ∂ϕj ∂ϕi ∂aj ∂ai ∂ϕj ∂ai

We prove that these conditions are satisfied if the force field f on R 3 is such that df = 0, which is also written in the usual way as curl f = 0. Condition (ii): For every i, j we have 

∂fj ∂fi Jij = ( − )(gy)dy = ∂xi  ∂xj

 (curl f )i,j (gy)dy. 

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289

Condition (iii): Kij =

 k

= =



 k



∂fi ∂xk ∂ (gy) − [(Ry × f (gy)).μE j ]dy ∂xk ∂ϕj ∂ai

[

∂fi ∂f E .(μE .μ ]dy j × Ry)k − [Ry) × ∂xk ∂xi j

[

∂fi ∂fk .(μE .(μE j × Ry)k − j × Ry)k ]dy. ∂xk ∂xi



 k

[

Therefore, finally, ∂αj ∂βi − = ∂ϕj ∂ai

 

[(curlx f ) × (μE j × Ry)]i dy.

(2.437)

Condition (i):  ∂ ∂ (Ry × f (Ry + a)).μE (Ry × f (Ry + a)).μE Iij = [ i ]−[ j ]dy ∂ϕi  ∂ϕj   ∂fk ∂xl ∂xl =− [ (Ry × μE (Ry × μE i )k − j )k ]dy ∂xl ∂ϕj ∂ϕi  k,l  ∂ ∂ f.[ (Ry × μE (Ry × μE + i )− j )]dy. ∂ϕ ∂ϕ j i  The last integral term is null, since at every point in , its integrand is null: [

∂ ∂ ∂ ∂ ∂ ∂ (Ry × μE (Ry × μE (Ry) − (Ry) = 0. i )− j )] = ∂ϕj ∂ϕi ∂ϕj ∂ϕi ∂ϕi ∂ϕj

Then the first term is  Iij = − 

E [(curlx f ).Ry][Ry.(μE i × μj )]dy = 0.

Thus we have proved that the Schwarz conditions are satisfied if the force field f on R 3 is the derivative of a potential, which completes the proof of the proposition. ¶ Remark 34 On axially symmetric rigid bodies. Then we can restrict ourself to consider the product space M˜ = S 2 × R 3 as the configuration space (it is sufficient to specify the body axes and a particular point of the body, for instance its center of mass). This space is also obtained as the quotient of the space D(3) by the rotation group around the axis of the system, with the identification S 2 = SO(3)/SO(2).

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The Riemannian metric of the unit sphere S 2 is obtained from that of SO(3) by passing to the quotient. Likewise for the covariant derivative and action of the exterior forces. The axial symmetry of the rigid body is taken into account when there is a fixed point, and a uniform force field (independent of x), so there exist in the configuration space D(3) two first integrals of the motion (see, for instance, [Arn1, ch. 6.3]). In the case that the force field derives from a potential, this is a consequence of the Noether theorem. Let K be a symmetry group of the body, that is, the domain M ⊂ E3 of the body and the mass density ρ are invariant under every s ∈ K. Let us specify the invariance under K of the (right-invariant) Riemannian metric gr on D(3) and of gˆ on E3 . The invariance of gr on D(3) by K is given by (ls )∗ gr = gr ,

∀s ∈ K.

Let φ0 be the map from E3 into D(3) such that φ0 (u) ˜ = (u˜ 0 )−1 (u), ˜ whose −1 −1 0 inverse φ0 is such that φ0 (g) = u˜ ◦ g ∈ E3 , ∀g ∈ D(3). The invariance of gr under K is also equivalent to gˆ = (φ0−1 ls φ0 )∗ gˆ ,

s ∈ K.

¶

2.8 Relative Motion We often have to know the motion of a system with respect to a frame that is time-dependent. Such is the case when there are two moving rigid bodies whose motions are to be compared. Many situations are possible, including that the two rigid bodies are independent, without interaction, or that we have a predominant body (for instance, the Earth with respect to any object on the Earth); then the relative motion is said to be a training motion.

2.8.1 Relative Motion of Euclidean Frames Besides the two previously used Euclidean frames, u˜ 0 = (u0 , O0 ), u˜ = (u, O), also denoted by u˜ S = (uS , OS ), let u˜ I be a Euclidean frame of E3 called intermediate, u˜ I = (uI , OI ) = ((e1I , e2I , e3I ), OI ) (for instance, one that is linked to a moving observer). This frame is a function of time with values in E3 ). Thus we have to consider, as in Remark 26, the pair (u˜ S , u˜ I ) ∈ E3 × E3 as a function of time. From the frames u˜ 0 , u˜ I , we define two diffeomorphisms φ0 , φI from E3 onto D(3):

2.8 Relative Motion

291

φ0 (u) ˜ = (u˜ 0 )−1 u, ˜

φI (u) ˜ = (u˜ I )−1 u. ˜

(2.438)

Thus we define the displacements g = g0,S , gI,S , and g0,I , by g0,S = φ0 (u˜ S ),

gI,S = φI (u˜ S ),

g0,I = φ0 (u˜ I ).

(2.439)

The change from one frame to another is made by the displacements g0,I , gI,S , and g0,S according to u˜ I = u˜ 0 ◦ g0,I ,

u˜ S = u˜ I ◦ gI,S = u˜ 0 ◦ g0,S ,

(2.440)

and the relation between these displacements is g0,S = g0,I ◦ gI,S ,

(2.441)

whence R0,S = R0,I RI,S , a0,S = g0,I aI,S = a0,I + R0,I aI,S .

(2.442)

The change of coordinates of points in E3 . Let M ∈ E3 , with coordinates xS , xI , x ∈ R 3 , respectively in the frames u˜ S , u˜ I , u˜ 0 : M = u˜ S (x S ) = u˜ I (x I ) = u˜ 0 (x),

(2.443)

thus with xS = xI = x, −1 x S = g0,S (xS ),

−1 x I = g0,I (xI ),

x I = (u˜ I )−1 u˜ S (x S ) = gI,S (x S ).

For the time-dependent frames u˜ S , u˜ I , the evolution of the coordinates (in R 3 ) of points in E3 is backward, and we write also −1 −1 −1 I x I = g0I (x) = gI 0 (x), x S = g0S (x) = gS0 (x) = g0I x ,

(2.444)

and thus x I = RI,0 x + aI,0 ,

x S = RS,0 x + aS,0 = RS,I x I + aS,I .

2.8.2 Composition of Velocities With the time-dependent frames u˜ S , u˜ I , we derive the relation of displacements (2.441) with respect to time, and we obtain the relation of composition of velocities:

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v = v0,S = g˙0,S = g˙ 0I ◦ gI,S + (g0,I )∗ g˙I,S = v0I ◦ gI,S + R0I vI,S .

(2.445)

Deriving relations (2.442) with respect to time, or using (2.445), we have ˙ 0,S = R˙ 0,I RI,S + R0,I R ˙ I,S , R ˙ 0,I aI,S + R0,I a˙ I,S . a˙ 0,S = a˙ 0,I + R

(2.446)

The velocities of the displacements of the frames u˜ S , u˜ I , can be computed at the origin e of the group D(3) by the Maurer–Cartan forms ωr , ωl , that is, by right or l left transport by the inverse of the displacement g0,S or gI,S . We denote by v0,S and r what was previously denoted by v S and v E , the transport of g˙ at the unit by v0,S 0,S l r the Maurer–Cartan forms ω , ω , denoted by ω; we simply write v0,S = ωg0,S (g˙0,S ),

vI,S = ωgI,S (g˙I,S ),

v0,I = ωg0,I (g˙0,I ).

(2.447)

From (2.445), by left or right multiplication by the inverse of g0,S , we deduce the relation of composition of velocities by left and right transport: l l l v0,S = vI,S + Adg −1 v0,I , I,S

r r r v0,S = Adg0,I vI,S + v0,I .

(2.448)

Consequences for the angular velocities and the translation velocities. The frame velocities may be given through their angular velocities and the velocities of their origin: v0,S = (μ0,S , α0,S )

v0,I = (μ0,I , α0,I ),

vI,S = (μI,S , αI,S );

we obtain the following relations (or by a straight calculation): μr0,S = μr0,I + R0,I μrI,S ,

l l μl0,S = R−1 I,S μ0,I + μI,S .

(2.449)

The relations for the velocities of translations are r r r = −(R0,I μrI,S ) × a0,I + R0,I αI,S + α0,I , α0,S l l l l α0,S = αI,S + R−1 I,S (α0,I + μ0,I × aI,S ).

Proof of (2.449). Indeed, we have ˙ 0,S )R−1 R−1 ˙ 0,S R−1 = (R ˙ 0,I R0,S + R0,I R Aμr0,S =R I,S 0,I 0,S r r = Aμr0,I + R0,I AμrI,S R−1 0,I = Aμ0,I + AR0,I μI,S .

(2.450)

2.8 Relative Motion

293

Likewise, Aμl

0,S

−1 −1 ˙ ˙ ˙ =R−1 0,S R0,S = RI,S R0,I (R0,I R0,S + R0,I R0,S )

= R−1 I,S Aμl RI,S + Aμl 0,I

I,S

= AR−1 μl + Aμl .¶ I,S

0,I

I,S

Remark 35 The relation between the Darboux differentials is π r,I = AdgI,0 π r,0 .

(2.451)

−1 Proof From u˜ I = u˜ 0 g0,I and gI,0 = gI,0 , we have φI = lgI,0 φ0 , and thus

π r,I = φI∗ (ωr ) = (lgI,0 φ0 )∗ (ωr ) = φ0∗ (lgI,0 )∗ (ωr ). Since ωr is right invariant, it follows that (rgI,0 )∗ (ωr ) = ωr , whence π r,I = φ0∗ (IntgI,0 )∗ (ωr ) = φ0∗ (AdgI,0 ωr ) = AdgI,0 φ0∗ (ωr ) = AdgI,0 π r,0 .

¶

2.8.3 Composition of Accelerations In deriving the composition of velocities (2.448), we obtain d l d l d d l l v0,S = vI,S + Adg −1 v0,I + ( Adg −1 ) v0,I . I,S dt I,S dt dt dt

(2.452)

Using (2.292) or (2.293), we have d Ad −1 = −adv l Adg −1 = −Adg −1 adv r . t t dt gt

(2.453)

Thus d l d l d l l v = vI,S + Adg −1 v0I − adv l Adg −1 v0,I . I,S I,S dt I,S dt 0,S dt

(2.454)

Then with (2.448), we obtain a formula for the composition of accelerations: d l d l d l l v0,S + adv l v0,S = vI,S + Adg −1 v0,I . I,S I,S dt dt dt

(2.455)

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l l Then the derivatives of v0,S = (μl0,S , α0,S ) are given by

d d l d l μ0,S + μlI,S × μl0,S = μlI,S + R−1 μ , I,S dt dt dt 0,I d l d l d l l l α0,S + μlI,S × α0,S (α + μl0,I × aI,S ). = αI,S + μlI,S × αI,S + R−1 I,S dt dt dt 0,I Remark 36 Relative motion of a rigid body. We assume that the intermediate frame u˜ I is not linked with the rigid body. Let g0 = gu˜ = (φ0−1 )∗ gˆ , 0

gI = gu˜ = (φI−1 )∗ gˆ , I

(2.456)

be the right-invariant metrics on D(3) induced by φ0 and φI from the right-invariant ˆ −1 Fu˜ in E3 , we deduce the metric gˆ on E3 . From the Euler equation ∇t,vˆ vˆ = m1 G u˜ Euler equations in d3 = Te D(3) by application of the Darboux differentials πur,0 ˜ = r,φ0 r,φI r,I πu˜ and πu˜ = πu˜ (see (2.451)). Then the Euler equation for the intermediate frame is πur,I ˜ (∇t,vˆ sˆ ) =

r dvI,S

dt

r r − B r,I (vI,S , vI,S )=

1 u˜ I −1 r,u˜ I (G ) Fe , m e

(2.457)

I ˆ −1 where the relative force is to be determined by Fer,u˜ = Geu˜ πur,I ˜ (Gu˜ Fu˜ ), thanks to (2.451). ¶ I

2.8.4 Relative Evolution of a Material Point On taking the derivative of the relation R˙ S,I = RS,I Aμl with respect to time, we S,I obtain ¨ S,I = RS,I S,I , with S,I = A l R μ˙

S,I

+ Aμl Aμl . S,I

S,I

(2.458)

And on taking the derivative of the relation (2.444) with respect to time, we successively obtain the velocity and the acceleration of a fixed material point in the moving frame u˜ S : ˙ S,I xI + a˙ S,I , (i) x˙ = RS,I x˙I + R ˙ S,I x˙I + R ¨ S,I xI + a¨ S,I . (ii) x¨ = RS,I x¨I + 2R

(2.459)

2.8 Relative Motion

295

˙ S,I xI = RS,I AμS,I xI = RS,I (μS,I × xI ), and thus Now R −1 (i) R−1 ˙ S,I , S,I x˙ = x˙ I + μS,I × xI + RS,I a −1 (ii) R−1 S,I x¨ = x¨I + 2μS,I × x˙ I + [ S,I xI + RS,I a¨ S,I ].

(2.460)

We sum up. We can still write (2.459)(i) in the form v = RS,I vI + vS,I xI .

(2.461)

The velocity of the material point is the sum of two terms. The first is its velocity in the frame u˜ I , and the second is the frame velocity u˜ I : ˙ S,I xI + a˙ S,I . vS,I xI = R Inertia force. Coriolis force. The evolution equation of the coordinates xt in the “fixed” or “absolute” frame u0 of a material point Mt in E3 with mass m subject to a given force field,= f (x, x) ˙ is given by the Newton equation mx¨t = f (xt , x˙t ). We will write the evolution equation of the coordinates xI of the point Mt in the moving frame u˜ I . The change of coordinates is given by xt = gt xI = Rt xI + at , with gt = gSI . Omitting the t-dependence, let F (xI , x˙I ) = R−1 f (x, x) ˙ = R−1 f (gxI , g∗ x˙I + gx ˙ I ). The equation of motion of the coordinates of Mt in the moving frame is (with μ = μS,I and (2.458), (2.460)) m[(μ˙ × xI + μ × (μ × xI )) + 2μ × x˙ I + x¨I ] = F r

(2.462)

(with F r = F − FaI and FaI = mR −1 a¨ I ), or also mx¨I = S I = F r − m[(μ˙ × xI + μ × (μ × xI )) + 2μ × x˙I ], = F r − m[ S,I xI + 2μ × x˙I ] = F I .

(2.463)

The complementary terms of Fr are called “inertial forces”: def

F I R = −m S,I xI = −m[μ˙ × xI + μ × (μ × xI )],

def

F C = −2mμ × x˙I , (2.464)

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with F C called the “Coriolis force” and F I R itself split into “inertial force of rotation” and “centrifugal force,” whence FI = F r + FIR + F C. Remark 37 Galilean frames. In the case of intermediate frames, Galilean frames are often used instead of Euclidean frames. A Galilean frame is the product of a Euclidean frame and a unit of time. The Galilean group is the set of affine maps g in E4 (or R 4 ) such that g(t, x) = (t + t0 , Rx + v(t + t0 ) + a),

∀(t, x) ∈ R × R 3 ,

for all parameters R ∈ SO(3), v and a ∈ R 3 , t0 ∈ R. From a Galilean frame U˜ 0 of E4 , we obtain every other Galilean frame U˜ by U˜ = U˜ 0 ◦ g. To these Galilean frames correspond respectively the Euclidean frames u˜ 0 (fixed) and u˜ I (moving) such that u˜ I = u˜ 0 ◦ gt , with gt (x) = Rx + v(t + t0 ) + a. Thus the displacement gt is also denoted by gS,I . Since the parameters of the Galilean group are time-independent, we have vS,I = g˙S,I = a˙ S,I = v, R˙ S,I = 0. Then we deduce μS,I = 0, a¨ S,I = 0 and thus the force F I in the equation of evolution (2.463) is identical to the force F r = F , that is, mx¨I = F I = F. ¶

2.9 Motion of Two Rigid Bodies Here we essentially specify the configuration space relative to two moving rigid bodies in the Euclidean space E3 with two different situations: the two bodies are independent (free) or have a link (a joint rotational axis, for instance).

2.9.1 General Free Case Let M10 , M20 be two (closed, bounded, connected) domains of E3 , with empty intersection, which are the reference positions of two rigid bodies S1 , S2 (for instance at the initial time). The first problem is to find the set of admissible displacements for the pair, then for each body. We choose a model of a situation in which the two rigid bodies are free, without contact point. First we define the following set of positions of the two bodies in the Euclidean space E3 :

2.9 Motion of Two Rigid Bodies def

(M10 ,M20 )S = {(M1 , M2 ) ⊂ E3 × E3 ,

297

M1 ∩ M2 = ∅,

∃(g˜1 , g˜2 ) ∈ D(E3 ) × D(E3 ), so that g˜1 M10 = M1 , g˜2 M20 = M2 }, (2.465) where D(E3 ) is the set of displacements in E3 . Let j be the map x ∈ E3 → (x, x) ∈ E3 × E3 , and  = j (E3 ). Then the sets M1 , M2 must be such that (M1 , M2 ) ∩  = ∅. To each rigid body we associate a Euclidean frame u˜ i that is related to an initial Euclidean frame u˜ 0i by u˜ i = u˜ 0i gi , with gi ∈ D(3), whence gi = (u˜ 0i )−1 u˜ i . Let u˜ 0 = (u˜ 01 , u˜ 02 ) be a pair of reference Euclidean frames on E3 × E3 such that ¯ 01 ,  ¯ 02 ) ⊂ R 3 × R 3 ,  ¯ 01 ∩  ¯ 02 = ∅. (u˜ 0 )−1 (M10 , M20 ) = ( Thus we have (g˜1 , g˜2 )u˜ 0 = u˜ 0 (g1 , g2 ), gi ∈ D(3), i = 1, 2. Let (D(3) × D(3))S = {(g1 , g2 ) ∈ D(3) × D(3),

¯ 01 ) ∩ (g2  ¯ 02 ) = ∅}. (g1 

This is an open space in the Lie group D(3) × D(3); this is not a group. It contains the diagonal D = {(g, g), g ∈ D(3)}, and D(3) operates on the right on (D(3) × D(3))S , that is, (g1 , g2 )g = (g1 g, g2 g) ∈ (D(3) × D(3))S if (g1 , g2 ) ∈ (D(3) × D(3))S . For the evolution problems, we must take the connected component of the identity for the set of pairs of admissible displacements, (D(3) × D(3))ad S , which ¯ 0 ) of the rigid bodies. ¯ 0,  depends on the initial positions ( 1 2 From a pair of displacements (g1 , g2 ) ∈ (D(3) × D(3))S , we can define the relative displacement gr = g1−1 g2 , which must be such that 1 ∩ gr 2 = ∅. The space (D(3) × D(3))S has the structure of a fibre bundle over D(3): for a given g1 , the set of relative displacements gr is identifiable with the fibre of (D(3) × D(3))S at g1 ∈ D(3); this is an open set of the Lie group D(3). Remark 38 The boundary of (D(3)×D(3))S corresponds to the situation in which the two bodies are in contact, and the dimension of this manifold in the general case is eleven if there is one contact point: we must have six parameters to give the position of the first rigid body, and two parameters to give the contact point on the boundary of this body, and also two parameters for this contact point on the second body, and one parameter to specify the position of this rigid body by rotation around the normal at the contact point, assuming that the domains are smooth. ¶

2.9.2 Rigid Bodies with an Axial Link We consider the case of two rigid bodies with a link such that the set of admissible relative displacements is the set of rotations around a common free axis with unit vector μ. Let S0(μ) be the rotation group with axis μ. Thus we give a model of

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situations such as a boat or a plane with a propeller. We can generalize to cases in which the set of admissible relative displacements is a part of S0(μ) (the case of a hinge, for instance). We will specify only the model of the relative motion and velocities with the mathematical framework and the corresponding Riemannian metric. Let M1 and M2 be two rigid bodies, linked by the common axis given by (μ, O) with a point O of the common axis. We again specify the space of the pairs of admissible Euclidean frames (u˜ 1 , u˜ 2 ) associated respectively with M1 and M2 . The configuration space may be given by seven variables, namely the axis of rotation μ ∈ S 2 and two unit vectors e1 and e2 that are orthogonal to μ, and a reference point O on the axis of rotation so that we can define two Euclidean frames u˜ 1 = (u1 , O), u˜ 2 = (u2 , O), each with μ and O, with the direct orthogonal frames u1 = (μ, e1 , e1⊥ ), respectively u2 = (μ, e2 , e2⊥ ). Thus the configuration space may be seen as a fibre product58 E3 ×S E3 , with S = S 2 × E3 if we take the same origin in the pair of Euclidean frames. Let  = 1 be the unit vector (1, 0, 0) in R 3 , and let π be the map u˜ = (u, O) ∈ E3 → (u(), O) ∈ S. Then we can define this fibre product by P = E3 ×S E3 = {(u˜ 1 , u˜ 2 ) ∈ E3 × E3 ,

π(u˜ 1 ) = π(u˜ 2 )}.

The relation (u˜ 1 , u˜ 2 ) ∈ P is equivalent to (u˜ 1 )−1 u˜ 2 ∈ SO(), that is, there exists θ ∈ R such that (u˜ 1 )−1 u˜ 2 = (Rθ , 0), denoted simply by Rθ . Proof From the definition we have u1 () = u2 (), then u−1 1 u2 () = , and thus −1 u u−1 u is a rotation of the -axis, and we obtain ( u ˜ ) ˜ = (Rθ , 0). The converse 2 1 2 1 is obvious. ¶ Note that (u˜ 1 , u˜ 2 ) ∈ P is equivalent to (u˜ 2 , u˜ 1 ) ∈ P . The tangent space of P is given by (see [Bour.var, 5.11.2]) T(u˜ 1 ,u˜ 2 ) P = {(v˜1 , v˜2 ), v˜i ∈ Tu˜ i (E3 ),

π∗ (v˜1 ) = π∗ (v˜2 )}.

We now want to find a pair of reference frames; the pair (uIG11 , uIG22 ) is not in P . But if the common axis is also a common principal axis of inertia μ for the two bodies, we can choose the frames u˜ 1 , u˜ 2 as the inertia frames, with μ = uI1 () = uI2 (), and with the center of mass G1 of M1 as common origin, so that we can choose (uIG11 , uIG21 ) as our pair of reference frames. The Euclidean frame uIG21 is −→

obtained from the inertial Euclidean frame uIG22 by translation of the vector G1 G2 , and the centers of mass are on the common μ axis. Transversal pair of displacements.

58 Or

also (0 ×S 2 0 ) × E3 with 0 , the space of Euclidean frames with a given origin.

2.9 Motion of Two Rigid Bodies

299

Definition 3 Let (g1 , g2 ) ∈ D(3) × D(3) and (u˜ 1 , u˜ 2 ) ∈ P . The pair of displacements (g1 , g2 ) is said to be (u˜ 1 , u˜ 2 )-transversal if (u˜ 1 g1 , u˜ 2 g2 ) ∈ P . Let θ0 ∈ R be such that (u˜ 1 )−1 u˜ 2 = Rθ0 , the condition that (g1 , g2 ) is (u˜ 1 , u˜ 2 )transversal is equivalent to g1−1 u˜ −1 ˜ 2 g2 = g1−1 Rθ0 g2 ∈ SO(), 1 u and thus there exists θ ∈ R such that g1−1 Rθ0 g2 = Rθ . This is a consequence of the definition u˜ 1 g1 (0) = u˜ 2 g2 (0),

u˜ 1 g1 () = u˜ 2 g2 (),

so that with gi = (Ri , ai ), i = 1, 2, we obtain a1 = Rθ0 a2 ,

θ0 R−1 1 R R2 ∈ SO().

(2.466)

Note that if (u˜ 1 , u˜ 2 ) ∈ P and (u˜ 01 , u˜ 02 ) ∈ P , then (g1 , g2 ) = (u˜ 01 , u˜ 02 )−1 (u˜ 1 , u˜ 2 ) = ((u˜ 01 )−1 u˜ 1 , (u˜ 02 )−1 u˜ 2 ) is (u˜ 01 , u˜ 02 )-transversal. Then (g1 , g2 )−1 = (g1−1 , g2−1 ) is (u˜ 1 , u˜ 2 )-transversal. The evolution of the system from the pair of reference frames (uIG11 , uIG21 ) will

be given by a pair of displacements (g1 , g2 ) , that is, (uIG11 , uIG21 )-transversal and time-dependent. Let gi = (Ri , ai ), i = 1, 2. Then the pair (g1 , g2 ) must satisfy uI1 (a1 ) = uI2 (a2 ),

uI1 R1 () = uI2 R2 ().

Thus the relation between the velocities v˜1 = g˙ 1 , v˜2 = g˙ 2 is such that uI1 (a˙ 1 ) = uI2 (a˙ 2 ),

˙ 2 (), uI1 R˙ 1 () = uI2 R

and thus 0 a˙ 2 = R−θ  a˙ 1 ,

0 ˙ ˙ 2 () = R−θ R  R1 ().

(2.467)

With R˙ i = Ri Aμi , we obtain −θ0 Aμ2 () = R−1 2 R R1 Aμ1 ().

(2.468)

Now with Rθ0 = (uI1 )−1 uI2 , there exists θ ∈ R (time-dependent) such that (g1 , g2 ) are given by a1 = Rθ0 a2 ,

θ0 θ R−1 1 R R2 = R .

(2.469)

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Thus (2.468) gives Aμ2 () = R−θ  Aμ1 (),

(2.470)

−θ that is, μ2 ×  = R−θ  (μ1 × ) = (R μ1 ) × , and thus there exists λ ∈ R such that

μ2 = R−θ  μ1 + λ.

(2.471)

We can also obtain this result by taking the derivative of (2.469), θ 0 ˙ ˙θ ˙ 2 = R−θ R  [R1 R + R1 R ],

0 a˙ 2 = R−θ  a˙ 1 ,

(2.472)

˙θ ˙ and thus with R−θ  R = θ A , we obtain 0 a˙ 2 = R−θ  a˙ 1 ,

˙ μ2 = R−θ  μ1 + θ ,

(2.473)

which corresponds to (2.471) with λ = θ˙ . Thus (2.473) gives the relations between the velocities v˜1 = g˙1 , v˜2 = g˙2 , a˙ 1 , a˙ 2 , μ1 , μ2 . Riemannian metric. The Riemannian metric on E3 ×S E3 is given by gu˜ 1 ,u˜ 2 (vˆ1 , vˆ2 ) = m1 g1u˜ 1 (vˆ1 ) + m2 g2u˜ 2 (vˆ2 ), with mi = metrics

gˆ i

mi m1 +m2 ,

(2.474)

i = 1, 2. The metric is made explicit thanks to the Riemannian

on the Euclidean frames E3 of Mi . Let αi = R−1 i .a˙ i , and

vˆ1 = uIG11 v˜1 ,

vˆ2 = uIG21 v˜2 ,

v˜iS = (lg −1 )∗ (v˜i ) = R−1 i v˜ i = (Aμi , αi ), i = 1, 2. i

Then with the inertia frame u˜ 01 = uIG11 , the metric for M1 is given by u˜ 0

gˆ 1u˜ 1 (v˜1 ) = ge 1 (v˜1S ) = < I1 μ1 , μ1 > + | α1 |2 .

(2.475)

With the inertia frame u˜ 02 = uIG21 , the metric for M2 is given by u˜ 0

gˆ 2u˜ 2 (v2 ) = ge 2 (v˜2S ) = < I2 μ2 , μ2 > + | α2 − a0  × μ2 |2 , −→

(2.476)

with a = a0  = (uI2 )−1 (G1 G2 )0 (that is, at the initial time), and τ−a the translate of the vector −a. This may be obtained with the relation

2.9 Motion of Two Rigid Bodies

301

I

uG2 ◦τ−a

ge

2

I

uG2

(v˜2S ) = ge

2

((Ad (I, −a))(μ2 , α2 )).

Recall that μ1 , μ2 , a˙ 1 , a˙ 2 are linked by (2.473), giving αi = R−1 i .a˙ i .

2.9.3 An Application with Nonholonomic Constraints Let a vehicle be moving on a flat surface, with the following 2D model. We have a framework with two axles. The front axle is moving with respect to this framework, while the back axle is fixed with respect to it. Let O1 (respectively O2 ) be the middle of the front (respectively back) axle, which also belongs to the framework, and let λ0 be the distance between them. Thus we consider two rigid bodies, the framework and the moving axle, so that we associate to them two Euclidean frames, with the point O1 as common origin, the first frame fixed within the framework defined by the unit vector αθ of the axis (O2 , O1 ). The front axle can make rotations in the plane with center O1 ,and the orthogonal direction to the axle is located by the unit vector αϕ , so that the corresponding Euclidean frame is given by ((αϕ , αϕ⊥ ), O1 ) ∈ E2 . Thus the position of the moving axle with respect to the frame is given by the angle ϕ˜ = ϕ − θ of the normal to that axle within the axis (O2 , O1 ). The position of the system is given by the common origin and two directions (O1 , αθ , αϕ ) ∈ M = E2 × S1 × S1 , or as above by the fibre product E2 ×E2 E2 , of Euclidean frames (u˜ 1 , u˜ 2 ) with common origin O1 . This is a 2D version of the previous example. The displacements are given by the translations τa and ϕ the rotations Rθe3 , Re3 (with e3 a unit vertical vector), corresponding to (in fact a submanifold of) (a, R1 , R2 ) ∈ G = R 2 × SO(2) × SO(2), with R1 = Rθe3 , R2 = Rϕe3 . But there is a supplementary constraint on the motions of O1 and O2 that is yet to be specified. A small displacement of O1 (respectively O2 ) must be along the direction αϕ (respectively αθ ), according to the corresponding wheels, so that the velocity v1 of O1 must be along αϕ , and the velocity v2 of O2 must be along αθ . Thus in a reference frame u˜ 0 , with the positions x 1 and x 2 of O1 and O2 in this frame, the velocities v1 and v2 must be such that v1 = αϕ

dxϕ1 dt

,

v2 = αθ

with αθ = (cos θ, sin θ ), αϕ = (cos ϕ, sin ϕ). We also have the relation x 1 − x 2 = λ0 αθ ,

dxθ2 , dt

(2.477)

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and thus the following: (i) The projection on αθ gives xθ1 − xθ2 = λ0 . (ii) The time derivative gives v1 − v2 = λ0

αθ , dt

vθ1 − vθ2 = 0.

(2.478)

Then from (2.477), (2.478), we have the relation for the velocity of O1 , the time derivative of θ , which defines the velocity of the rigid body S1 (the framework), depending on the parameter ϕ: αϕ

dxϕ1 dt

− αθ

dxθ1 dαθ − λ0 = 0. dt dt

(2.479)

From this relation, we define a differential form ω on M (or E2 ×E2 E2 ) with values in R 2 by ω = αϕ dxϕ1 − αθ dxθ1 − λ0 dαθ ,

(2.480)

d that is, such that < ω, dt > = 0. Then we are led to determine whether the kernel of ω is integrable, that is, whether the constraints on the motion of S1 are holonomic. From the Frobenius theorem, this is equivalent to

ker ωζ ∧ ker ωζ ⊂ ker((dω)ζ ) with ζ = (x 1 , θ, ϕ), which is also equivalent to ∀X1 , X2 ∈ Tζ M such that < ωζ , Xi > = 0, one has < (dω)ζ , X1 ∧ X2 > = 0. Now the components of ω are given by (i) ω1 = cos ϕ dxϕ1 − cos θ dxθ1 − λ0 sin θ dθ, (ii) ω2 = sin ϕ dxϕ1 − sin θ dxθ1 + λ0 cos θ dθ.

(2.481)

Let X = a1

∂ ∂ ∂ + a2 1 + aθ . ∂xϕ1 ∂θ ∂xθ

Then the condition < ωζ , X > = 0 implies (i) < ω1 , X > = cos ϕ a1 − cos θ a2 − (a sin θ )aθ = 0, (ii) < ω2 , X > = sin ϕ a1 − sin θ a2 + (a cos θ )aθ = 0. The solution of these equations is, up to a multiplicative factor,

(2.482)

2.10 Incompressible Fluid in a Fixed Domain

X = λ0

303

∂ ∂ ∂ + λ0 cos(ϕ − θ ) 1 + sin(ϕ − θ ) . ∂xϕ1 ∂θ ∂xθ

Moreover, dω = αϕ dϕ ∧ dxϕ1 − αθ dθ ∧ dxθ1 , where αϕ = αϕ+ π2 = αϕ⊥ , and αθ = αθ+ π2 = αθ⊥ are the derivatives of αϕ , αθ . ∂ Let Y = ∂ϕ . Then < ω, Y > = 0. Moreover, the interior product of Y on dω gives i(Y )dω = αϕ dxϕ1 , and then we have < X, i(Y )dω > = < X ∧ Y, dω > = λ0 αϕ = 0. Thus the kernel of ω is not integrable, which means that the constraints on the motion of S1 are nonholonomic.

2.10 Incompressible Fluid in a Fixed Domain We consider a perfect incompressible fluid in the fixed domain U in R 3 (U has the structure of a manifold with closed smooth boundary in R 3 ). The configuration space is the set M = SDiff (U ) of diffeomorphisms of U that leave the dx measure invariant. This set M (which is open in the manifold C r (U, U )) has the structure of an infinite-dimensional manifold (see [Bour.var, 15.3.6], here considering diffeomorphisms of class C r ) and that of a topological group, but not of a Lie group. Let us specify the tangent space to the manifold M˜ = Diff r (U ) at a point φ of M˜ (see [Bour.var, 15.3.6]). We first define the fibre bundle   φ ∗ T U = (x, v), x ∈ U, v ∈ Tφ(x)U , which is the pullback of T U by φ. Let x be the mapping g ∈ Diff (U ) → g(x) ∈ U . The image of ξ ∈ Tφ M˜ by Tφ (x ) is denoted by ξx ∈ Tφ(x) U. The mapping x ∈ U → ξx ∈ Tφ(x) U is a lifting from φ into T U and is identified with a section of class C r of the fibre bundle over U . Thus we obtain an isomorphism from Tφ M˜ onto the (Banach) space of sections of φ ∗ T U (of class C r ), which allows us to ˜ and we can identify the identify these two spaces. The set M is a submanifold of M, tangent space Tφ M where φ ∈ M = SDiff (U ) with the set of vector fields on U , with null divergence, and tangent on the boundary ∂U of U . Observe that this set is independent of φ. We can define a Riemannian structure on these vector fields by  gφ (ξ1 , ξ2 ) =

ξ1 (x).ξ2 (x) ρ(x) dx, U

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where ρ is a positive function that represents the volumetric mass of the fluid, which is taken constant (equal to 1) for a homogeneous incompressible fluid. This Riemannian metric is right invariant (for the composition of diffeomorphisms). We verify that if rψ (with ψ ∈ M) is the map of right composition φ ∈ M → φ ◦ ψ ∈ M, we have (T (rψ )ξ )x = ξψ(x) , ∀ξ ∈ Tφ M. Indeed, for all ξ ∈ Tφ M, (Tφ (rψ )ξ )x = Trψ (φ) (x )Tφ (rψ )ξ = Tφ (x ◦ rψ )ξ = Tφ (ψ(x) )ξ = ξψ(x) . Furthermore, we have gφ◦ψ (ξ1 ◦ ψ, ξ2 ◦ ψ) = gφ (ξ1 , ξ2 ),   since gφ◦ψ (ξ1 ◦ ψ, ξ2 ◦ ψ) = U ξ1 (ψ(x)).ξ2 (ψ(x)) dx = U ξ1 (y).ξ2 (y)dy. Now we calculate the commutator [v, w] of two vector fields v and w on U of null divergence and tangent on the boundary ∂U . In a coordinate system of U , we have by summing on repeated subscripts and superscripts, [v, w] = v j ∂j (wk ∂k ) − wk ∂k (v j ∂j ) = (v j (∂j wk )∂k ) − wj (∂j (v k ))∂k , and we verify that (curl (v × w))k = ((wj ∂j )v k − (v j ∂j )wk , k = 1, 2, 3, whence [v, w] = −curl (v × w). This vector field is of null divergence and is tangent to the boundary ∂U . By duality, we obtain the bilinear map B (see (2.539)) that defines the covariant derivative in a group, (B(v, w), u) g = (v, [w, u])g :  (B(v, w), u) g = −

v.curl (w × u)dx U





curl v.(w × u)dx −

=− U

n.((w × u) × v) d .

The last term is null, since v, w, u are tangent on the boundary = ∂U . Hence B(v, w) = w × curl v − grad α, where α is a real-valued function on U such that B(v, w) is a vector field with null divergence, and tangent to the boundary of U , and thus α is defined by div B(v, w) = div (w×curl v)−α = 0, n.B(v, w) = n.(w×curl v)− Then the covariant derivative ∇v v of the velocity field v is (see (2.541)) ∇v v = −B(v, v) = (curl v) × v + grad α.

∂α | = 0. ∂n

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305

The sign difference in this expression is due to the right-invariant metric. A  k 2 calculation by components gives ((curl v) × v)j = v ∂k v j − ∂j v2 , which is 2

written (curl v) × v = (v.∇) v − grad v2 . As a consequence, we have the relation ∇v v = (v.∇) v + grad (α −

v2 ). 2

We assume that the fluid is subject to a force f (t) = f E (t), here identified with a vector field, such that div f = 0. The velocity field v = v E must satisfy the Euler incompressible equation ∇t,v(t )v =

∂v + ∇v v = f (t), ∂t

whence

v2 ∂v + (v.∇) v + grad (α − ) = f, ∂t 2

(2.483)

with div v = 0, v.n| = 0. In this equation (with Euler coordinates) the term (α − (v 2 /2) does not represent a physical pressure (see [Piro8, ch. I.4, Rem. 3, p. 23]). We emphasize the difficulty due to the infinite dimension of M: the mapping G is no longer an isomorphism from the Banach space of velocity fields Tφ M on its dual Tφ∗ M. The requirement that the last term of (2.483) be a section of φ ∗ T U of class C r is an important restriction on admissible force fields. For a study of moving (stationary) properties of a fluid in differential geometry, we refer, for instance, to [Arn1, App. 2, pp. 333–344]. To solve the mechanical equations explicitly is rarely possible, with the exception of integrable systems that depend on the Liouville theorem. We have first to find the first integrals of motion, which allow us to reduce the dimension of the integral manifold. A main tool for that is the Noether theorem, which associates a first integral with each 1-parameter Lie group of symmetry for the problem. We recall essential notions on Lie groups and Lie algebras in Appendix 1 of this chapter.

2.11 Action of a Lie Group on a Manifold Let G be a Lie group with right (respectively left) differential action on a manifold M. This action is denoted by x.s (respectively s.x) for all x in M and s in G. def

We define a right action from a left action by x.s = s −1 .x. Depending on the application, we are led to a right or left action. For studies of properties on differential operators on a manifold M, it is usual to have a right action (see, for instance, [Dieud3, ch. 19.3])

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2.11.1 Action of G by Differential Operators Let σx be the mapping s ∈ G → x.s ∈ M with a right action of G on M. With every differential operator P left-invariant on G we can associate a differential operator PM on M (with order at most equal to that of P ) by < PM (x), f > = < P (e), f ◦ σx >,

∀f ∈ C ∞ (M),

(2.484)

or even PM (x) = (σx )∗ (P (e)), e being the identity of G. Now let A be an element of the Lie algebra g = Ge and let A˜ be the left-invariant field on G corresponding to A: A˜ s = s.A = (ls )∗ A ∈ Ts G, ∀s ∈ G. Notation. Let x.A = Te (σx )(A) = (σx )∗ (A) ∈ Tx M, and let A∗ = σ (A) be the vector field on M defined by A∗x = x.A, ∀x ∈ M. As a consequence of the formula (2.484), the Lie derivatives along A˜ in G and along A∗ in M are linked by (LA˜ )M = LA∗ . Indeed, for every differentiable function f on M, we have (LA˜ )M .f (x) = LA˜ .(f ◦ σx )(e) = < de (f ◦ σx ), A > = < dx f, Te (σx ).A > = < dx f, x.A > = LA∗ f (x). The mapping A → A∗ = σ (A) is a Lie algebra homomorphism from g into the vector fields on M: A∗1 , A∗2 = [A1 , A2 ]∗ ; A∗ = σ (A) is called the Killing field (or fundamental field) associated with A. We recall the main definitions of the action of a group G on a set M, of course valid for a left or a right action. Definition 4 We say that the group G acts effectively (respectively freely) on M if x.s = x for all x ∈ M (respectively for some x ∈ M) implies that s = e (the identity of G). We have the following properties (see [Kob-Nom, ch. I, 4, Prop. 4.1]): Proposition 7 If G acts effectively on M, then the map σ : A → A∗ is an isomorphism from g onto σ (g) ⊂ X (M). If G acts freely on M, then σ (A) never vanishes on M for A = 0, i.e., A∗x = 0, ∀x ∈ M, ∀A ∈ g, A = 0. Let rs be the right multiplication by s in M, and as usual we denote by (rs )∗ X the pushforwards of the tangent vector X ∈ Tx M of M by rs , i.e., (rs )∗ X(f ) = X(f ◦ rs ) = X(f (x.s)),

∀f ∈ C 1 (M).

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307

Then for all A ∈ g, the fundamental field A∗ associated with A satisfies the relation (rs )∗ A∗ = (Ads −1 .A)∗ ,

∀s ∈ G,

with ((rs )∗ A∗ )x = T (rs ).A∗(r

s)

−1 x

.

¶

2.11.2 Lie Algebra of Functions Let (M, ω) be a manifold M equipped with a nondegenerate 2-form ω, called a symplectic form. From ω, we define an isomorphism J from Tx∗ M onto Tx M by ω(X, Y ) = −J −1 (X)(Y ),

∀X, Y ∈ Tx M.

To every real differentiable function F on M there corresponds the vector field XS such that XS = J (dF ). The vector field XS is a Hamiltonian field (see (2.31)): i(XS )ω = −dF. Definition 5 The Poisson bracket of two differentiable functions F1 and F2 on a symplectic manifold (M, ω) is defined by {F1 , F2 } = ω(J (dF1 ), J (dF2 )).

(2.485)

The Poisson bracket of the functions F1 and F2 depends only on their differentials, and thus does not change if we add a constant CF1 to F1 and CF2 to F2 . Then we have the following essential property: Proposition 8 Let F1 and F2 be two differentiable functions on (M, ω). The Lie derivative of the function F2 along the vector XF1 is equal to the opposite of the Lie derivative along the vector XF2 of the function F1 , so that XF1 F2 = < XF1 , dF2 > = {F1 , F2 } = ω(J (dF1 ), J (dF2 )) = ω(XF1 , XF2 ) = −XF2 F1 .

(2.486)

(Observe that if F1 (or F2 ) is locally constant, then the associated field XF1 (or XF2 ) is null, and then (2.486) is trivially satisfied.)

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Let us take the Lie derivative of the differential dF2 along the Hamiltonian field η = XF1 . We have Lη dF2 = dLη F2 and also Lη dF2 = −Lη i(XF2 )ω = −i(Lη XF2 )ω (since Lη ω = 0), which implies Lη XF2 = XLη F2 , whence for η = XF1 , [XF1 , XF1 ] = X{F1 ,F2 } .

(2.487)

Moreover, with X3 = XF3 , we have LX3 {F1 , F2 } = {F3 , {F1 , F2 }} = LX3 LX1 F2 , and using the equality (LX3 LX1 − LX1 LX3 )F2 = L[X3 ,X1 ] F2 , we see that the Poisson bracket satisfies the Jacobi identity. Thus the space C ∞ (M, R) is equipped with a Lie algebra structure by the Poisson bracket.59 Let H0 (M) be the space of Hamiltonian vector fields on M. The space X (M) of vector fields on M has the structure of a Lie algebra, and the space H0 (M) is a Lie subalgebra, which is a consequence of (2.487). j

The mapping j : F ∈ C ∞ (M, R) → XS ∈ H0 (M) ⊂ X (M) is a Lie algebra homomorphism by (2.487), and if M is connected, it has the space of constant functions as kernel. This corresponds to the following sequence of Lie algebras being exact: j

0 → R → C ∞ (M, R) → H0 (M) → 0. Example 9 When M = T ∗ N is a phase space, the relation XS = J (dF ) with variables (q, p) is given by XS =

 ∂F ∂ ∂F ∂ ( − ) for ω = dp ∧ dq. ∂pj ∂qj ∂qj ∂pj

(2.488)

j

Then the Poisson bracket is defined by {F1 , F2 } =

 ∂F1 ∂F2 ∂F1 ∂F2 ( − ), ∂pj ∂qj ∂qj ∂pj

∀F1 , F2 ∈ C ∞ (T ∗ N).

(2.489)

note that if X1 = XF1 , X2 = XF2 are vector fields for the Hamiltonian functions F1 , F2 , then {F1 , F2 } is not necessarily the Hamiltonian function of the field [X1 , X2 ].

59 But

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309

With the canonical form θ = pdq in T ∗ N, we have θ (XS ) = priori different from F . ¶



∂F pj ∂p , which is a j

2.11.3 Noether Theorem Here we assume the left action of groups on a manifold M. We consider once again the situation of the previous proposition with (2.486). By the definition of the Lie derivative, if (gt1 ) is the (local) group with one parameter generated by XF1 , we have {F1 , F2 } (x) =

d F2 (gt1 (x))|t =0 dt

∀x ∈ M.

In the case of a conservative system whose evolution is obtained by a Hamiltonian function H on a symplectic manifold (M, ω), a function F that is constant along the system evolution is said to be a first integral. We see that F is a first integral (of the motion) if and only if {F, H } = 0, that is, if and only if the Poisson bracket of the functions F and H is null. (Observe that from the Jacobi identity, the set of first integrals is a Lie subalgebra of C ∞ (M, R).) The notion of first integral is associated with that of invariance, thanks to the following theorem, due to Emmy Noether. Theorem 2 (The Noether theorem) Let (gλ ) be a one-parameter group of symplectic transformations of (M, ω) generated by a vector field XS that is associated with a Hamiltonian function F . If (gλ ) keeps invariant the Hamiltonian H of the system, H ◦ gλ = H , then F is a first integral of the motion. Proof Since H ◦ gλ = H and gλ∗ iXH ω = iXH ω, this implies gλ∗ XH = XH , and hence gλ commutes with the evolution. Then LXH (F ) = (XH , dF ) = {XH , A∗ } = < A∗ , dH > = 0.

¶

When M = T ∗ R n and (gλ ) is a 1-parameter group of transformations of the configuration space R n (which induces a group of symplectic transformations of the phase space T ∗ R n keeping invariant the canonical form θ = p.dx), then the invariance of H with respect to this group also implies the invariance of the Lagrangian. Indeed, L(x, x) ˙ = px˙ − H (x, p), and thus L = < XH , θ > −H. Furthermore, if A∗ is the generator of the group, then the first integral is also expressed (up to a constant) by F = θ (A∗ ). This is due to the relation −dF = iA∗ dθ = LA∗ θ − diA∗ θ , and thus dF = diA∗ θ , because LA∗ θ = 0. Generalization of the Noether theorem for a Lie group.

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Let G be a connected Lie group with a differentiable left action on a symplectic manifold (M, ω) such that the map x ∈ M → g.x ∈ M is a symplectic diffeomorphism for all g ∈ G. Let σx be the map g ∈ G → g.x ∈ M, with x ∈ M, and we still denote by σ the map A ∈ g → A∗ ∈ X (M) (see [Kob-Nom, ch. I, 4, p. 42]). Also, we will use the notation X g (M) = σ (g), and for all x ∈ M,   Xxg (M) = A∗x = (σx )∗ A ∈ Tx (M), A ∈ g . g

(2.490) g

The map x ∈ M → Xx (M) is an r-field if the dimension r of the space Xx (M) is independent of x. Moreover, if this r-field is of class C 1 , then it is said to be g completely integrable (see [Mall, ch. II.5.4]), and we have Xy (M) = Ty (Gx), y ∈ Gx. We assume that for every A ∈ g, the vector field A∗ ∈ X (M) is Hamiltonian, thus that there exists a function, denoted by FA (or FA∗ ), such that A∗ = XS = J (dFA ). Then the action of the group G on M is said to be Hamiltonian. Note that the group acts by symplectic diffeomorphisms on M. Theorem 3 (The Noether theorem for a Lie group) Let G be a Lie group whose action on (M, ω) is Hamiltonian and that keeps invariant the Hamiltonian H of the system. Then the functions FA for all A ∈ g are first integrals of the system.     Proof We have  only to prove that if H,  FA1 =  0 and H, FA2 = 0, then H, F[A1 ,A2 ] = 0. Now we have  H, FA1 , FA2 = 0 by the Jacobi identity. Moreover, F[A1 ,A2 ] and FA1 , FA2 differ only by some constant, as indicated below, which implies the theorem.¶ Of course, these first integrals are not all independent: the number of independent first integrals is equal to at most the dimension of the Lie algebra. Moreover, note that the groups (gλ1 ), (gμ2 ) of symplectic diffeomorphisms on (M, ω) with Hamiltonian generators XF1 , XF2 commute if and only if {F1 , F2 } is constant.

2.11.4 Momentum Map, Poisson Action of a Lie Group The mapping by J between vector fields and functions is not bijective, each function being defined up to a constant. For every pair (A1 , A2 ) ∈ g × g, there exists a constant (A1 , A2 ) (bilinear and skew-symmetric function on g × g) such that   F[A1 ,A2 ] = FA1 , FA2 + (A1 , A2 );

2.11 Action of a Lie Group on a Manifold

311

is such that (A1 , A2 ) = − (A2 , A1 ), and from the Jacobi identity, ([A1 , A2 ], A3 ) + ([A2 , A3 ], A1 ) + ([A3 , A1 ], A2 ) = 0. We say that is a 2-dimensional cocycle on g. Example 10 (cf. [Land, p. 187]). Let M = T ∗ (R n ) = R n × R n , ω = dp ∧ dq. Let (q0 , p0 ) ∈ R n × R n , and let F1 , F2 be such that F1 (p, q) = q0 .p, F2 (p, q) = ∂ −p0 .q. Then we have {F1 , F2 } (p, q) = p0 .q0 and X1 = XF1 = q0 ∂q , X2 = ∂ , and the groups (gt1 ) and (gt2 ) generated by X1 and X2 are the XF2 = p0 ∂p commuting translation groups in R n × R n such that

(gt1 )(p, q) = (p, q + q0 t),

(gt2 )(p, q) = (p + p0 t, q).

Let Pj and Qk with j, k = 1, . . . , n be the generators of the translation groups along the coordinates pj and qk . Then (Pj , Pk ) = (Qj , Qk ) = 0,

(Pj , Qk ) = δj,k , ∀j, k.

Thus the cocycle is not null, although here the group G is commutative.

¶

Definition 6 Momentum map. Let G be a Lie group whose action on (M, ω) is Hamiltonian, and let g∗ be the dual space of the Lie algebra g of G. The mapping P from M into g∗ defined by < P (x), A > = FA (x),

x ∈ M, A ∈ g,

is called the momentum map. Thus we have dFA = −iA∗ ω. In the case that M = T ∗ (N) and ω = dθ , and when the Lie group G acts in N, then we can write FA = θ (A∗ ). We generally assume that the map σ : A → A∗ is injective, which corresponds to the fact that the group G acts effectively on M, but we do not assume that the action is free. Let   gx = A ∈ g, A∗x = 0 . We assume in the following that the mapping P is of class C 1 . Then we have the following result. Proposition 9 The rank r of the mapping P at x is equal to the codimension of gx . Proof We identify Tp g∗ with g∗ for all p ∈ g∗ . For all f ∈ C ∞ (g∗ , R) and Y ∈ Tx (M), we have P∗ Y (f ) = Y (f ◦ P (x)), and thus for f = A ∈ g, we have P∗ Y (A) = Y (P (x)A) = Y (FA (x)) = < Y, dFA > = −ω(A∗x , Y ).

(2.491)

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Since ω is not degenerate, the relation P∗ Y (A) = 0, ∀Y ∈ Tx M, implies A∗x = 0. Thus Im P∗ is orthogonal to gx in g∗ , hence the Proposition.¶ Note that gx is the Lie algebra of the isotropy group Gx of x. Let k be its dimension. Then the rank r of P is r = nG − k, where nG is the dimension of G. Let Ex be a vector subspace of Tx M, and let Ex⊥ be the vector subspace of Tx M such that Ex⊥ = {X ∈ Tx M,

ω(X, Y ) = 0, ∀Y ∈ Ex } .

(2.492)

Let us specify the kernel ker P∗ of P∗ . Let Y ∈ ker P∗ . Then (P∗ Y )(A) = 0, ∀A ∈ g, g and thus ω(A∗x , Y ) = 0, ∀A ∈ g, whence Y is in Xx (M)⊥ , with the notation of (2.490). Thus ker P∗ = (Xxg (M))⊥ .

(2.493)

Corollary 1 If the action of the group G on M is free, then the momentum map P is a submersion from M into g∗ . Proof Indeed, if the action of G on M is free, we know (see [Kob-Nom]) that the field A∗x satisfies A∗x = 0, ∀x ∈ M, ∀A ∈ g, A = 0. Then the map P∗ = P  (x) from Tx M is onto TP (x) g∗ . We deduce the foliation of M. ¶ We will see in (2.496) that P is not constant on the orbits of G. The codimension of gx is constant for x ∈ OM , i.e., on an orbit of G in M. From Proposition 9, the mapping P has constant rank (P is a subimmersion), and then we can apply the theorem of constant rank,60 and thus Mp = P −1 (p) is a closed submanifold of M of dimension 2n − r (with dim M = 2n), and at every x ∈ Mp we have Tx (Mp ) = ker P∗ = ker P  (x). Let Ai , i = 1, . . . , r, be a basis of g, and pi ∈ R, i = 1, . . . , r. Let p ∈ P (M) be such that < p, Ai > = FAi (x) = pi . Then Mp = {x ∈ M, FAi (x) = pi , i = 1, . . . , r}. Furthermore, if P is a submersion, the family Mp , p ∈ g∗ , is a foliation of M, and we denote by MP the corresponding foliated manifold. Under fairly general conditions that will be specified below, we can split these leaves and define a reduced phase space. ¶ Remark 39 Notice that the orbit of x is a submanifold of M under fairly general conditions, for instance, see [Dieud3, ch. 16.10.7], if G.x is locally closed. Also notice that the set of orbits G.x forms a partition of M, but not necessarily a foliation; notably, the quotient space M/G is not necessarily a manifold. In order that this space be a manifold, then called an orbit manifold, and that the canonical

60 See,

for instance, [Mall, ch. I, 6.3, 6.5, 7.6].

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313

projection π : M → M/G be a submersion, it is necessary and sufficient that the following property be satisfied (see [Dieud3, ch. 16.10.3]): The set61 EG = {(x, y) ∈ M × M, ∃g ∈ G, y = gx} is a closed submanifold of M × M. ¶ Let us specify the kernel of the restriction to ker P∗ of the symplectic form ω. We have the following properties (see [Arn1, p. 380]) with the orbit OM = Gx of x ∈ M: ker P∗ = Tx Mp = (Tx (OM ))⊥ ,

(Tx Mp )⊥ = Tx (OM ),

∀p = P (x). (2.494)

Proof We have the following equivalences by (2.491): X ∈ (Tx (OM ))⊥ ⇐⇒ ω(X, A∗x ) = 0, ∀A ∈ g ⇐⇒ P∗ X(A) = 0, ∀A ∈ g, i.e., P∗ X = 0. Hence the first property. The second is a result of duality. ¶ We deduce the property for x ∈ OM ∩ Mp : Tx (OM ) ∩ Tx Mp = (ker P∗ )⊥ ∩ ker P∗ = ker(ω|Tx Mp ). g

Indeed, the common elements of the two tangent spaces are of the form A∗x ∈ Xx such that ω(A∗x , X) = 0, ∀X ∈ Tx Mp . Then the question is to know whether this space is a Lie algebra. This will be implied under a supplementary hypothesis indicated below. Definition 7 The Poisson action . The Hamiltonian action of the group G on the symplectic manifold (M, ω) is said to be Poisson if it is possible to choose the functions FA in order to eliminate the constants , thus if   F[A1 ,A2 ] = FA1 , FA2 ,

∀(A1 , A2 ) ∈ g × g.

(2.495)

In this framework, the momentum map has the following property: Theorem 4 Let G be a Lie group with a Poisson action on a symplectic manifold (M, ω). The momentum map P transforms the action of group G on M into the contragredient62 of the adjoint action of G on g, P (gx) = t Adg −1 (P (x)),

∀x ∈ M, g ∈ G.

(2.496)

is also the graph of the equivalence relation x ∼ y when y ∈ Gx. that we should have P (xg) = t Adg (P (x)) for an action on the right of G on M, and then P transforms the action on the right of G into the coadjoint action; P must be a homomorphism for the various actions of G, on the left as well on the right.

61 It

62 Note

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Let (gt ) be the group generated by the vector field A∗ , with A ∈ g. The theorem is a consequence of the following lemma. Lemma 4 For all B ∈ g and x ∈ M, we have equivalence between the following: (i) FB (gt (x)) = FAd

gt−1

B (x),

(ii) F[A,B] = LA∗ FB = < A∗ , dFB > = {FA , FB } .

(2.497)

Proof From (i), by time derivation (at t = 0), we obtain the point (ii). Conversely, by integration, (ii) implies (i). ¶ Also observe that for all Y ∈ Tx M and all B ∈ g (the generating the group (gt ) in M), we have < dx FA , Y > = ωx (A∗ , Y ) = (gt∗ ω)x (A∗ , Y ) = ωgt x ((gt )∗ A∗ , (gt )∗ Y ),

∀A ∈ g.

But we still have ωgt x ((gt )∗ A∗ , (gt )∗ Y ) = < dgt x Fgˆt A , (gt )∗ Y >, with gˆt A = Adgt A ∈ g.63 Then we obtain < dx FA , Y > = < gt∗ dFgˆt A , Y > = < dx (Fgˆt A ◦ gt ), Y >, thus the property FA = Fgˆt A ◦ gt , which is point (i). ¶ We deduce from this theorem some important consequences. Under the hypothesis of the Poisson action, the isotropy group Gp of a point p ∈ g∗ keeps stable the manifold Mp , and we have Gp x = Gx ∩ Mp . The Lie algebra gp of the group Gp is identified with the space Tx (OM ) ∩ Tx Mp , ∀x ∈ Mp (by the mapping A → A∗x ). Note that gp may be defined by gp = {A ∈ g, < p, [A, B] > = 0,

∀B ∈ g} ,

and we also have, ∀A ∈ gp , ∀B ∈ g, < p, [A, B] > = < P (x), [A, B] > = F[A,B] (x) = {FA , FB } (x) = ωx (A, B) = 0. If the quotient space Fp = Mp /Gp has a manifold structure,64 then the symplectic form (which is degenerate on T Mp ) gives, by passage to the quotient, a symplectic

it is important to distinguish the action of the group (gt ) in g from that in T M (notably, see [Kob-Nom, ch. I, 4, Prop. 4.1]). 64 See the previous remark and [Dieud3, ch. 16.10.3], which is realized, for instance, if the action of Gp on Mp is proper; see [Bour.var, 6.2.3]. 63 Here

2.12 Appendix 1. Lie Group and Lie Algebra

315

form on Fp (see [Arn1, App. 5]). Then this space is called a reduced phase space.65 If the canonical projection of Mp on Fp = Mp /Gp is a submersion, then the set of orbits Gp x with x ∈ Mp constitutes a foliation of Mp . Remark 40 For a commutative Lie group G with symplectic  and Hamiltonian action on a space M such that the Poisson brackets FAi , FAj are null for all Ai , Aj ∈ g, then the action of this group is Poisson  with the  choice F0 = 0. But if there exist two elements Ai , Aj ∈ g such that FAi , FAj is a nonzero constant, then the action of this group cannot be Poisson: such is the case of the Heisenberg group of translations of the phase space T ∗ (R n ). ¶

2.12 Appendix 1. Lie Group and Lie Algebra 2.12.1 Definitions A Lie group G is a set equipped with a topological group structure and with a differentiable manifold structure such that (g1 , g2 ) ∈ G × G → g1 g2−1 ∈ G is a differentiable mapping. Then the product and inverse mappings are differentiable. Examples R, T , C, Gl(n, R), Gl(n, C), SO(n), . . .. Every closed subgroup of a Lie group is also a Lie group. If H is an invariant subgroup of a Lie group G, then G/H is a Lie group. A Lie algebra g is a vector space equipped with a mapping denoted by [, ]: (X, Y ) ∈ g × g → [X, Y ] ∈ g that is bilinear, skew-symmetric, and satisfies the Jacobi identity: [[X, Y ] , Z] + [[Y, Z] , X] + [[Z, X] , Y ] = 0,

∀X, Y, Z ∈ g.

The space of vector fields X on a Lie group G that are left invariant, i.e., such that (denoting by la the left multiplication by a ∈ G) (la )∗ X = X, ∀a ∈ G, i.e., X(f (as)) = X(f (s)),

∀f ∈ C 1 (G, R), ∀a, s ∈ G,

is naturally equipped with a Lie algebra structure, the mapping [, ] being the commutant of the differential operators LX (where LX is the Lie derivative along the vector field X). This vector space, here denoted by G (or g), is called the Lie algebra of the Lie group G. For these notions and their main properties, we refer to [Chev], and to [Kir].

65 This space may be notably used for the study of relative equilibrium and stability, for the study of stationary rotations and bifurcations [Arn1], and for the search for irreducible representations of a group G by its orbits in g [Kir].

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Remark 41 Maurer–Cartan form on G, see [Mall, ch. III, 1.4]. A left-invariant vector field X is defined only from the value Xe at e, and then the Lie algebra G is identified with the tangent space Te G, denoted by g (or also Ge ). Indeed, the derivative of the left multiplication by the inverse of g, i.e., by Tg (lg −1 ), is an isomorphism from Tg G onto the fixed vector space g. This isomorphism defines a differential form, denoted by ω, on G with values in g, which is called the Maurer– Cartan form of G. This differential form is characterized by the following two properties (see [Mall, ch. III, 1.4]): • ω is left invariant, i.e., (lg0 )∗ ω = ω, • ωe is the identity. ¶

∀g0 ∈ G,

2.12.2 Adjoint Representation of a Lie Group Let G be a Lie group with unit element e. We consider the map in G, denoted by Int s (with s ∈ G), defined by g → (Int s)g = sgs −1 . The Lie group G acts on itself by interior automorphisms. Naturally, (Int s)e = e. Thus the mapping Ads = Te (Int s), which is the derivative of Int s at e, is a linear mapping in the Lie algebra g. We set (if there is no risk of confusion) Ads v = s.v.s −1 , for all v ∈ g. We have the following three essential properties: (i) Ads is a Lie algebra automorphism, and thus satisfies ∀v1 , v2 ∈ g.

Ads [v1 , v2 ] = [Ads v1 , Ads v2 ] ,

(2.498)

(ii) The mapping s ∈ G → Ads in the set L(g) of linear maps on g satisfies Ads.˜s = Ads ◦ Ads˜ .

(2.499)

Thus this is a representation of the Lie group G in the Lie algebra g, called the adjoint representation. (iii) The derivative of the map s → Ads at the identity of the group G is a linear map, denoted by ad = Te (Ad), from g into the space of linear operators on g such that adv1 v2 = [v1 , v2 ] ,

∀v1 , v2 ∈ g.

(2.500)

Furthermore, we have the derivation formula d Adgt = Adgt adv S = adv E Adgt , dt

(2.501)

2.12 Appendix 1. Lie Group and Lie Algebra

with v =

dg dt

317

= g, ˙ and v S = (lg −1 )∗ v, v E = (rg −1 )∗ v. It is very easy t

d dt Adgs+t |t =0

to verify from the derivative that d Adgs dt Adgt )|t =0.

=

t

d dt (Adgs .Adgt )|t =0

=

2.12.3 Coadjoint Representation From the adjoint representation of a Lie group G in the Lie algebra g (see (2.498), (2.499)), we naturally define the coadjoint representation,66 denoted by Adg∗ , from G into the dual space g∗ of g by < Adg∗ p, v > = < p, Adg v >,

∀g ∈ G, p ∈ g∗ , v ∈ g,

(2.502)

where < ., . > means the natural duality of g∗ with g. The property of the representation of G relative to the product (g, h) → h.g is given by ∗ Adgh = Adh∗ ◦ Adg∗ ,

∀g, h ∈ G.

We also define by duality with (2.500) (or by derivation of this representation) the mapping, denoted by adu∗ in g∗ , < adu∗ p, v2 > = < p, adu v2 > = < p, [u, v2 ] >, ∀p ∈ g∗ , u, v2 ∈ g. (2.503) We also write (see, for instance, [Arn1, App. 2]) adu∗ p = {u, p} ,

∀u ∈ g, p ∈ g∗ .

(2.504)

d Ad ∗ = Adg∗t adv∗E = adv∗S Adg∗t , dt gt

(2.505)

Moreover, we have

still with v =

66 Do

dg dt

= g, ˙ and v S = (lg −1 )∗ v, v E = (rg −1 )∗ v. t

not confuse this usual notation with a pullback.

t

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2.12.4 Splitting of the Tangent and Bitangent Bundles We use the fact that every Lie group is a parallelizable manifold, that is, the tangent bundle is identifiable with a product (thus trivializable).67 Indeed, the mapping ω associated with the Maurer–Cartan form, ω:

def

(g, v) ∈ T G → (g, v S ) = (g, ωg (v)) = (g, T (lg −1 )v) ∈ G×g,

(2.506)

is a diffeomorphism from T G onto G × g, with inverse ω−1 , ω−1 (g, ξ ) = T (lg )ξ (often denoted by g.ξ ), with ξ = v S ∈ g. (We also obtain a diffeomorphism ωr from T G onto G × Ge with the right multiplication (g, v) → (g, ωgr (v)) = (g, v E ) = (g, T (rg −1 )v).) The space T ∗ G is also identifiable with a product by the diffeomorphism def

ω˜ : (g, p) ∈ T ∗ G → (g, pS ) = (g, t T (lg −1 )p) ∈ G × g∗ .

(2.507)

Due to the trivialization of bundles T G and T ∗ G, the bundles T (T G), T (T ∗ G), and T ∗ (T G),T ∗ (T ∗ G) are also parallelizable manifolds; thus T ω is a diffeomorphism from T (T G) onto T (G × g). The fibre Tg,ξ (G × g) is identified with the product Tg (G) × Tξ (g), and the vector bundle T (G × g) is identified with the sum π1∗ T G ⊕ π2∗ T g of the pullback tangent bundles π1∗ T G and π2∗ T g with π1 (respectively π1 ) the canonical projection from G × g onto G (respectively g) (see [Bour.var, 5.6])). Now g is a vector space, and thus its tangent space is identified with the product g × g. Then the vector bundle T (T G) is identified with the product (G × g) × (g × g). The vector bundle T (T ∗ G) is identified68 with the bundle π1∗ T G ⊕ π2∗ T g∗ , identifiable (by ω) with (G × g∗ ) × (g × g∗ ). Then we identify every vector S ∗ ∗ X(g,p) ∈ T(g,p)(T ∗ G) with an element X(g,p S ) ∈ (G × g ) × (g × g ). Then let S S S S X(g,p) = ((g, p), (v, f )); we set X(g,p S ) = ((g, p ), (v , f )), with

pS = (lg −1 )∗ p, v S = (lg −1 )∗ v˜ = ωg P1 ω˜ ∗ (v, f ), and f S = P2 ω˜ ∗ (v, f ), where P1 (respectively P2 ) is the canonical projection from Tg G × Tp∗ g∗ onto Tg G (respectively Tp g∗ ). Furthermore, the group G naturally acts in T G, and thus in T (T G) by (double derivation of) the left multiplication T (T (lg )) (respectively right multiplication T (T (rg ))). We also have an action of G in T ∗ G by gˆ0 (g, p) = (g0 g, t T (lg −1 )p), also denoted gˆ0 (g, p) = (g0 g, (lg −1 )∗ p), 0

0

and thus in T (T ∗ G) by derivation of this mapping. 67 See

[Arn1, ch. 6.28], [Mall, ch. III, 1.4]. is identified with Tg G × Tp g∗ , thus by the Maurer–Cartan form with g × g∗ .

68 T ∗ g,p (T G)

2.12 Appendix 1. Lie Group and Lie Algebra

319

Remark 42 Group with an invariant Riemannian metric. Using the fact that the Riemannian metric g is left invariant, we verify the commutativity of the following diagram: ω˜

(s, p) ∈ T ∗ G −→ (s, pS ) ∈ G × g∗ ↑G ↑ I × Ge ω (s, v) ∈ T G −→ (s, v S ) ∈ G × g By derivation of the diffeomorphisms ω, ω, ˜ we obtain the corresponding diffeomorphisms between tangent spaces, which leads to the following diagram: T ω˜

T (T ∗ G) −→ T (G × g∗ ) ↑ TG ↑ T (I × Ge )

Diagram

¶

Tω

T (T G) −→ T (G × g) Expression of Canonical and Symplectic Forms on T ∗ G We can transport the canonical form θ on T ∗ G to the space G × g∗ , which gives, in a chart of G, with coordinates (q, p), < θ(g,p), X(g,p) > = < pS , v S >, S = pS dq S , the transported form. and we define θ(q,p) Likewise, we transport the symplectic 2-form = dθ to obtain the 2-form S = dθ S = dpS ∧ dq S ; θ and have the following property: The canonical form θ and the symplectic 2-form = dθ on T ∗ G are invariant under left multiplication.

Proof We have to prove that for all g0 ∈ G, we have (gˆ0 )∗ θ = θ . Let π be the canonical projection from T ∗ G onto G; we have θ(q,p)(X) = p(π∗ (X)), for all X ∈ T(q,p)(T ∗ G) by definition, and ((gˆ0 )∗ θ )(q,p) (X) = θgˆ0 (q,p)((gˆ0 )∗ X) = (t lg −1 )p(π∗ ((gˆ0 )∗ X))) = p((lg −1 )∗ (π∗ ((gˆ0 )∗ X)))) 0

0

(2.508)

= p([lg −1 ◦ π ◦ gˆ0 ]∗ X) = p(π∗ X) = θ(q,p)(X), 0

hence the desired property. ¶ Symplectic form on T ∗ G (or S on G × g∗ ). From the invariance of the form under left multiplication, we have for all X, X˜ ∈ T(g,p)(T ∗ G),

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˜ (g,p) ) (g,p) (X(g,p) , X

=

S S ˜S (X(g,p S ) , X(g,p S ) ), (g,p S )

with S S S S X(g,p S ) = ((g, p ), (v , f )),

Then the symplectic form

S S X˜ (g,p ˜ S , f˜S )). S ) = ((g, p ), (v

is expressed by

˜ (g,p)) (g,p) (X(g,p) , X

=

S S ˜S (X(g,p S ) , X(g,p S ) ) (g,p S )

= < f S , v˜ S > − < v S , f˜S > .¶

2.13 Appendix 2. Covariant Derivative The modelling of a physical system very often uses manifolds called fibre bundles. A fibre bundle, denoted by (E, M, π) (or simply E), consists of the manifolds E and M called the base space, and a mapping π called the canonical projection from E onto M, with the following property: For every x ∈ M, there exist an open neighborhood U of x, a manifold F , and an isomorphism φ from π −1 (U ) onto U × F with π(φ −1 (x, e)) = x, ∀x ∈ U, e ∈ F. The sets Ex = π −1 ({x}), x ∈ M, are called the fibres of E. If all fibrers are identifiable (diffeomorphic) with one manifold F , we say that E is of standard fibre F . This is the case of the trivial fibre bundle (E = M × F, M, π), where π is the projection from E onto M. In a fibre bundle E, in order to define the derivative of a function f on a manifold M with values in a vector bundle E, we have to link the fibre Ex to another Ey . Then we can define the covariant derivative of a section of a vector bundle f thanks to the notions of connection of parallel transport and horizontal lift.

2.13.1 Connection, Horizontal Lift To define a covariant derivative, there are several points of view with various spaces: (i) Use of a frame bundle P over the manifold M (linear, affine, or orthonormal frames) or more generally use of a principal bundle over M with a Lie group G acting on each fibre of P . In the case of a frame bundle, the Lie group G corresponds to the changes of frames. Then we can define a structure with two differential forms on P : the connection form69 ω (its kernel is the set of 69 See

[Kob-Nom, ch. II.1, ch. III.7]. We also call its pullback on M by a section σ of P a connection form.

2.13 Appendix 2. Covariant Derivative

321

horizontal vectors on P , and its image is the Lie algebra of G) and the canonical form70 θ (which corresponds to the identity on Tx M, ∀x ∈ M). (ii) Use of the bitangent bundle T (T M) (or T (T ∗ M)), or more generally of a vector bundle E (i.e., whose standard fibre F is a vector space) over the manifold M. We first consider this case, following [Dieud3].

Vertical and Horizontal Tangent Vectors on a Vector Bundle E Let M be an n-dimensional manifold, and F a k-dimensional vector space. Consider the tangent space Te E at a point e of E. This (n + k)-dimensional vector space contains a natural k-dimensional subspace, denoted by Ve , of vertical tangent vectors, that is, of tangent vectors at e to the fibre Ex with π(e) = x. Let 0x be the origin of the fibre Ex ; the set of vertical vectors is defined by def

Ve = {γ ∈ Te E, T π(γ ) = π∗ (γ ) = 0x } . A linear connection on E is defined by giving to every e ∈ E, an n-dimensional vector subspace He of Te E that is transverse to the space Ve , that is, such that Te E is the direct sum (for each e ∈ E) Te E = H e ⊕ V e ; He is the space of horizontal tangent vectors. We assume that this space is a differentiable function of e, that is, for every differentiable vector field X on E, its horizontal and vertical components are also differentiable. We denote by P h the horizontal projection from Te E onto He , and by P v the vertical projection from Te E onto Ve , and thus I = P h + P v .

Horizontal Lift on T E of a Vector X on M The restriction to the space He of the map T π : T E → T M is bijective from He onto Tx M. Its inverse map, denoted by Rhe = π∗ |−1 He , associates with every vector v ∈ Tx M a unique vector Rhe v = v ∗ ∈ He called the horizontal lift of v in T E. Therefore, we can define a map (called a connection map) C : T M ⊕ E → T E, by Cx (v, e) = v ∗ = Rhe v ∈ Te E,

∀v ∈ Tx M, e ∈ Ex , π(e) = x ∈ M,

with π∗ (Cx (v, e)) = v, and πE (Cx (v, e)) = e, (2.509)

70 See

[Kob-Nom, ch. III.2].

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where πE is the canonical projection from T E onto E. Let X be a vector field on M. Then the map e ∈ Ex → Xe∗ = Cx (Xx , e) ∈ He ⊂ Te E is a section of the bundle T E, called a horizontal lift of X in E. Conversely, giving a linear connection map C, injective with respect to each variable v ∈ Tx M, and e ∈ Ex (the other variable being fixed) and satisfying (2.509), allows us to define a horizontal projection P h by P h (γ ) = Cx (π∗ (γ ), e) for all γ ∈ Te E, and a horizontal lift Rhe by Rhe v = v ∗ = Cx (v, e). The notion of horizontal lift may also be applied to a curve in M, which allows us to define the parallel transport of a vector of T E along a curve τ = (xt ), t ∈ [0, 1] in M. Let u0 ∈ Ex0 and τ ∗ = (ut ), t ∈ [0, 1], ut ∈ Ext be the unique horizontal lift of τ in E, with initial point u0 . Let u1 be the final point of τ ∗ . The mapping u0 ∈ Ex0 → u1 ∈ Ex1 is the parallel transport of the fibre Ex0 along the curve τ from x0 to x1 . Descent of a Vertical Vector γ ∈ Ve on the Fibre Ex Along e The vertical space Ve being the tangent space at e ∈ Ex to the fibre Ex , which is a vector space, there naturally exists an isomorphism, denoted by τ e , from Ve onto Ex , defined in the following way: let u = (ej )j =1,...,k be a frame of Ex and let ξ = (ξ j )j =1,...,k ∈ R k and u(ξ ) =  ξ j ej ∈ Ex be the decomposition of a generic element of Ex . Then (ξ j )j =1,...,k is a coordinate system of Ex , and the family U = (∂/∂ξ j )j =1,...,k is a basis of the tangent space: every element of Ve is written γ˜ = U (γ ) =  γ j ∂/∂ξ j with γ = (γ j )j =1,...,k ∈ R k . Then the map τ e from Ve into Ex defined by τ e (γ˜ ) =  γ j ej ∈ Ex , thus τ e = uU −1 , is an isomorphism giving the descent of the vertical vector γ˜ ∈ Ve on Ex at e. This map is independent of basis. Thus we have τ e (∂/∂ξ j ) = ej . This map, which allows us to identify a vector space with its tangent space, is naturally used for the derivative of a vector function t → ξ(t): we write ξ˙ (t) =  ξ˙ j (t)ej by identifying ej with ∂/∂ξ j for all j .

Vertical and Horizontal Differential Forms A differential form ω on E such that < ω, γ > = 0,

∀γ ∈ Ve (respectively He ), ∀e ∈ E,

is said to be horizontal (respectively vertical). The decomposition Te∗ E = He⊥ ⊕Ve⊥ of the space Te∗ E, where Ve⊥ (respectively He⊥ ) is the set of null covectors on Ve (respectively on He ), gives a decomposition of the differential form space on E. The ∂ ∂ canonical form θ on T ∗ M is horizontal, since < θp , w ∂x + γ ∂v > = p.w.

2.13 Appendix 2. Covariant Derivative

323

Application with E = T M Then the vertical and horizontal spaces have the same dimension n = k. An element γ in Te E = Tx,v (T M) (with e = (x, v)) is written in a chart U with coordinates (x j , v j ): γ =  (wj ∂/∂x j + γ˜ j ∂/∂v j ), which is also ((x, v), (w, γ˜ )) in T (T M), with these coordinates. Then the connection is given by Cx ((x, v), (x, w)) = Cx (v, w) = ((x, w), (v, − x (v, w))) ∈ Tx,w (T M), (2.510) where x is a bilinear map from R n × R n into R n . The vertical elements of Tv (T M) are such that wj = 0, j = 1, . . . , n, thus such that γ˜ = γ˜ j ∂/∂v j , and the j j 71 descent map τ v is such that τ v (∂/∂v ) = ∂/∂x . Let v ∈ Tx M. The horizontal lift of v at w ∈ Tx M is given by ∗ = Rhw v = Cx (v, w) = ((x, w), (v, − x (v, w))) ∈ Tw (T M), vw

(2.511)

with x (v, w) =



xk (v, w)

∂ , ∂wk

xk (v, w) =



v i wj ijk .

(2.512)

The coefficients jki are called the Christoffel symbols of the connection. With every element γ ∈ Tw (T M), we associate (using the vertical projection P v and the descent map τ w ) an element Qw γ ∈ Tx M: Qw γ = τ w (P v (γ )) = τ w (γ − P h (γ )) = τ w (γ − Cx (π∗ (γ ), w));

(2.513)

Qw is called the vertical descent map, and Qw γ is the vertical descent of γ on Tx M. With π∗ (γ ) = v, (2.513) reads Qw γ = τ w (γ − Cx (v, w)) =

71 But

 (γ˜ k + v i wj ijk ) ∂/∂x k .

a priori, vectors in Tv (T M) such that γ˜ j = 0, j = 1, . . . , n, are not horizontal.

(2.514)

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2.13.2 Covariant Derivative Covariant Derivative of a Section of a Vector Bundle E Let s be a section of a vector bundle E (over the base M, with projection π): s is a differentiable map from M into E such that π(s(x)) = x, ∀x ∈ M. Let C be a linear connection on E (see (2.509)). The covariant derivative of s along the vector v ∈ Tx M is defined by def

(∇v s)x = τ s(x) [Tx (s)v − Cx (v, s(x))] = τ s(x)P v (γ ) = Qs(x)γ ∈ Ex , with γ = Tx (s)v ∈ Ts(x)E, (2.515) using the vertical projection P v and the descent map τ e at e = s(x). Indeed, we have π∗ (γ ) = T (π) ◦ T (s).v = T (π ◦ s).v = v and πE (γ ) = s(x) = e, π(e) = x. Another (inequivalent) definition of the covariant derivative can be given thanks to the parallel transport. Let τ = (xt ) be a curve in M such that x0 = x, and let τtt +h be the (inverse) parallel transport of the fibre Ext+h to the fibre Ext along τ from xt +h to xt . Then the covariant derivative of s along v is also defined by (see [Kob-Nom, ch. III.2, pp. 114, 123]) def

∇v s = ∇x˙t s = lim

h→0

 1  t +h τt (s(xt +h )) − s(xt ) . h

(2.516)

Now if v is a vector field on M, then (2.515) (or (2.516)) defines a section of E by def

(∇v s)(x) = ∇v(x)s,

(2.517)

which is called the covariant derivative of s along the vector field v. We have thus defined a differential operator in the section space of E with the following properties: 

∇v1 +v2 s = ∇v1 s + ∇v2 s ∇σ v s = σ ∇ v s



∇v (s1 + s2 ) = ∇v s1 + ∇v s2 ∇ v (σ s) = (Lv σ ) s + σ ∇ v s,

(2.518)

where s, s1 , s2 (respectively v, v1 , v2 ) are sections of E (respectively of T M), σ is a real differentiable function on M, and Lv σ is the Lie derivative of σ along v. Conversely, a differential operator that satisfies the properties (2.518) is the covariant derivative with respect to a unique (linear) connection.

2.13 Appendix 2. Covariant Derivative

325

Covariant Derivative for E = T M The formulas (2.518) show that it suffices to define the covariant derivative for E = T M using local frames. Let (X1 , . . . , Xn ), be a local frame72 for E (that is, a section of the frame bundle LE). The covariant derivatives of these vector fields is defined by ∇Xj Xi =



jki Xk ,

(2.519)

with the Christoffel symbols jki of this family. This allows us to determine the connection and the covariant derivative of any section s of T M, or more generally of the tensor space Tsr (M), along any tangent vector. The covariant derivative is a derivation that commutes with the contractions and is identical to the Lie derivative on functions. Let (U, ϕ) be a chart of an atlas, with coordinates (x j ), and its associated frame of vector fields (Xj = ∂x∂ j , j = 1, . . . , n). Then (2.519) applied to these vector fields allows us to define the covariant derivatives of the differential forms (dx j ) by ∇Xj dx i = −



ji k dx k = −ωji ,

(2.520)

with the components ωji of the pullback of the connection form ω on P = L(M) the linear frame bundle, by the section x ∈ M → (Xj ) ∈ P = L(M). Proof Since < dx i , Xk > = δki , we have ∇Xj (< dx i , Xk >) = < ∇Xj dx i , Xk > + < dx i , ∇Xj Xk > = 0. Thus (2.519) gives (2.520).

2.13.3 Extension of the Covariant Derivative Extension of the covariant derivative of a differentiable map from a manifold N into a vector bundle E (see [Dieud3, ch. XVII.17]). For the applications in view, we give only the cases in which N = R (or N = J × M, J an open interval of R), then N = R × M with E = T (M).

72 See

[Gilk, ch. 2.1].

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Time-Covariant Derivative of a Vector Field v on M τ∗

v∗

∂t ∈ T R −→ Tx(t )M −→ γ (t) ∈ Tx(t ),v(x(t ))T M Diagram ↑↓ ↑v↓ ↓ πT M ↓ T π τ v t ∈ J ⊂ R −→ x(t) ∈ M −→ v(x(t)) ∈ Tx(t )M The vector field v defines an evolution pseudogroup in M, and thus an embedding t ∈ J → x(t) ∈ M. Let τ = (x(t)), t ∈ J ⊂ R, be a path (of class C 2 ) in M with x(0) = x ∈ M. Then we have (i) a path τ˜ = (x(t), v(x(t))), t ∈ J in T M, with v(x(t)) = x(t) ˙ ∈ Tx(t )M, with ∂t = (∂/∂t)t the unit tangent vector of R; (ii) a path τ˜˜ = (x(t), v(x(t)), γ (t)) in T (T M), with γ (t) = (x(t), ˙ x(t)) ¨ γ (t) = (v(x(t)), γ˜ (t)) = v∗ v(τ (t)) = v∗ τ∗ (∂t ) = (v ◦ τ )∗ (∂t ). The covariant derivative of the vector field v along the tangent vector v(x(t)) ∈ Tx(t )M, denoted by ∇v(x(t ))v (or (∇v v)x(t ) ), is defined by   def (∇v(x(t ))v) = τv(x(t )) γ (t) − Cx(t )(v(x(t)), v(x(t))) = Qv(x(t ))(γ (t)). (2.521) Geodesics are curves in M of a velocity vector field v defined by ∇v v = 0. They are also the projections on M of integral curves (called geodesics) of the spray (also called geodesic field) on T M, denoted by S, defined by S(v) = Cx (v, v),

∀v ∈ T M, π(v) = x.

(2.522)

k ), Q Remark 43 Recall that in the frame v(t ) (γ (t)) is given by (2.514).  k (∂/∂x k j Using the connection form ωi = j i dx , we have in this frame, with a velocity vector field v, at Tx(t )M,

(∇v v)k =

dv k  + < ωik , v > v i . dt

(2.523)

We can also write this, using components with the differential forms σ and ω for a vector field X = v, by (∇v v)k =

dv k  k dv k  + ωi (v) σ i (v) = + < ωik ⊗σ i , v⊗v > . ¶ dt dt

(2.524)

Let w be a vector field on M; we define its time-covariant derivative ∇vx (t )w along τ˜ by def

∇vx (t ) w = τw(x(t ))



 dw(x(t)) − Cx(t )(vx (t), w(x(t))) . dt

¶

2.13 Appendix 2. Covariant Derivative

327

Time-Covariant Derivative of a Vector Field Let w(t, x) ∈ Tx M, wt = w(t, .) be a vector field on M with t ∈ J ⊂ R. On τ , we have w(t) = w(t, x(t)) ∈ Tx(t )M. Let v(t) ˜ = (∂t , vx (t)). The covariant derivative ∇v(t ˜ is defined by ˜ ) w ∈ Tx(t )M of w along v(t) 

def

∇v(t ˜ ) w = τw(t ) with components

dw dt (t)

=

 dw (t) − Cx(t )(vx (t), w(t)) , dt

 dwi dt

(t) ∂x∂ i = [



i

v j ∂w + ∂x j

∂wi ∂ ∂t ] ∂x i , ∂i

(2.525) =

∂ : ∂x i

 ∂wi  ∂i + ∇vx (t ) w(t, .), with (wi ∂i ) = ∂t i i   k ∇vx (t ) w(t, .) = [(v j ∂j wk ) + v j wi j,i ] ∂k . ¶

∇v(t ˜ ) w = ∇v˜

j

i

Furthermore, if u is a differentiable mapping from a manifold N1 into a manifold N, then the covariant derivatives of a differentiable mapping φ from N into E and of φ ◦ u are linked through (see, for instance, [Dieud3, ch. XVII.17]) ∇h1 (φ ◦ u) = ∇h φ,

h1 ∈ Tz1 (N1 ), h = Tz (u)h1 ∈ Tz (N), z = u(z1 ). (2.526)

2.13.4 Riemannian Covariant Derivative When M has a Riemannian structure g, we determine a connection, called a Riemannian connection or Levi-Civita connection, with the property LX (g(Y, Z)) = g(∇X Y, Z) + g(Y, ∇X Z)

(2.527)

of the covariant derivative for all differentiable vector fields X, Y, Z on M. The left term of (2.527) is the derivative along X of the function x ∈ M → gx (Yx , Zx ) ∈ R. With (2.518), (2.527), the associated covariant derivative is such that ∇X Y − ∇Y X = [X, Y ].

(2.528)

Let X, Y, Z be vector fields on M. From (2.527), we obtain the fundamental formula [Kob-Nom, ch. IV.2, Prop. 2.3]

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2g(∇X Y, Z) = g1 (X, Y, Z) + g2 (X, Y, Z),

with

g1 (X, Y, Z) = Xg(Y, Z) + Y g(Z, X) − Zg(X, Y ),

(2.529)

g2 (X, Y, Z) = g([X, Y ], Z) + g([Z, X], Y ) + g([Z, Y ], X). Then we have to consider the following situations. (i) A coordinate system (x i ) is locally given, with corresponding vector fields Xi = ∂x∂ i such that [Xi , Xj ] = 0, ∀i, j . Hence g2 (Xi , Xj , Xk ) = 0, ∀i, j, k, and thus 2g(∇Xi Xj , Xk ) = g1 (Xi , Xi , Xi ) = Xi g(Xj , Xk ) + Xj g(Xk , Xi ) − Xk g(Xi , Xj ). (ii) An orthonormal frame (ei ) is given, thus g(ei , ej ) = δi,j , ∀i, j . With Xi = ei , we have g1 (Xi , Xj , Xk ) = 0, ∀i, j, k, and thus 2g(∇Xi Xj , Xk ) = g2 (Xi , Xi , Xi ) = g([Xi , Xj ], Xk ) + g([Xk , Xi ], Xj ) + g([Xk , Xj ], Xi ). (2.530) Moreover, if we have the structure equations [Xi , Xj ] =



k ci,j Xk ,

then j

k i 2g(∇Xi Xj , Xk ) = ci,j + ck,i + ck,j . k , with ∇Xi Xj = Thus the Christoffel symbols i,j k i,j =



k i,j Xk , are such that

1 k j i ). (c + ck,i + ck,j 2 i,j

(iii) Let g be a left-invariant Riemannian metric on a Lie group; we have g1 (X, Y, Z) = 0, and at the identity e of the group, let (Xi ) be an orthogonal frame such that g(Xi , Xj ) = 0, i = j, and g(Xi , Xi ) = Ii , ∀i. Then we obtain j

k i 2g(∇Xi Xj , Xk ) = ci,j Ik + ck,i Ij + ck,j Ii , k are such that and the Christoffel symbols i,j k i,j =

1 k j i (c Ik + ck,i Ij + ck,j Ii ). 2Ik i,j

2.13 Appendix 2. Covariant Derivative

329

Thus, in the case of a rigid body with a fixed point, the Lie group is the rotation 3 = c 2 = c 1 = 1; we obtain, group, where the constants of structure are c1,2 3,1 2,3 for instance, 3 1,2 =

I2 − I1 1 (1 + ). 2 I3

Remark 44 Let G be the metric map (see (2.15)) of a Riemannian metric. As a direct consequence of the definition, the covariant derivative commutes with the metric map, that is, for all vector fields v, X on M, ∇v (GX) = G(∇v X).

(2.531)

Now let ωn be the natural volume element on an oriented Riemannian manifold M, that is, an n-form such that for every (direct) orthonormal basis (Xj ) of Tx M, we have ωn (X1 , . . . , Xn ) = 1; ωn is given in a coordinate system by ωn = G1/2 dx 1 ∧ · · · ∧ dx n ,

with G = det (gi,j ).

Then we have ∂ ∂ , . . . , n ) = G1/2 , ∇X ωn = 0, ∀X, ∂x ∂x 1  and thus, with (2.520) and X = Xj Xj , Xj = ∂x∂ j , ωn (

∇X (G1/2dx 1 ∧ · · · ∧ dx n ) = [X(G1/2) −



Then for all Xj , we obtain Xj (G1/2) = G1/2 G−1/2 Xj (G1/2) =



i j,i ,



i Xj j,i ](G1/2 dx 1 ∧ · · · ∧ dx n ). i j,i , and

thus Xj (G) = 2G



i j,i .

(2.532)

The Christoffel symbols of the Riemannian manifold are given by l j,i =

∂gj k ∂gj i 1  lk ∂gki − ). g ( j + i 2 ∂x ∂x ∂x k

(2.533)

Thus we obtain 

i = j,i

1  ik ∂gik . g 2 ∂x j

(2.534)

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The Lie derivative of ωn with respect to the vector field X is given by LX ωn = (div X) ωn = (G−1/2 X(G1/2) + =(

 ∂Xj

1  j ik ∂gik X g + ) ωn . j ∂x 2 ∂x j

 ∂Xj

∂x j

) ωn (2.535)

¶

2.13.5 Connection on a Principal Fibre Bundle Intrinsic definitions of connection and the covariant derivative may be given using the notion of principal fibre bundles of frame bundles and associated fibre bundle; we briefly recall these notions, referring to [Kob-Nom, ch. I.5]. The frame bundles have the structure of a principal fibre bundle: Definition 8 A principal fibre bundle P (M, G) or P (M, G, π) is a fibre bundle π P → M, with π the differentiable canonical projection from the manifold P onto the manifold M, with a Lie group G that acts freely73 on the right on P , on every fibre π −1 (x), ∀x ∈ M. Moreover, P is locally trivial, i.e., every x ∈ M has a neighborhood U and a diffeomorphism ψ : π −1 (U ) → U × G such that ψ(ua) = (π(ua), φ(ua)) = (π(u), φ(u)a), ∀u ∈ π −1 (U ), a ∈ G. Therefore, the manifold M may be identified with the quotient P /G. Let M be a Riemannian manifold; then we can define the bundle of orthonormal frames O(M) (with the structure of a principal fibre bundle, with Lie group O(n)), that is, the set of frames u = (e1 , . . . , en ) of Tx M, with g(e i , ej ) = δi,j , and we identify the frame with the map λ = (λ1 , . . . , λn ) → u(λ) = λj ej . Differential forms on a Riemannian manifold. There naturally exists on O(M) a canonical differential form θ with values in R n such that θu (X) = u−1 (π∗ X), ∀X ∈ Tu (O(M)). With the Riemannian structure we can associate on O(M) a differential form ω, called a Riemannian connection form, with values in the Lie algebra o(n), that allows us to define the notion of covariant derivative of vector fields on M. These two differential forms are linked by the structure equation dθ + ω ∧ θ = 0. Here in order to be self-contained, we recall the following notion [Kob-Nom, ch. I.5], which we have often used.

73 Recall

that ua = u for some u ∈ P , implies a = e.

2.13 Appendix 2. Covariant Derivative

331

Definition 9 Let F be a manifold on which the Lie group G acts on the left: (a, ξ ) ∈ G × F → aξ ∈ F , and let P (M, G) be a principal fibre bundle. Then G acts on the product space P × F by (u, ξ )a = (ua, a −1 ξ ) ∈ P × F . The quotient space of P × F by the group action is denoted by E = P ×G F , and the image of (u, ξ ) by the projection πE on E is denoted by u.ξ , so that ua.ξ = u.aξ . Let π be the projection of P (M, G) on M; with the map (u, ξ ) → π(u) = x ∈ M, the space E is a fibre bundle of base M. It is called the fibre bundle associated with P and denoted by E(M; F, G, P ).

2.13.6 Christoffel Symbols, Riemann–Christoffel Tensor Differential forms of orthonormal frames on a Riemannian manifold. Let Ax = (A1 , . . . , An )x in Tx M, x ∈ U ⊂ M, be a “moving” orthonormal  j frame that defines the map Ax : x = (x 1 , . . . , x n ) ∈ R n → ux (x) = x Aj in Tx M, that is, a section of the orthonormal frame bundle O(M) (thus such that g(Ai , Aj )x = δi,j ). With the existence of the frame Ax is associated a differential form ω ∈ (M, R n ) (from T M into R n ), with ωx the inverse of the map Ax . Thus ωx (X) = (x i ) ∈ R n ,

∀X =



x i Ai ∈ Tx M.

Then the Riemannian metric g is such that on U , gx = ωx ⊗ ωx ,

x ∈ U.

(2.536)

The differential form ω allows us to define the covariant derivative of the vector field X on M along the vector field Y by ω(∇Y X) = LY ω(X) + γ (Y )ω(X),

(2.537)

with the Lie derivative LY , and with the differential form called a Riemann– Christoffel form γ ∈ (M, Mn,a ), with values in the set of skew-symmetric matrices on R n ; see [Mall, ch. III, 8.4]. Then we have ωi (∇Aj Ak ) = γki (Aj ), with the differential form ω = (ω1 , . . . , ωn ) associated with the orthonormal frame A = (Aj ). For every x ∈ M, ωx is an isomorphism from Tx (M) onto R n , and thus ker ωx = {0}. From the Frobenius theorem, there exists a differential form γ ω = γ = (γj k ) on M with matrix values (called the Riemann–Christoffel form associated with the orthonormal field ω) such that dω = −γ ∧ ω.

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2 Classical Mechanics

Moreover, in a coordinate system (ϕ j ), with Xj = ∂/∂ϕ j , the covariant derivative in this frame is given by the Christoffel symbols ji k ; see (2.519). ¶

2.13.7 Time-Covariant Derivative on a Riemannian Group Invariant Riemannian Connection on a Lie Group We use the duality of g∗ with g due to the Riemannian metric that is invariant on the left. Let G be a metric mapping of duality from g onto g∗ (see the example of a rigid body). We then define a map, denoted by adG,v , in g by def

(adG,v v1 , v2 )g = (v1 , adv v2 )g = (v1 , [v, v2 ])g ,

∀v, v1 , v2 ∈ g,

and also the mapping B, g × g → g by def

B(v, v1 ) = adG,v v1 ,

thus g(B(v, v1 ), v2 ) = g(v1 , [v, v2 ]).

(2.538)

We easily verify the relation B(v, v1 ) = G−1 {v, Gv1 } = G−1 adv∗ Gv1 .

(2.539)

Indeed, we have ([v, v2 ] , v1 )g = < G [v, v2 ] , v1 > = < [v, v2 ] , Gv1 > = < adv v2 , Gv1 > = < v2 , adv∗ Gv1 > = < v2 , GG−1 adv∗ Gv1 > = (v2 , G−1 adv∗ Gv1 )g . The relation [v, v1 ] = − [v1 , v] implies (B(v, v2 ), v1 )g + (B(v1 , v2 ), v)g = 0, for all v, v1 , v2 in g, which is again written

∗ v ,v adG,v 1 2

 g

  = − v1 , adG,v v2 g .

Now we define a connection on G. Recall first that a connection C : T G ⊕ T G → T (T G) is said to be left invariant if it satisfies Cg.h (g.v1 , g.v2 ) = g.Ch (v1 , v2 ),

∀v1 , v2 ∈ Th G, ∀g, h ∈ G,

(2.540)

where the actions T (lg ) and T (T (lg )) of g ∈ G successively in T G and T (T G) by derivation of the left multiplication by g are simply denoted by g.h and g.Ch . The relation (2.540) is equivalent to Ch (v1 , v2 ) = h.Ce (h−1 v1 , h−1 v2 ),

∀v1 , v2 ∈ Th G, ∀h ∈ G.

2.13 Appendix 2. Covariant Derivative

333

Thus it is enough to know the map (v1 , v2 ) ∈ g × g → Ce (v1 , v2 ) to define the connection.

Covariant Derivative on a Riemannian Lie Group Let v and w be left-invariant vector fields on G; then for a left-invariant connection, the corresponding covariant derivative ∇v w is also a left-invariant vector field on ˜ e = G. Let X ∈ g, and let X˜ ∈ G be the left invariant vector field on G with (X) −1 ˜ X; thus X = ω (X). The mapping B of (2.538) is linked to the left-invariant Riemannian connection associated with the Riemannian metric g of G, and to the covariant derivative ∇X Y˜ at e, of the left-invariant vector field Y˜ , in the direction of X ∈ g = Te G, by 1 (∇X Y˜ )e = ([X, Y ] − B(X, Y ) − B(Y, X)). 2

(2.541)

The right-hand term of (2.541) is a bilinear map from g × g into g. Note that we also have (∇v˜0 v˜1 )e = (∇v0 v˜1 )e . Proof We still denote Y˜ simply by Y . We return to (2.529), and as previously indicated, g1 (X, Y, Z) = 0, whence 2g(∇X Y, Z) = g2 (X, Y, Z) = g([X, Y ], Z) + g([Z, X], Y ) + g([Z, Y ], X). Using the definition of B, we have 2g(∇X Y, Z) = g([X, Y ], Z) − g(B(X, Y ), Z) − g(B(Y, X), Z), and hence (2.541).

¶

Proposition 10 The time-covariant derivative of the velocity field v is given by transport to the identity of the group by the (left) Maurer–Cartan form, with v S (t) = (lg −1 )∗ vgt = ωg(t )(vgt ) ∈ g, and s(gt ) = vgt (or s(t, gt ) = vgt ), by t

ωg(t )(∇(t,v)s) =

dv S − B(v S , v S ). dt

(2.542)

Proof The covariant derivative of a section s of the bundle E = T G with respect to the vector v = hg ∈ Tg G is given by (2.515) (with G = M), i.e., ∇t,v s = τ s(g)[s∗ (v) − Cg (v, s(g))].

(2.543)

Since the Riemannian metric of the group is left invariant by multiplication, the Riemannian connection C is also left invariant (see (2.540)), and then with the left

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2 Classical Mechanics

Maurer–Cartan form ωg = (lg −1 )∗ , and its derivative as a map from T (T G) on T (g), denoted by (lg −1 )∗∗ , we have (lg −1 )∗∗ (Cg (v, v)) = Ce (ωg (v), ωg (v)) = Ce (v S , v S ), with v S = ωg (v) ∈ g. Moreover, we have the relation τ v ◦ (lg )∗∗ = (lg )∗ ◦ τ ωg (v) = (lg )∗ ◦ τ v S . Then (2.543) becomes   ωgt (∇t,v s) = τ v S ((lg −1 )∗ s)∗ (v) − Ce (v S , v S ) .

(2.544)

S

Since ((lg −1 )∗ s)∗ (vt ) = ( dv dt ) (in Tv S (T G)), we obtain t

ωgt (∇t,v s) = τ v S [

dv S dv S − Ce (v S , v S )] = − B(v S , v S ). dt dt

¶

(2.545)

Remark 45 Time-covariant derivative with the right Maurer–Cartan form ωr . In a similar way, we s have ωgr t (∇(t,v)s) =

dv E − B r (v E , v E ). dt

¶

(2.546)

Chapter 3

Fluid Mechanics Modelling

3.1 Kinetic Modelling Modelling fluid mechanics generally begins with defining a fixed framework with a fixed domain M of the fluid, a space of diffeomorphisms of M, so that the time intervenes as an exterior parameter as a last resort. When the domain of the fluid is time-dependent, we must change the strategy. The time must be taken into account in the framework from the beginning, and we have to use the basic notions of Euler and Lagrange variables. The point of view is in part different from that of Chapter 1, but there will be some repetitions, and the notation does not always agree with the previous notation. From a smooth velocity field (known or assumed to be known) v(x, t), x ∈ M ⊂ E3 of a fluid we obtain the local evolution of the fluid by the integral curves of v, following classical results of differential geometry. We then define the evolution of the metric and the strain tensor. The first main point is to determine the velocity field from the fluid equations, but this field is often not smooth, so that this implies that the definition of the flow is not easy. Then we have to determine the strain tensor and the metric, which is not trivial, even if the initial metric tensor is assumed to be the identity.

3.1.1 Integral Flow and Time Foliation Integral Flow We first recall some notions of evolution in a fixed domain. A deterministic modelling of the evolution of a medium that is represented by a manifold M is made by a family (φt ) of 1-parameter transformations in M, so that φt (x) = φ(x, t) ∈ M, with t ∈ R, x ∈ M, or more generally by a 2-parameter transformation family © Springer International Publishing AG, part of Springer Nature 2018 M. Cessenat, Mathematical Modelling of Physical Systems, https://doi.org/10.1007/978-3-319-94758-7_3

335

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3 Fluid Mechanics Modelling

(φt,s ) (then φt,s (x) = φ(x, t, s), x ∈ M, t, s ∈ R), giving the state of the fluid at time t from its state at time s < t: xt = φ(x, t, s). Recall that a 1-parameter transformation group of a manifold M is a map (x, t) ∈ M × R → φt (x) ∈ M such that (i) for every real t, φt : x ∈ M → φt (x) ∈ M is a differentiable transformation, i.e., a diffeomorphism of M, namely φt is a homeomorphism such that φt and its inverse are differentiable (or φt admits an inverse φt−1 such that φt and φt−1 ∈ C 1 (M)); (ii) for every real s and t, φt +s (x) = φt (φs (x)), ∀x ∈ M. In the case of a transformation family φt,s , (ii) becomes (ii) for every real s, t, φt +s,0(x) = φt +s,s (φs,0(x)), ∀x ∈ M. Then φ0 (x) = x, and φs,s (x) = x, ∀x ∈ M, and thus φ0 = IM , φs,s = IM . Let M be a manifold, and X a tangent vector field on M of class C k , with k ≥ 1. An integral curve of X with x as origin is a function φ = φ (x) from an open interval I in R, with 0 ∈ I , with values in M, such that dφ (t) = Xφ(t ), dt

φ(0) = x,

x ∈ M.

(3.1)

Let Ix be the largest open interval in R containing 0 on which φ (x) may be defined; this interval is called the lifetime interval of the fluid particle at x, for the field X. Then we obtain a function (still denoted by) φ defined in R × M with values in M by φ(x, t) = φ (x) (t), called an integral flow of X, such that (i) ∀t ∈ Ix ,

φt : x → φ(x, t) is a transformation in M;

(ii) with s ∈ Ix , and t ∈ Iφs x we have t + s ∈ Ix , φt +s (x) = φt (φs (x)). (3.2) The family (φt ) is called a 1-parameter local group of local transformations1 generated by X (X is then called a generator of the group). The case of a time-dependent vector field; 2-parameter flow. When the velocity vector field is time-dependent, we have to generalize the previous situation. Let Xt be a time-dependent vector field on M that defines a map of class C k , k ≥ 1, from M × R into T M, thus such that Xxt = Xx,t ∈ Tx M, ∀t. Then X generates a timedependent flow φt,s , called a local pseudogroup, such that d φt,s (x) = Xφt t,s (x) , dt

with φs,s (x) = x.

(3.3)

This flow satisfies φt,s ◦ φs,r (x) = φt,r (x)

1 Or

also a pseudogroup; see [Mall, ch. II.2.4]).

(thus φt,s ◦ φs,t (x) = x).

(3.4)

3.1 Kinetic Modelling

337

We specify the definition domains (as in (3.2)) by Is,x , the largest open time interval t ∈ R that contains s on which we can define φt,s (x).

Time Foliation When the fluid domain M is time-dependent, we still have to generalize the previous situation. Let M˜ ⊂ E3 × R be the spacetime domain such that its section M˜ s = Ms × {s}, for every s ∈ R, is the fluid domain at time s.2 The natural foliation of the space E3 × R for the canonical projection πR on R induces a foliation, denoted ˜ The map π from M˜ onto R, π(y, s) = s, called by M˜ π = ∪s∈R Ms × {s}, of M. the canonical projection, is a submersion, and the sets π −1 (s) = M˜ s = Ms × {s} ˜ identified with Ms are the leaves of a foliation of M. We recall these definitions and properties, following [Bour.var]. The set M˜ π is equipped with the unique structure of a manifold on M˜ for which the leaves (Ms ) are open submanifolds; this structure is said to be obtained by gluing together the domains Ms , s ∈ R. The topological space M˜ π is the sum of the spaces Ms . The canonical map M˜ π → M˜ is a bijective immersion. Then M˜ π is called the foliation of M˜ defined by the canonical projection π. The kernel of T (π): ker T (π) is a vector subbundle of T M˜ called the subbundle ˜ M˜ π ). We can identify the of T M˜ tangent to the foliation Mπ and denoted by T (M, ˜ ˜ ˜ ˜ vector space Tx˜ Mπ = Tx˜ (M, Mπ ) with Tx Ms for x˜ = (x, s) ∈ M. s s The vector field family (X ) on Ms such that Xx = Xx,s , see (3.3), allows us to define the vector field X˜ on M˜ by X˜ = (X, 1), or X˜ x˜ = Xx˜ + ∂/∂s, x˜ = (x, s),

(3.5)

˜ M˜ π ). Thus π∗ X˜ x˜ = ∂/∂s. with Xx˜ ∈ Tx˜ (M, Two-parameter flows on M˜ and on M˜ π . We assume that the fluid evolution in M˜ is expressed for all times t, s by a family (φt,s ) of diffeomorphisms from Ms onto Mt . This family is changed into a 1-parameter family (φ˜ t ) acting in M˜ π by φ˜ t (x, s) = (φt +s,s (x), t + s).

(3.6)

We easily verify the relation of a local 1-parameter group (with (3.2)): φ˜ t1 +t2 = φ˜ t1 φ˜ t2 ;

(3.7)

2 If the domain M is time-independent, then M ˜ is identified with M × R. Notice that M˜ may be unknown, giving a free boundary problem. In usual models, the closure of the domain Ms occupied at time s by the fluid is the support of the volume mass ρ(s).

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(φ˜ t ) preserves the foliation (i.e., transforms the leaf Ms into Mt +s , ∀s) and satisfies π ◦ φ˜ t = τt ◦ π, where τt is the translation of t in R that corresponds to the commutativity of the following diagram: φ˜ t

x ∈ Ms −→ φt +s,s (x) ∈ Mt +s ↓π ↓π τt s ∈ R −→ s+t ∈R Let (φt,s ) be the flow generated by the family of vector fields (Xs ). Then the ˜ vector field X˜ is the generator of the flow (φ˜ t ), or (φ˜ t ) is the integral flow of X. Indeed, by (3.3), d d φ˜ t (x, s) = ( φt +s,s (x), 1) = (Xφt +s , 1) = X˜ x˜ , with x˜ = φ˜ t (x, s). t+s,s (x) dt dt

3.1.2 Euler and Lagrange Coordinates The time-dependent velocity field that is to be determined by solving the fluid mechanics equations must lead to the existence of a family of functions (φt,s ) giving the evolution of a fluid particle in a domain M of R 3 (time-dependent or not) by xt = φt,s (xs ).

(3.8)

We first make the essential hypothesis of embedding of the set M˜ into R 3 × R. We denote by j this embedding, by j t its restriction to the leaf Mt . We also assume that the lifetime interval of any fluid particle is R. We say that the velocity vector field is complete, that is, that vt generates, at each time t, a global “pseudogroup.” We will assume in a first step that the velocity vector field is of class C 2. Euler coordinates. The embedding j t of Mt into R 3 defines a coordinate system, called Euler coordinates. We denote by x (or (x, t)) the Euler coordinates of x˜ ∈ Mt ; thus (x, t) = j t (x). ˜ The map j E , which is the family (j t ), t ∈ R, is a morphism from the foliation M˜ π of M˜ into the natural foliation denoted by R 3+1 that is the set sum of (R 3 × {t}), t ∈ R, of R 4 : jE

x˜ ∈ M˜ π −→ ↑ jt

R 3+1 ↑

x˜ ∈ Mt −→ (x, t) ∈ R 3 × {t}

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339

Lagrange coordinates. The leaf M0 is a “standard” leaf to which every leaf Mt is diffeomorphic by φ0,t , which allows us to define the Lagrange coordinates of every point of Mt . We identify M0 with a domain of R 3 with the embedding j 0 of M0 into R 3 . Then the map j L,t = j 0 ◦ φ0,t from Mt into R 3 is a coordinate system for Mt , called a Lagrange system. This defines the map j L from M˜ π into R 3 by x˜ ∈ Mt → j L,t (x) ∈ j 0 (M0 ) ⊂ R 3 × {0}. The Lagrange coordinates of x˜ ∈ Mt are usually denoted by ξ : ξ = j L,t (x). ˜ A main interest of the Lagrange coordinates is to keep the same fixed domain M0 along the evolution. ¶ Thus we have associated the transformation φt,0 with a coordinate system that R represents the fluid evolution in R 3 is given by the following diagram, where φ0,t t 0 (thanks to the embeddings j , j ): R t R t j L,t = φ0,t j = φ0,t j = j 0 φ0,t :

x ∈ Mt ↓ jt

φ0,t

−→

(3.9)

a ∈ M0 ↓ j0

R φ0,t

x˜ ∈ t ⊂ R 3 × {t} −→ ξ ∈ 0 ⊂ R 3 × {0} With respect to the evolution group (φ˜ t ), we have j L ◦ φ˜ t = j L

and j E ◦ φ˜t = φ˜ tR ◦ j E , ∀t ∈ R,

(3.10)

where (φ˜ tR ) represents the evolution group in R 3 × R deduced from φ˜ by the family (j t ), t ∈ R. Thus the map j L is invariant under the evolution group. Indeed, j L (φ˜ t (x, s)) = j L (φt +s,s x, t + s) = j 0 (φ0,t +s φt +s,s x) = j 0 (φ0,s x) = j L (x, s). Transverse derivative of functions. Let f be a function from M˜ into R. The derivative of f ◦ φt,s at t = s is given by the Lie derivative LXs applied to f , with d d Xs = dt φt,s |t =s : dt (f ◦ φt,s )|t =s = LXs f. The time derivative of the function f (φt,0 (a), t) = f (φ˜ t (a, 0)) at t = 0 is then the Lie derivative Lv˜ f of f , with v˜ ∈ T(a,0)M˜ defined according to (3.5). Thus (Lv˜ f )(a, 0) =

 d ∂f ∂f f (φ˜ t (a, 0))|t =0 =< v, )(a, 0). ˜ df (a, 0) > = ( vj . j + dt ∂x ∂t

A differential form. Let ωx,t = dφ t (x) = (j L,t )∗ dξ be the differential at x ∈ M of the map φ t = j L,t = j 0 ◦ φ0,t from Mt into R 3 × {0}; ωx,t is an isomorphism from Tx Mt onto R 3 , ∀x ∈ Mt , which is a parallelism [Mall, ch. II, 1.1] of Mt . We also have R R ∗ ◦ j t )∗ dξ = (j t )∗ (φ0,t ) dξ = (j t )∗ d x. ˜ ωx,t = (φ0,t

(3.11)

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3 Fluid Mechanics Modelling

3.1.3 Euler and Lagrange Metrics Let g0 be the usual Euclidean metric on R 3 , which gives a metric on each leaf Rt3 of R 3+1 , and we also denote by g0 this (degenerate) metric on R 3+1 . Let g0 be a Riemannian metric on M0 that is the pullback of g0 by j 0 : g0 = 0 (j )∗ g0 . Thus in a Cartesian coordinate system of M0 , g0 is identifiable with the unit matrix g0x (X, Y ) = g0 (j∗0 X, j∗0 Y ) = (j∗0 X).(j∗0 Y ),

∀X, Y ∈ Tx M0 .

(3.12)

We define two different metrics on the foliation M˜ π , the first one denoted by gE , called an Euler metric, the second one gL called a Lagrange metric, as in rigidmedium mechanics. (1) Euler metric. We define the (degenerate) Euler metric gE on M˜ π , which induces the metric gE,t on Mt , by the pullback of the (degenerate) metric g0 of R 3+1 by jE: gE = (j E )∗ g0 ,

gE,t = gE |Mt = (j t )∗ g0 .

(3.13)

 Thus with v ∈ Tx Mt , gE,t (v, v) = g0 (j∗t v, j∗t v) = g0 (v, ˜ v) ˜ = v. ˜ v˜ = (v j )2 . (2) Lagrange metric. We define a metric gL,t (or gt ) on Mt that is the pullback of g0 on M0 by φ0,t = φt with (3.11): ∗ (g0 ) = (j L,t )∗ (g0 ), gL,t = φ0,t

and thus gtx (X, Y ) = ωx,t (X).ωx,t (Y ), (3.14) for all X, Y ∈ Tx Mt . Thus φt is an isometry from (Mt , gt ) onto (M0 , g0 ): gtx (X, Y ) = (φt∗ g0 )x (X, Y ) = g0a ((φt )∗ X, (φt )∗ Y )

with a = φt (x).

The Riemannian metric family (gt ) on the leaves (Mt ) is such that ∗ 0 ∗ ∗ 0 ∗ s gt +s = φ0,t +s (g ) = φs,t +s (φ0,s (g )) = φs,t +s (g ).

(3.15)

Therefore, the map φs,t +s (thus φ˜ t ) is isometric from (Ms , gs ) onto (Mt +s , gt +s ). We have defined a metric denoted by gL on the foliation M˜ π of M˜ by gL |Mt = gt , which is degenerate.  i dx ⊗ dx i , the metric on Mt is such With the Euclidean initial metric g0 = t ∗ 0 0 that gxt = φ0,t (g )xt = gx (φ0,t )∗ ; thus gt =



dxti ⊗ dxti =

 ∂x i ∂x i t t dx k ⊗ dx l . ∂x k ∂x l

(3.16)

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3.1.4 Associated Connection Form and Covariant Derivative ∂x i

Let J = ( ∂xtl ) be the Jacobian matrix of the change of Euler–Lagrange coordinates. Let t J be the transpose of J . Then the metric is given by gL = t J.J. The connection form ω0 on the frame bundle L(M0 ) is flat,3 and if σ0 is the section of this bundle so that σ0 (x) = σ0 ((x j )) = (Xj ) = ( ∂x∂ j )j =1,2,3 , then the differential form σ0∗ (ω0 ) on M0 is null. Thus the connection form ωt on the linear frame bundle L(Mt ) is flat, and the corresponding differential form σt∗ (ωt ) on Mt is given by (see [Kob-Nom, ch. III.7, p. 142]) σt∗ (ωt ) = (J −1 dJ )k = j

 ∂x j ∂ 2 x i  j t dx l = i,l dx i , i k l ∂xt ∂x ∂x i,l

j

j

with the Christoffel symbols i,l , so that i,l =



they are the Christoffel symbols of the Riemannian 

(3.17)

i

2 i ∂x j ∂ xt i ∂x i ∂x k ∂x l . We can verify t metric gt ; see (3.16):

gtj,k i,l dxti = ((t J.J ).(J −1 dJ ))k,l = (t J dJ )k,l . j

Then the covariant derivative of a vector field Y = Xi = ∂ i is given by



that

(3.18)

Y k Xk in the direction of

∂xt

∇Xi Y =



Xi (Y k ) Xk +

k



Y l ωlk (Xi )Xk .

k,l

˜ or a We still have to find a connection form ω on the linear frame bundle L(M), ˜ that “induces” the connection form ωt on L(Mt ). Recall that subbundle of L(M), M˜ is not a Riemannian manifold. We assume that M˜ is a Euclidean manifold with the global variables (xti , τ ), so that the change with respect to the variables (x i , t) is given by the Jacobian matrix, if we assume that τ is a function of t only:  J v J˜ = , 0λ ∂x i

with the column vector v = (v i ), v i = ∂tt , and λ = ∂τ ∂t , and the previous Jacobian ∂ ∂ ∂ , ∂t∂ ) is given by matrix J , so that the change of frames u˜ t = ( ∂xt , ∂τ ), and u˜ = ( ∂x u˜ = (

∂ ∂ ∂ ∂ , ) = u˜ t J˜ = ( , )J˜. ∂x ∂t ∂xt ∂τ

L(M0 ) trivial, which is identified with the product M0 × Gl(3), the connection form ω0 is the pullback of the Maurer–Cartan form θMC on Gl(3) by the projection p defined by p(x, g) = g ∈ Gl(3), ω0 = p∗ θMC . This implies that the torsion and the curvature are null; see, for instance, [Kob-Nom, ch. II.9, p. 92].

3 With

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3 Fluid Mechanics Modelling

˜ is flat, and the Then the connection form ω on the linear frame bundle L(M) corresponding differential form ω˜ = σ ∗ ω on M˜ is given by4 

ω˜ = J˜−1 d J˜ =

J −1 −J −1 v 0 λ−1



dJ dv . 0 dλ

Thus ω˜ =

j

j

ω˜ k ω˜ 4 ω˜ k4 ω˜ 44



 =

J −1 dJ J −1 dv − J −1 vdλ . 0 λ−1 dλ

With the decomposition dJ = dx J + dt J, dv = dx v + dt v, dλ = dt λ, and the  expressions ω˜ βα = γα,β dx γ , with α, β, γ = 1, . . . , 4, we first obtain the relations with i, k = 1, 2, 3, 4 = 0, i,k

4 4 and 4,j = 0, i,4 = 0,

(3.19)

and then summing on the indices m, m = 1, 2, 3, we obtain j

i,k = j i,4

∂x j ∂ 2 xtm , ∂xtm ∂x i ∂x k

∂x j ∂v m = , ∂xtm ∂x i ∂x i

∂x j ∂ 2 xtm ∂λ 4 , 4,4 = λ−1 m k ∂xt ∂x ∂t ∂t  m j ∂v ∂λ ∂x − vm . = m ∂xt ∂t ∂t

j

4,k =

j 4,4

j

(3.20)

j

α with α, β, Since v i = ∂tt , we have 4,k = k,4 , and more generally γα,β = β,γ γ = 1, . . . , 4. We have to recall some definitions (see [Kob-Nom, Vol. 2, ch. VII.8, p. 54]) in ˜ Mt , M˜ π . the framework of the manifolds M,

Definition 1 Adapted frames. A linear frame u˜ of M˜ at a point x˜ = (x, t) ∈ Mt is said to be adapted if it is of the form u˜ = (u, X4 ), with u a linear frame of Mt , that is, u = (X1 , X2 , X3 ), with Xi ∈ Tx Mt , i = 1, 2, 3. The set of these adapted linear ˜ Mt ). frames is denoted by L(M, Note that the set of column vectors of the Jacobian matrix J˜ is an adapted linear ˜ Mt . frame for M, Let GL(3 + 1, 3) be the subgroup of GL(4) with elements of the form  a˜ =

4 We ∂ ∂x 4

aw 0μ

,

˜ such that σ0 (x) = (Xj ) = ( ∂ j )j =1,2,3,4 , with assume that if σ0 is the section of L(M) ∂x ∂ ∗ = , then the differential form σ ω on M˜ is null. ∂t

0

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343

with a ∈ GL(3), μ ∈ R, μ = 0, and w ∈ R 3 a column vector. Thus a(R ˜ 3 × {0}) = 3 3 R × {0}, and the space R is invariant under GL(3 + 1, 3). Note that the affine group A(3) is a subgroup of GL(3 + 1, 3), with μ = 1. The group GL(3 + 1, 3) operates freely on the right in the space of adapted ˜ Mt ) is a principal fibre bundle over Mt with group GL(3+ frames. Moreover, L(M, 1, 3) ( [Kob-Nom, Vol. 2, ch. VII.8, p. 54]). & ˜ M˜ π ) = ˜ We define the set sum L(M, t ∈R L(M, Mt ), which is a principal fibre 5 ˜ ˜ bundle over both M and Mπ with group GL(3 + 1, 3), and which has the structure of a foliated manifold. We have the following diagram, with the map h such that h(u) ˜ = u, which is a ˜ Mt ) onto L(Mt ): homomorphism6 of the principal bundles L(M, j j h ˜ Mt ) −→ ˜ M˜ π ) −→ ˜ M˜ π ) L(Mt ) ←− L(M, L(M, L(M, ↓ ↓ ↓ ↓ Mt −→ M˜ π −→ M˜ Mt ←−

˜ Mt )) in many ways. We identify the We can define a connection form ω (on L(M, ˜ ˜ principal bundle L(M, Mπ ) with the trivial bundle M˜ ×GL(3+1, 3), since for every ˜ M˜ π ), we have u˜ = u˜ 0 a˜ with u˜ 0 = I , the identity, corresponding to the u˜ ∈ Lx˜ (M, frame (Xi ) with Xi = ∂x∂ i , i = 1, 2, 3, X4 = ∂t∂ in a chart with x˜ = (x 1 , x 2 , x 3 , t). Thus we can identify (x, ˜ u) ˜ with (x, ˜ a). ˜ Let p be this identification; the connection ˜ M˜ π ) is the pullback by p of the Maurer–Cartan form of GL(3 + form ω on L(M, 1, 3): ω = p∗ θMC = u˜ −1 d u, ˜ which we also denote by ω = J˜−1 d J˜. Then we obtain the differential form ω˜ on M˜ by the following section σ˜ from the velocity field vt and its flow ψt,0 , which is defined by σ˜ (x) ˜ = J˜x˜ = (Jx˜ , X4 ),

with Jx˜ = (ψt,0 )∗ (

 ∂ ∂ ∂ ) , X = vj j + . j =1,2,3 4 j ∂x ∂x ∂t

Thus we can identify J with the Jacobian matrix as previously. Notice that X4 may d be denoted by v˜ or dt . Then we obtain the differential form ω˜ on M˜ by ω˜ = σ˜ ∗ ω = σ˜ ∗ p∗ θMC = σ˜ −1 d σ˜ .

˜ over M; ˜ see [Kob-Nom, Vol. 2, ch. VII.8, p. 53]. may be viewed as a reduced bundle of L(M) recall that we have h(u˜ a) ˜ = ua = h(u)h ˜ 0 (a), ˜ with h0 the projection from gl(3 + 1, 3) onto gl(3).

5 It

6 We

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3 Fluid Mechanics Modelling

We recall the following properties, from [Kob-Nom, Vol. 2, VII, Prop. 8.2; 8.6]: ˜ M˜ π ) (i) The relation between the connection forms ωMt on L(Mt ) and ω on L(M, is h∗ ωMt = h0 ω, where h0 is the projection from gl(3+1, 3) onto gl(3), where gl(3+1, 3) (respectively gl(3)) is the Lie algebra of the Lie group GL(3+1, 3) (respectively GL(3)). ˜ and is totally geodesic. (ii) Mt (and M˜ π ) is an auto-parallel submanifold of M, For these notions and properties, we refer to [Kob-Nom, Vol. 2, ch. VII.8, pp. 54–58]. ˜ Let X, Y be two vector fields on (iii) Let ∇ be the covariant differentiation on M. Mt . Then ∇ X Y is tangent to Mt at every point x˜ = (x, t) of Mt . Furthermore, ∇XY  = ∇X Y if ∇ is the covariant differential on Mt or on M˜ π . Indeed, let X = j =1,2,3 wj Xj . Then the covariant differential of X is 

∇X =

Xk (dwk +



ω˜ jk wk

k with ω˜ jk = ωjk + 4j dt.

j

k=1,2,3

Then < ω˜ jk , Y > = < ωjk , Y >, and we have 

∇Y X =

Xk (< dwk , Y > +

 j

k=1,2,3

∇∂X= ∂t



< ωjk , Y > wk ) = ∇Y X,

Xk (

∂wk

k=1,2,3

∂t

+



(3.21) k 4j wj ),

j

k with < ωjk , ∂t∂ > = 4j . Then with X = v,

(∇ v v)k =

∂v k j  k j j v + i,j v v , ∂x j i,j

(∇ ∂ v) = k

∂t

 m,j =1,2,3

∂x k ∂v m j ∂v k , v + ∂xtm ∂x j ∂t

(3.22)

giving ∇ v˜ v with v˜ = (v, 1), (∇ v˜ v)k = (

∂v k ∂v k k k + v j j ) + v j [v i i,j + 4,j ], ∂t ∂x

(3.23)

with the Christoffel symbols given by (3.20). ˜ M˜ π ) and ξ˜ = (ξ, ξ 4 ) ∈ R 4 . We have that Remark 1 Let u˜ = (u, X4 ) ∈ Lx˜ (M, 4 ˜ is not in Tx˜ (M, ˜ M˜ π ), whereas u.ξ ∈ Tx˜ (M, ˜ M˜ π ). ¶ u. ˜ ξ˜ = u.ξ + X4 ξ ∈ Tx˜ (M)

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345

3.1.5 Metric and Strain for a Continuum Medium The terminology “continuum medium” refers to a medium between a rigid medium and a fluid, and we generally want to know the strain in the medium. The geometric description of a continuum medium naturally includes a distance notion and a Riemannian structure linked with the medium itself in its motion. This allows us to define the mechanical quantities of the medium, and notably its deformations according to frame deformations. Here we have to use metrics and frames on the medium itself, and not only on its tangent space; that is possible by the modelling of the continuum medium as an open subset of a Euclidean space if we identify the space of vectors in T M with the space V of (translation) vectors of E3 . As in the case of a rigid medium, we have to use frames, but the frames must be more general, by deformation of the Euclidean frames. We first begin with a more general setting. Let M be a 3-dimensional manifold that is embedded into the Euclidean space E3 . Let f be an embedding from M into E3 ; to f we can associate a section x ∈ M → ux ∈ Lx M of the linear frame bundle L(M)7 with u−1 x = ωx the isomorphism defined by Xx ∈ Tx M → f  (x)(Xx ) = < df (x), Xx > = f∗ Xx = < ωx , Xx > ∈ R 3 . L on To the embedding j L,t = j 0 ◦ φ0,t we associate the differential form x → ωx,t M whose inverse is a section x → ux,t of L(M): L Xx ∈ Tx M →< ωx,t , Xx > = φt (x)(Xx ) = < dφt (x), Xx > .

Elimination of the rigid-body motions. The study of a continuum medium is often reduced to studying deformations only, so that we have to eliminate the global set motions, which are the displacements. We can do that by passing to a quotient space (see [Mall, ch. III, 8.1, 8.4]). Two maps f, f˜ ∈ C 1 (M, E3 ) are said to be equivalent if there exists a displacement u in E3 such that f˜ = u ◦ f . We recall the following fundamental property (see [Mall, ch. III, 8.1]): Let f and f˜ be two embeddings of the manifold M in E3 . Let f ∗ (g0 ) and f˜∗ (g0 ) be the pullback Riemannian metrics of the metric g0 on M by f and f˜. Then f and f˜ are equivalent if these Riemannian metrics are identical: f ∗ (g0 ) = f˜∗ (g0 ) = g. Then the pullback metric g is independent of the rigid displacement of the medium. We eliminate the rigid displacements with the quotient space A(3)/D(3) of affine transformations by the displacement group,8 which is also the quotient space 7 The

space Lx M is identified with the space of linear bijective mappings from R 3 onto Tx M (denoted by Isom (R 3 , Tx M)), by associatingto each frame u = (e1 , e2 , e3 ) of Tx M the λi ei ∈ Tx M. isomorphism λ = (λ1 , λ2 , λ3 ) ∈ R n → u(λ) = 8 Here the displacement group is R 3 × O(3) instead of R 3 × SO(3) in Chapter 2.

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3 Fluid Mechanics Modelling

GL(3)/O(3), thus the set of left classes for O(3): a¯ = {au, u ∈ O(3)}, for every a ∈ GL(3). This quotient space is identifiable with the set, denoted by GL+ (3), of symmetric positive definite matrices. Indeed, every element g of GL(3) may be written in a unique way in the form (called polar decomposition) g = h.R, with R ∈ O(3), h ∈ GL+ (3). The element h is the symmetric positive definite matrix h = (g.T g)1/2 . Therefore, we can identify the quotient space GL(3)/O(3) with the space GL+ (3), which is the positive cone of a 6-dimensional Euclidean space. Notice that we can also consider right classes, with the polar decomposition g = ˜ h, ˜ h˜ = (T g.g)1/2 ; the two decompositions are changed by passing to the inverse R. or to the transpose. Then we define the strain tensor from the metric tensor.

Green–Lagrange Strain Tensor Field The “strain tensor,” called the Lagrange tensor (or Green–Lagrange tensor), on = gtx , gE,t = ((j t )∗ g0 )x (viewed as Mt ⊂ E3 is defined from the metrics gL,t x x quadratic forms on Tx Mt ) by xt =

1 L,t (g − gE,t x ) 2 x

x ∈ Mt .

(3.24)

This is a symmetric tensor field x ∈ Mt → xt ∈ S 2 Tx∗ Mt , independent of the rigid displacement motions. Then we define a tensor field ˆ t , of type (1, 1), with the unit matrix I , by ˆ t = (GE,t )−1  t =

1 ((GE,t )−1 GL,t − I ). 2

We can also define an isomorphism Uxt in Tx Mt linked to the trace of the quadratic E,t 0 form gL,t x with respect to the quadratic form gx or gx ( [Bour.int, ch. IX Annexe]), by 0 t gL,t x (X, Y ) = gx (Ux (X), Y ),

∀X, Y ∈ Tx Mt ,

with 0 tr Uxt = tr (gL,t x /g ) =

 i=1,2,3

gL,t x (ei ),

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for every frame u = (ei ) in Tx Mt such that g0 (ei , ej ) = δi,j . Then we have Uxt = T (φ ) (φ ) , and 0,t ∗ 0,t ∗ 0 tr (gL,t x /g ) =



| (φ0,t )∗ ei |2 =

 (ei , T (φ0,t )∗ (φ0,t )∗ ei ) = tr (T (φ0,t )∗ (φ0,t )∗ )

with xt (X, Y ) =

1 L,t 1 1 (gx (X, Y ) − g0 (X, Y )) = g0 ((Uxt − I )X, Y ) = (Uxt − I )X.Y, 2 2 2

so that we can identify the strain tensor field  t with 12 (U t − I ). To the Lagrange metric gL,t , defined in (3.14), we can associate a frame utx = (ωx,t )−1 in Lx Mt , so that with X = utx ξ, Y = utx ξ  ∈ Tx Mt , ξ, ξ  ∈ R 3 , we have 0 t −1 t −1  gL,t x (X, Y ) = g ((ux ) (X), (ux ) (Y )) = ξ.ξ . t

Conversely, from the frame utx (up to rotations), we define the metric gL,t = gu . We have utxt .ξ = (φt,0 )∗ (u0x .ξ ), ∀ξ ∈ R 3 ; thus since ut : x → utx is a section of the fibre bundle L(Mt ) of the linear frames, we can write ut = (φt,0 )∗ (u0 ) and ut ◦ φt,0 = L(φt,0 )u0 . The previous definition of the Green–Lagrange tensor uses the evolution family φ0,t . In fact it seems better to define this tensor from the tensor field ˜v (see below, which may be defined from the velocity field only) by ˆ

L,t

= (G

E,t −1 L,t

)



1 = ((GE,t )−1 GL,t − I ) = 2



t



t

˜v ds =

0

0

1 Lv˜ gE ds. 2

3.1.6 Rate of Strain Tensor Field for a Fluid Let v (respectively v) ˜ denote the velocity field that generates the evolution φt,s (respectively φ˜ t ), which is assumed to be of class C 1 . We first define the tensor field called the rate of strain tensor field ˜v on M˜ π from the Green–Lagrange strain ˜ tensor field  by (the limit being taken at each point x˜ = (x, t) ∈ M) 1 ˜v = − lim (φ˜ t∗  − ) = −Lv  L , and t →0 t 1 ˜vs = ˜v |Ms = − lim (φt∗+s,s  t +s −  s ). t →0 t In the following, we simply denote ˜v by v . Then we directly define v from the metric gL by

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3 Fluid Mechanics Modelling

1 1 1 L lim (gL − φ˜ t∗ gL x˜ ) = (Lv˜ g )x˜ , 2 t →0 t x˜ 2 1 1 1 vs = lim (gL,s − φt∗+s,s gL,s ) = Lvs (gL,s ); 2 t →0 t 2

(v˜ )x˜ =

(3.25)

vs is (up to the factor 1/2) the Lie derivative of the metric tensor gL,s with respect to the vector field v. Then vt : x ∈ M →  t (x) ∈ T20 (Mt ) is a symmetric tensor field that is independent of the rigid displacement motions. An explicit expression of the tensor field v in a chart with coordinates x j . With the sum convention on repeated indices in this section, an explicit expression of the tensor field vs with a vector field vs , simply denoted by v, is the following: 2vs = Lv gL,s = v(gij )dx i ⊗ dx j + gij (Lv dx i ) ⊗ dx j + gij dx i ⊗ Lv dx j . i

∂v k With v = v i ∂i , we have v(gij ) = v k ∂k gij , Lv dx i = dv i = ∂x k dx . With the 1 initial metric g0 =  dx i ⊗ dx i , we have that  s = vs = 2 Lv (gL,s ) is the following symmetric tensor field of type (0, 2):

s =

1 j (dv ⊗ dx j + dx j ⊗ dv j ). 2

Thus we can write9 s either  s = i,j dx i ⊗ dx j , j

s = ( s )i = i,j

j

or  s = ( s )i dx i ⊗ dx j , (3.26)

1 ∂v i ∂v j ( j + ). 2 ∂x ∂x i

More generally, if g0 is a Riemannian metric on M0 , then  s is given by s =

1 1 j Lv g = (i )s dx i ⊗ dx j = (v k ∂k gij + gkj ∂i v k + gik ∂j v k ) dx i ⊗ dx j . 2 2

Thus with the usual notation vj = gj k v k , 1 1 (∂i vj + ∂j vi ) + v k (∂k gij 2 2 1 = (∂i vj + ∂j vi ) − v k k,ij = 2

ijs =

− ∂i gkj − ∂j gik ) (3.27)

1 (∂i vj + ∂j vi ) − vl ijl ), 2

with the Christoffel symbols l;j k = gli ji k = 12 (∂j glk + ∂k gj l − ∂l gik ). j

the notation i,j we refer to the components of the tensor  ∈ T20 (M), whereas i refers to a tensor in T11 (M), giving, by taking the product with dx i ⊗ dx j and contraction, the tensor .

9 With

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349

Remark 2 The chosen (deterministic) model leads to (locally) Euclidean structures of continuous media: we have assumed that the medium at a given time, taken as initial time, is equipped with a Euclidean structure; then at all times t, by pullback of g0 by the pseudogroup of evolution, the medium is equipped with a Euclidean structure (see [Mall, ch. III, 8, thm. 8.3.2]). We can also assume a Riemannian structure. ¶

3.1.7 Strain Rate with Covariant Derivative In this section we consider a fixed domain M = Mt , with a Riemannian metric g on M, and its Riemannian connection, giving a covariant derivative such that ∇g = 0. Let X be a vector field on M that is of class C 2 . We will write, in an essential way, the properties of a derivation operator AX that is identified with a tensor field of type (1, 1) and is a central tool in the study of transformations of a manifold (see [Kob-Nom, ch. VI]): AX = LX − ∇X ;

(3.28)

LX is the Lie derivative with respect to the vector field X, and ∇X the covariant derivative with respect to X. Let us first give some easy to prove properties (for a time-independent vector field X), using some classical results of differential geometry (see [Kob-Nom, ch. III.3.7, ch. VI.2]): (i) Since the torsion of the Levi-Civita connection is null, we have AX Y = −∇Y X.

(3.29)

(ii) From (3.29) we obtain an expression for the tensor AX in a coordinate system (x j ). Using the Christoffel symbols jik , we have j

∇∂i (dx j ) = − ik (dx k ), and ∇∂i (∂j ) = ijk ∂k , and with X = Xi ∂i , Y = Y i ∂i , we have ∇X Y = (∇X Y )i ∂i ,

(∇X Y )i = Y;ji Xj

and Y;ji =

∂Y i + jik Y k . ∂x j

Finally, we have i i (AX Y )i = −X;j Y j , thus (AX )ij = −X;j = −(∂j Xi + jik Xk ).

(3.30)

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3 Fluid Mechanics Modelling

(iii) We can also calculate the action of AX on the differential forms, especially on the 1-forms dx i , since LX (dx i ) = dXi , AX (dx i ) = LX (dx i )−∇X (dx i ) = dXi −∇X (dx i ) = ∂j Xi dx j −Xj ∇∂j (dx i ). Thus AX (dx i ) = (Aˆ X )ij dx j = (∂j Xi + Xk kji ) dx j . def

¶

For other interesting properties of AX in differential geometry, see [Kob-Nom, ch. VI, App6]. Let us give some properties. A vector field X is said to be an infinitesimal isometry if the local 1-parameter group of local transformations generated by X in a neighborhood of each point of M preserves the metric g. The vector field X is an infinitesimal isometry if and only if the tensor AX is skew-symmetric for g, which is equivalent to (see (3.34) and (3.32)) X = 0. In the present case in which the space is flat (that is, the torsion and curvature tensors are null), the set of infinitesimal isometries is a 6-dimensional Lie algebra. This set corresponds to the “rigid” displacements (translations and rotations). Thus the strain tensor is null if and only if the velocity field is an infinitesimal isometry. We can obtain this result as well in a Cartesian coordinate system from the identity (see [D-L19, ch. VII.2.7.3]) ∂ 2 vi /∂xj ∂xk = ∂ik /∂xj + ∂ij /∂xk − ∂j k /∂xi . Thus  = 0 implies that v = a + MS x + MA x, a first-order polynomial at most, with MS (respectively MA ) a symmetric (respectively skew-symmetric) matrix. But we have (v) = MS = 0, and thus v is given by v = a + b × x. ¶ Notice that the rate strain tensor is such that (for X = v) 2v = Lv g = Av g.

(3.31)

Let A∗v be the adjoint of Av (with respect to the Riemannian metric g): g(Av Y, Z) = g(Y, A∗v Z).

(3.32)

The adjoint is linked to the transpose (by duality) by GA∗v = t Av G. Hence we have def

(GAv )a = GAv − t (GAv ) = GAv − t Av G = G(Av − A∗v ). Then we have the following result.

(3.33)

3.1 Kinetic Modelling

351

Proposition 1 The tensor field ˆv = G−1 v of type (1, 1) is such that 1 ˆv = − (Av + A∗v ). 2

(3.34)

Proof We use the derivative formula ( [Kob-Nom, ch. VI.3]) Av (g(Y, Z)) = (Av g)(Y, Z) + g(Av Y, Z) + g(Y, Av Z).

(3.35)

The left-hand side is null, since Av maps every function to zero, and thus 0 = 2v (Y, Z)+ < (Av + A∗v )Y, Z >, which is (3.34).

¶

Proposition 2 If we identify (GAv )a = G(Av − A∗v ) with a 2-form,10 we have G(Av − A∗v ) = − dωv ,

(3.36)

with ωv = Gv. Corollary 1 The tensor fields Av and A∗v of type (1, 1) are such that 1 GAv = −v − dωv , 2

1 GA∗v = −v + dωv , 2

(3.37)

with dωv being taken as a map from T M into T ∗ M, or 1 Av = −ˆv − G−1 (dωv ), 2

1 A∗v = −ˆv + G−1 (dωv ). 2

(3.38)

Proof of the corollary It is sufficient to add and remove the formulas (3.34) and (3.36), or (3.39) of Propositions 1 and 2. ¶ Proof of Proposition 2 In a coordinate system, the differential form ωv is given by ωv = Gv = G(v i ∂i ) = v i G(∂i ) = v i gij dx j = vj dx j . Using (3.30), we have (with X = v) AX Y = (AX Y )i ∂i = (AX )i;j Y j ∂i = −(∂j Xi + ji k Xk )Y j ∂i .

10 This

is allowed by the fact that (GAv )a (f Y ) = f (GAv )a (Y ) for every function f .

(3.39)

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3 Fluid Mechanics Modelling

Thus GAX Y = G((AX Y )i ∂i ) = (AX Y )i G(∂i ) = (AX Y )i gil dx l = gil (AX )i;j Y j dx l = (GAX )l,j Y j dx l = −gil (∂j Xi + ji k Xk )Y j dx l . Therefore, (GAX )l,j = gli (AX )ij = −(∂j Xl + l;j k Xk − Xk ∂j glk ), whence (GAX )al,j = −(∂j Xl − ∂l Xj ) − ( l;j k − j ;lk − ∂j glk + ∂l gj k )Xk .

(3.40)

Moreover, l;j k =

1 (−∂l gj k + ∂j glk + ∂k glj ), 2

j ;lk =

1 (−∂j glk + ∂l gj k + ∂k glj ). 2

Thus the last term of the right-hand side of (3.40) is null, and we have (GAX )al,j = −(∂j Xl − ∂l Xj ), and therefore G(AX − A∗X )(∂l ) = (GAX )a (∂l ) = (GAX )aj,l dx j = −(∂l Xk − ∂k Xl )dx k . (3.41) The inner product of dωX = dXj ∧ dx j and ∂l is (i(∂l )dωX )(Y ) = (dXj ∧ dx j )(∂l , Y ) = [∂l Xj − ∂j Xl ] Y j . Thus (i(∂l )dωX ) = [∂l Xj − ∂j Xl ] dx j , hence (3.39), (3.36).

¶

Proposition 3 The covariant and the Lie derivative of a vector field v in a Riemannian manifold are such that with v 2 = g(v, v), ∇v v = −grad

v2 v2 + G−1 Lv Gv = −grad + 2ˆv .v, 2 2

or

v2 1 (Lv G − ∇v G)v = d g(v, v) = d , 2 2 with ˆv = G−1 v , ˆv .v = 12 G−1 Lv G.v, since Lv (G.v) = (Lv G).v.

(3.42)

3.1 Kinetic Modelling

353

Proof Let w be a vector field, and we define (v, w) = g(v, w) = < Gv, w >. We apply Lv to (v, w). Note that we have a priori, Lv (G.w) = (Lv G).w if w = v. Thus using the properties of the Lie derivative, we obtain Lv (v, w) =Lv < Gv, w > = < Lv (Gv), w > + < Gv, Lv w > = (G−1 Lv Gv, w) + (v, Lv w) = (G−1 Lv Gv, w) − (v, Lw v), hence Lv (v, w) + (Lw v, v) = (G−1 Lv Gv, w).

(3.43)

From the basic property Lv (v, w) = (∇v v, w) + (v, ∇v w),

(3.44)

we have, using the fact that the torsion of a Riemannian metric is null, Lv (v, w) + (Lw v, v) = (∇v v, w) + (v, ∇v w) + (v, [w, v]), = (∇v v, w) + (v, ∇w v).

(3.45)

Furthermore, (v, ∇w v) = Lw (

v2 v2 ) = < d( ), w > . 2 2

(3.46)

Thus (3.43), (3.45), and (3.46) give, for every w, (∇v v, w)+ < d( and hence (3.42).

v2 ), w > = (G−1 Lv Gv, w), 2

¶

Remark 3 In the Euclidean case, with Cartesian coordinates, the covariant derivative ∇v v of v in the direction of v is identical to (v.∇)v, and it can be written, using the vector product ×, as ∇v v = −v × curl v + grad

v2 . 2

(3.47)

Proof This expression is obtained by its components: (v × curl v)1 = v2 (∂1 v2 − ∂2 v1 ) − v3 (∂3 v1 − ∂1 v3 ) = v.∂1 v − (v.grad )v1 .

¶

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3 Fluid Mechanics Modelling

We deduce from (3.47), (3.42) the formula v × curl v = grad v 2 − 2ˆv .v.

(3.48)

From the formulas (3.47), (3.42), we can give the usual expressions for the time ˜ which is also denoted derivative of a vector field v (with v(x(t), t) ∈ T(x(t ),t )(M)), by ∇t,v v: ∂v ∂v v2 dv = + ∇v v = − v × curlv + grad . dt ∂t ∂t 2

¶

3.1.8 Trace Calculus In order to obtain the expression for the energy, we will calculate the traces: tr (ˆv ◦ ˆv ) and tr (ˆv ◦ ˆw ). First we use the following trace property (see [p. 282, App. 6]Koba): tr ˆv = −tr Av = div v,

with Lv v g = (div v) v g

(3.49)

for any vector field v. If v has compact support  in R 3 or E3 , then using Green’s theorem, if the measure v g is Lebesgue measure (see, for instance, [D-L19, ch. VII.2.7.3]), we prove 

   1 2 2 (div v) + |curl v| v g . tr (ˆv ◦ ˆv ) v = 2  g



(3.50)

This corresponds to a formula using the divergence of a vector field Y v to specify tr(ˆv ◦ ˆv ) = (div v)2 +

1 |curl v|2 + div Y v . 2

In order to make the last term explicit, we will use the following lemma. Lemma 1 Let X and Y be vector fields on M ⊂ R 3 . We have 1 1 tr ((AX − A∗X )(AX − A∗X )) = − |curl X|2 4 2 1 1 = tr ((AX − A∗X )(AY − A∗Y )) = − (curl X, curl Y ). 4 2 def

J0 = J0X = J0X,Y

(3.51)

Proof Let gˆ be the determinant of the matrix G. Recall that the curl of a vector field X is expressed in a chart with coordinates x m by (see [Ces, App. A.6])

3.1 Kinetic Modelling

curl X =

355

 (curl X)m ∂m ,

with (curl X)m =



gˆ − 2  mj k Xj,k , 1

with  mj k = 0 if two exponents are identical,  m,j,k = 1 if the permutation of m, j, k with respect to 1, 2, 3 is even,  mj k = −1 if it is odd. Then Xj,k = ∂j Xk − ∂k Xj with Xj = gj k Xk , and we have 4J0 = tr ((AX − A∗X )(AX − A∗X )) =



< dx j , (AX − A∗X )(AX − A∗X )∂j >,

and using (3.41) twice, with G−1 dx k = gkj ∂j , we have 4J0 =



Xj k g km Xml g lj =



Xj k Xml g km g j l .

Note that j = k, m = l from the skew-symmetry of Xj k and Xml , and therefore 4J0 =

1 Xj k Xml [g km g j l − g j m g kl ]. 2

The [g km g j l −g j m g kl ] are the determinants of the minors of the matrix G−1 . If N is the matrix of the cofactors of G−1 , i.e., N = (−1)i+j det N j,i with N j,i the minor of G−1 corresponding to the term with superscripts (j, i), and if t N is the transpose matrix of N, then we have the relation11 G = gˆ t N. We deduce −2J0 =

  1 1 (curl X)m (curl X)n gmn (ˆg) 2 gˆ 2 gˆ −1 = (curl X)m (curl X)n gmn .

Thus (3.51). The relation between X and Y is a consequence of the polarization property. ¶ Now we can calculate the desired traces for the vector fields v and w: Proposition 4 If the initial metric g0 of the fluid is Euclidean, then the bilinear symmetric form a(v, w) = tr (ˆv ◦ ˆv ),

(3.52)

with the vector fields v, w, is related to the bilinear form 1 (curl v).(curl w) + div v.div w 2

(3.53)

a(v, w) = a0 (v, w) + div (∇w v − (div v) w),

(3.54)

a0 (v, w) = by

11 See,

for instance, [Bour.alg0, ch. AIII]

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3 Fluid Mechanics Modelling

and thus a(v, v) = tr (ˆv .ˆv ) =

1 |curl v|2 + (div v)2 + div Yv = a0 (v, v) + div Yv , 2

Yv = ∇v v − (div v) v. (3.55) Proof From (3.34), we have 1 tr ((Av + A∗v )(Aw + A∗w )) 4  1 tr (Av Aw + A∗v A∗w ) + tr (Av A∗w + A∗v Aw ) . = 4

tr (ˆv .ˆw ) =

But tr (Av A∗w ) = tr (A∗v Aw )

and tr (Av Aw ) = tr (A∗v A∗w ),

whence tr (ˆv .ˆw ) =

 1 tr (Av Aw ) + tr (Av A∗w ) . 2

(3.56)

Likewise, the calculus of J0v,w gives J0v,w =

 1 tr (Av Aw ) − tr (Av A∗w ) . 2

(3.57)

Adding (3.56) and (3.57), we obtain tr (ˆv .ˆw ) + J0v,w = tr (Av Aw ).

(3.58)

The Yano formula is (see [Dieud3, ch. XX.10]) tr (Av Aw ) = div v.div w− < K  , v ⊗ w > −div (Av .w + (div v)w),

(3.59)

where K  is the Ricci tensor on M. Using (3.58) and (3.51), we obtain tr (ˆv .ˆw ) =

1 (curl v.curl w) + div v.div w− < K  , v ⊗ w > +div Y v 2

(3.60)

with Y = −(Av .w + (div v)w) = ∇w v − (div v) w. v

We assume that at time t0 , the fluid is represented by the manifold M equipped with a Euclidean metric g0 . Then at all times t, the manifold is locally Euclidean (the map φt,t0 is an embedding), see [Mall, ch. 8, Thm. 8.5.3, Thm. 8.7.3], and the Ricci tensor K  is null, whence (3.52). This result is easily obtained in a Cartesian

3.1 Kinetic Modelling

357

coordinate system (which is implicitly admitted by K  = 0). The expression (3.55) is obtained with (3.47). ¶ Differential operators associated with the bilinear forms a(u, w), a0 (u, w). (i) From the bilinear form a0 (v, w), see (3.53), we can define a linear differential operator P0 on the sections of T M and a linear operator B0 on the sections of T , so that by integration on M, we have (formally) 





a0 (v, w) dx = M

(v, P0 w) dx + M

< v, B0 w > d .

(3.61)



We have div v. div w = −v. grad div w + div (v div w), curl v. curl w = v. curl curl w + div (v × curl w).

(3.62)

Then 1 1 a0 (v, w) = v.( curl curl w − grad div w) + div (( v × curl w) + v div w). 2 2 Thus the operator P0 is defined on the sections of T M by P0 w =

1 curl curl w − grad div w, 2

(3.63)

and the operator B0 is defined on the sections of T , with the (external) normal n to , by < v, B0 w > = n.v div w + 12 n.(v × curl w),, and thus 1 B0 w = − n × curl w + n div w, 2

(3.64)

which we can write for v = w in matrix form (with the orthogonal projections vn on n and v on the tangent plane of the surface ) as  B0

vn v



 =

div v − 12 n × curl v

;

(3.65)

the decomposition of div v is given below, see (3.80), and the decomposition of the last term is given by (see [Ces, App. 7, p. 350]) 1 ∂ − n × curl v = −grad vn + v + 2Rm v , 2 ∂n where Rm is the mean curvature of .

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3 Fluid Mechanics Modelling

Remark 4 Intrinsic expression. The curl and div of v are expressed with the Hodge ∗ transformation, the differential dωv of ωv , and the codifferential δ (see [Ces, App. A.1]) by curl v = G−1 ∗ dωv ,

div v = −δωv .

(3.66)

Then we have (curl v, curl w) = (G−1 ∗ dωv , G−1 ∗ dωw ) = (∗dωv , ∗dωw ) = (dωv , dωw ) (div v, div w) = (δωv , δωw ). Thus by integration on the domain M with the Lebesgue measure dx, we have 

 1 1 [ (curl v, curl w) + (div v, div w)]dx = [ (dωv , dωw ) + (δωv , δωw )]dx M 2 M 2   1 1 = [(ωv , ( δd + dδ)ωw ) − [ (t ωv , n dωw ) + (n ωv , t δωw )], 2 M 2

if t ωv and n dω correspond to the (tangential and normal) values of ω and dω, on the boundary of M. ¶ (ii) From the bilinear form a(v, w) we can define a linear differential operator P on the sections of T M and a linear operator B on the sections of T , so that by integration on M, we have (formally) 





a(v, w) dx = M

(v, Pw) dx +

< v, Bw > d .

M

(3.67)

M

These operators are defined by (Pw)i = −

∂ (w )ij , ∂x j

< v, Bw > = n.(w v).

(3.68)

j

Note that we give an intrinsic sense to ∂j (v )k dx k by j

j

< ∂j (v )k dx k , w > = ∂j (v )k wk = div (v .w) − tr (v .w ).

(3.69)

(iii) We can prove that the operators P and P0 are identical as differential operators from the Sobolev space H 1 (M)3 into H −1 (M)3 , where H 1 (M)3 (respectively H01 (M)3 ) is the space of the velocities with square integrable derivatives (and respectively null at the boundary), and H −1 (M)3 is the dual space of H01 (M)3 . Let  and 0 be the differential operators respectively from H 1 (M)3 and H01 (M)3 into L2 (M, S 2 (M)) so that v = v, respectively v = 0 v. Then the dual operator ∗0 of 0 is an operator from L2 (M, S 2 (M)) into H −1 (M)3 , and we can write the differential operator P as P = ∗0  (which is different from the operator ∗  from H 1 (M)3 into its dual space (H 1 (M)3 )∗ ). Then we have

3.1 Kinetic Modelling

359

< v, Pw > = < v, P0 w >,

∀w ∈ H 1 (M)3 , v ∈ H01 (M)3 ,

since < v, Bw >, < v, B0 w > and n.(∇w v − (div v) w) are null. But if we consider the operators P and P0 to be unbounded operators in L2 (M, S 2 (M)) with their “natural” boundary conditions Bw = 0, respectively B0 w = 0, then these operators are different, since their domains are different, D(P0 ) = {v ∈ H 1 (M)3 , P0 v ∈ L2 (M)3 , B0 v = 0}, D(P) = {v ∈ H 1 (M)3 , Pv ∈ L2 (M)3 , Bv = 0},

(3.70)

and they are positive self-adjoint operators. Boundary values. Let us make explicit the boundary term n.v w of (3.69) on the boundary of M, with n the outside unit normal to . We agree that the metric on is the induced metric from M. We first note that when the vector fields v and w are tangent at the boundary, we have, thanks to the Gauss formula (see, for instance, [Kob-Nom, vol. 2, ch. VII.2]), g(n, ∇v w) = n.∇v w = h(v, w), with h(v, w) the second fundamental form. At each x ∈ , hx is a symmetric bilinear function on Tx × Tx . Furthermore, to h(v, w) is associated a linear mapping Aˆ n in the space of the vector fields S( ) on defined by g(Aˆ n v, w) = h(v, w).

(3.71)

We recall that the eigenvalues ρi of h are the principal curvatures of at x ∈ , and the mean curvature of is Rm = 12 (ρ1 + ρ2 ). In an orthogonal coordinate system (e1 , e2 , e3 ) with e3 = n, where the metric tensor is diagonal, with eigenvalues g1 , g2 , 1, the second fundamental form is given by 1 h=− 2



0 g1−1 ∂n g1 0 g2−1 ∂n g2

.

(3.72)

Using the relations (3.34) and12 An v = −∇v n, since the torsion of the Riemannian metric is null, we have 2n.v w = 2v (n, w) = −[g(n, Av w) + g(Av n, w)] = g(n, ∇w v) − g(Av n, w).

12 See

[Kob-Nom, ch. VI.2, Prop. 2.5].

(3.73)

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3 Fluid Mechanics Modelling

(i) With the  decomposition of v into tangential and normal parts v = v + vn n, thus v = vj ∂j + vn n, in a chart of with coordinates xj , j=1,2, we have Av n = [v, n] − ∇v n = −

 ∂vj ∂n

∂j −

∂vn n − ∇v n. ∂n

Then ∇v n = vn ∇n n + ∇v n, and we have g(n, ∇n n) = 0, since g(n, n) = 1. Furthermore, since g(w , n) = 0, we have for every vector field w to that Ln (g(w , n)) = g(∇n w , n) + g(w , ∇n n) = 0; then ∇n n = 0, since g(∇n n, w ) = −g(∇n w , n) = −g(∇w n, n) − g([n, w ], n) = 0  ∂wj and since [n, w ] = ∂n ∂j and ∇w n are in the tangent space to . Furthermore, the Weingarten formula13 implies, with the map Aˆ n defined by (3.71), that ∇w n = −Aˆ n (w ). Thus we obtain, for every vector field v at , − Av n =

∂vn ∂v + n − Aˆ n (v ), ∂n ∂n

with

 ∂vj ∂v = ∂j . ∂n ∂n

(3.74)

(ii) For g(n, ∇w v), we will again use the decomposition w = w + wn n; we have g(n, ∇n v) = g(n, ∇n (v + vn n)) =

∂vn + g(n, ∇n v ), ∂n

and g(n, ∇n v ) = g(n, ∇v n + [n, v ]) = 0, since ∇v n and [n, v ] are tangent vectors to . Thus g(n, ∇n v) = we have only to evaluate the last term g(n, ∇w v) = g(n, ∇w v ) + g(n, vn ∇w n) + Lw (vn ).

13 See

[Kob-Nom, Vol. 2, ch. VII.3, p. 15].

∂vn ∂n .

Then

3.1 Kinetic Modelling

361

The first term is given by the second fundamental form14 h(w , v ), and the second term is null, so that we have g(n, ∇w v) = h(w , v ) + Lv (vn ). Thus we obtain g(n, ∇w v) = wn

∂vn + h(w , v ) + Lw (vn ). ∂n

(3.75)

Let g(Bv, w) = g(n, ∇w v), with Bv ∈ Tx ( )⊥ × Tx ( ). Then Bv = grad vn + Aˆ n (v ) +

∂vn n. ∂n

(3.76)

Then with (3.74) and (3.76), we obtain for (3.73), 2n.v w = g(Bv − Av n, w), Bv − Av n = grad vn +

with

∂vn ∂v +2 n. ∂n ∂n

(3.77)

Let C1 be the linear map in the space of vector fields on with values in T ( )⊥ × T ( ) given by C1 v = Bv − Av n, so that g(C1 v, w) = 2n.v w. Then C1 is given by  C1

vn v



 =

n 2 ∂v ∂n grad vn +

∂v ∂n

.

(3.78)

We will have to decompose the divergence of v at the boundary . This is easy to do using the fact that it is the trace15 of −Av . Let (Xi ) = (n, e1 , e2 ) be an orthonormal frame of T M at x ∈ . Then we have   < Xi , Av Xi > = < Xi , ∇Xi v > div v = −tr Av = −  = < n, ∇n v > + < ei , ∇ei v >  ∂vn  + < ei , ∇ei v > + < ei , ∇ei (vn n) > ∂n  ∂vn = + div v + vn < ei , −Aˆ n ei > . ∂n

=

14 See 15 See

[Kob-Nom, Vol. 2, ch. VII.3]. [Kob-Nom, Appendix 6].

(3.79)

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3 Fluid Mechanics Modelling

 Then < ei , −Aˆ n ei > is the trace of −Aˆ n in . Since tr (−Aˆ n ) = 2Rm , with Rm the mean curvature of , we obtain the decomposition of the divergence of the vector field v at the boundary: div v = div v +

∂vn + 2Rm vn . ∂n

(3.80)

3.1.9 Functional Study of Existence of a Flow We insist on the difference between the notions of local group (or pseudogroup) and of semigroup or group, with respect to certain questions of reversibility. For a 1parameter local group (φt ) generated by a vector field X, there exists at every x ∈ M a lifetime interval Ix (for X = v, this is the lifetime interval of the particle fluid). For every x, we can define two values of time, t0 = −α− (x), and t1 = α+ (x), such that Ix = ]t0 , t1 [ ; t0 corresponds to the birth of the particle, t1 to the death of the particle. A priori, t0 = −t1 , and there is no reversibility, even local, with respect to time. The local group defines a semigroup g(t), t > 0, in M if α+ (x) = +∞ for every x, and a group g(t), t ∈ R, in M if moreover, α− (x) = −∞, i.e., Ix = R, ∀x ∈ M, for a “global reversibility.” Recall that we say that the vector field X (or v) is complete if Ix = R for every x. We recall the following properties:16 α− and α+ are lower semicontinuous; if α− (respectively α+ ) is bounded from below on M by a constant C > 0, it is constant and equal to +∞. Recall also that if M is a compact manifold (without boundary), then every regular vector field is complete. If the velocity field X = v is not complete, then passing from Lagrange to Euler coordinates and conversely can be done only locally. We are led to define the sets t and , t = {x ∈ M, t ∈ Ix } and  = {(x, t) ∈ M × R, t ∈ Ix }, which are open sets in M and M × R respectively. The map φ (see (3.2)) is an integral flow of X with domain , and φt is a diffeomorphism from t onto −t (see [Mall, ch. II.2.4]). If Ix = ]a, b[ , then φ(x, t) tends to infinity as t → a (or b), which means that for every compact set K in M, there exists  such that when t < a +  (respectively t > b − ), φ(x, t) ∈ / K. We insist on the hypothesis of regularity. We have two main “regular” cases: (i) v is of class C 1 or C 0,1 . This implies the existence and uniqueness of a local group (or pseudogroup), but not necessarily of a group. This regularity of v is often globally out of reach. (ii) v is of class C 0 , and then we must exclude the points such that v = 0, which are generally isolated points. Globally we can hope, with “reasonable” conditions of regularity (reasonable with respect to thermodynamics, excluding shocks),

16 See,

for instance, [Bour.var, 9.1.4].

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363

that v will be in the Sobolev space H 1 (M), but that is still not sufficient (M in R 3 ) for v to be continuous. We should have a regularity result such as v ∈ H s (M), a Sobolev space of index s > 3/2, to obtain from the Sobolev theorems that v is continuous. Such a global regularity result is very generally out of reach. But we can apply some finer regularity results of the following type,17 given in the simple case  = R n . Let l, m be positive integers with m ≤ l, (l − m)p < n,  a nonempty open set in R n . Then ∀u ∈ W l,p (),18 and ∀ > 0, there exists v ∈ C m () such that if F = {x ∈ , u(x) = v(x)} , then we have &u − v&m,p < , and Rl−m,p (F ) <  (i.e., F is a set of Rl−m,p -capacity less than , with Rl−m,p (F ) the Riesz capacity of F ; see [Ziem, ch. 2.6.1, pp. 52, 50]). Thus u is a function of class C m outside of an open set U of capacity less than . The application of these regularity theorems to the framework of the Sobolev space H 1 () allows us to reach the regularity C 1 by eliminating sets of capacity less than . Moreover, an interesting relation between a flow φt that is a Lipschitz mapping in R n with its inverse, and the Sobolev space W 1,p (),  ⊂ R n , p ≥ 1, is the following (see [Ziem, Thm.2.2.2]: if u ∈ W 1,p (), then w = u ◦ φt is in W 1,p (φt−1 ), and we have the relation (for a.e. x ∈  and all ξ ∈ R n ) Du(φt (x)).dφt (x, ξ ) = Dw(x).ξ, with dφt (x, ξ ) the directional derivative of φt at x for ξ . In other notation, we can  ∂w  ∂u ∂y j (φt (x)) ∂x ξ . This results shows the equivalence, with write k ξk = ∂y j ∂x k k respect to the Sobolev spaces, between Euler and Lagrange coordinates if the flow is (bi)-Lipschitz. Conversely, for a velocity field v in the Sobolev space H 2 () = W 2,2 (), with  ⊂ R 3 , v is continuous (and even such that |v(x) − v(y)| < C|x − y|1/2) and is of class C 1 (and thus Lipschitz) for v ∈ H 3 (), which is a regularity result that is generally not reached. Under fairly weak regularity conditions (to be specified below, without Lipschitz condition on the velocity!), we can prove the existence of a unique generalized flow,, thanks to a theorem of [DiP-PLL1, DiP-PLL2] given below in the case that the domain is R n . The case in which the domain is a bounded open set  in R n rests on similar results. Theorem 1 Let v be a vector field on R n such that 1,1 v ∈ (Wloc (R n ))n , with div v ∈ L∞ (R n ) and v/(1 + |x|) ∈ L1 (R n )n + L∞ (R n )n .

There exists a unique generalized flow X(x, t), such that:

17 See 18 This

[Ziem, ch. 3.10, Thm. 3.10.4, Thm. 3.10.5, Thm. 3.11.6] or [Eva-Gar]. is the space of p-integrable functions with their derivatives up to the order 1.

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3 Fluid Mechanics Modelling

• ∀β ∈ C 1 (R n , R) ∩ L∞ (R n ), ∂t β(X) = Dβ(X).v(X), β(X)|t =0 = β(x), • a.e. x ∈ R n , X(x, t1 + t2 ) = X(X(x, t2 ), t1 ), ∀(t1 , t2 ) ∈ R 2 , • ∃C > 0, exp(−Ct) λ ≤ X(t) ◦ λ ≤ X(t) exp(Ct) λ, with X(t) ◦ λ the image measure of the Lebesgue measure λ by the flow X. Moreover: • a.e. x ∈ R n , • a.e. x ∈ R n ,

t X(x, t) = x + 0 v(X(x, s))ds, X(x, .) ∈ C 0 (R)n .

This theorem concerns specifically the case of “stationary” equations in fluid mechanics, for n = 3 or 2, which is also the case for an evolution problem when the velocity v does not explicitly depend on time. This theorem rests on properties relative to the linear transport equation (of the mass density ρ) associated with v, with given v and the initial conditions of weak regularity ∂t ρ + v.grad ρ = 0,

ρ|t =0 = ρ0 ,

whose solution in the regular case is given by ρ(x, t) = ρ0 (X−1 (x, t)). For the equation of mass conservation in fluid mechanics, ∂t ρ + v.grad ρ + ρdiv v = 0, the notion of renormalized solution is used (with ρ0 measurable on R n ), such that for every bounded β ∈ C 1 (R, R), we have ∂t β(ρ) + div (β(ρ)v) + (ρβ  (ρ) − β(ρ)) div v = 0,

β(ρ)|t =0 = β(ρ0 ).

According to this notion, with that of solution in the sense of distributions in a more q 1,p regular framework where ρ ∈ Lloc (R n ), and v ∈ Wloc (R n ), with p1 + q1 ≤ 1, the following regularization result (by convolution ∗) is: r = v.grad (ρ ∗ φ ) − (v.grad ρ) ∗ φ → 0 in L1loc (R n ),  for any radial positive function φ such that φ ∈ C ∞ (R n ), R n φ(x)dx = 1, and φ (x) =  −n φ(x/). The theorem of R. Di Perna and P.L. Lions may be improved (see [Desj]) by changing the condition on div v for div v ∈ L1w (R n ) + L∞ (R n ), with (if dx is the Lebesgue measure) L1w (R n )

1 = {f ∈ L (R ), such that sup ( n ω(|A|) A⊂R 1

 |f |dx) < ∞},

n

A

3.2 Thermodynamic Modelling

365

the sup being taken over the family of measurable sets with positive measure, and ω 1 1 a nondecreasing function from R+ into R+ such that 0 ω(u) du = +∞. For another notion of generalized flow, see, for instance, [Arn3, ch. IV.7.G] due to Brenier.

3.2 Thermodynamic Modelling 3.2.1 Introduction. Fibre Product of Bundles We first study an evolution of a fluid occupying a fixed domain M (an open connected subset of R 3 ) in a time interval J (for instance J = R + ). The main problem in a deterministic model is to describe the evolution of characteristic parameters of the medium from what we know at the initial time t = 0. Of course, the velocity field allows us to describe the evolution of the points in the configuration domain M, and thus to have a notion of fluid particle that does not agree with the notion of elementary particle (nor of point particle in Newtonian physics). But the notion of fluid particle is a macroscopic notion, to which we can assign a velocity and a thermodynamic state, thanks to the section of a vector bundle FM , called a thermodynamic bundle, whose base space is M and the fibres are diffeomorphic to the space F (as introduced in the first chapter); a section ζ F of FM is a field giving at each point x ∈ M the thermodynamic state ζ F (x) ∈ F. The space FM is identified with the (trivial) product M × F. Describing the fluid particle evolution at a fixed time t requires one to determine a family of sections (ζt ) of the fibre product bundle T M ×M FM with fibre Tx M × F and base M and with projection denoted by π. To the velocity field (under smoothness conditions) is associated a flow (φt,s ), t > s, in M, a solution of the differential equation dx dt (t) = vt (x(t)), with given x(s) = xs in M. The velocity field depends on time, and thus it will be necessary to change M for the spacetime M˜ = M × R in the previous spaces, notably to use the fibre product bundle T M˜ ×M˜ FM˜ . We insist on the following essential facts, linked to the notion of bundle section of T M˜ ×M˜ FM˜ : • The thermodynamic variables explicitly depend only on (x, t) and not on the velocity v: we notably write T (x, t) for the temperature and not T (x, v, t). • The velocity field does not depend explicitly on the thermodynamic variables:19 we write v(x, t) and not v(x, ρ, . . . , T , t).

19 Note that we have to consider separately the incompressible fluid case depending on the Navier– Stokes equation, where the trajectories are independent of the pressure, but thermodynamics is not really present in this model.

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3 Fluid Mechanics Modelling

We can give many other variants, notably the case of several species of fluid particles (at a given point). Then we have to specify the thermodynamic variables of each species and use a thermodynamic fibre bundle for each species or for the entire system.¶

3.2.2 Some Reminders We will have to use some subsets D of a fixed domain M of the Euclidean space E3 for which we can apply the Green formula and that have the structure of a manifold with oriented boundary. More precisely, for an n-dimensional manifold, we have (see [Bour.var, 11.1]) the following definition. Definition 2 Piece. Let M be a manifold of class C r , and D a subset of M; D is said to be a piece of M if for every x ∈ D, there exists a chart c, c = (U, φ, R n ), of M at x such that φ(D ∩ U ) is an open set of a closed half-space of R n . The set D is naturally equipped with the structure of a “manifold with boundary,” with ∂D = D − D ◦ (where D ◦ is the interior of D), which is then a submanifold of D. A closed subset D of M is a piece of M if and only if D ◦ is dense in D and ∂D is a submanifold M of codimension 1 at each of its points. If D is a piece of M and if b is a closed part of ∂D, then D − b is a piece of M. Thus D ◦ is a piece of M. The dimension of a piece is that of M. The k-chain notion generalizes the notion of piece to the case in which the dimension of D is k. Remark 5 The usual hypotheses on the domain D, i.e., that its boundary is of class C k,α and D is locally on one part of its boundary, are such that D is a piece. We will assume that the domain M itself is a piece of E3 . ¶ We can define the set of outgoing (respectively incoming) vectors on the boundary, denoted by Tx+ D (respectively Tx− D) and Tx (∂D) = Tx+ D ∪ Tx− D. If M is an oriented manifold, then the same is the case for every piece D and its boundary. Recall the following notions. Let μ be a form volume (or element volume) on a 3-dimensional oriented manifold M, i.e., an odd 3-form such that μ(∂1 , ∂2 , ∂3 ) > 0 (with ∂i = ∂/∂x i ) for every oriented local coordinate system (x 1 , x 2 , x 3 ). We can identify μ with a measure on M (see, for instance, [Bour.var, 10.1.6, p. 38]) and  define the integral M f μ for every μ-integrable function f . Moreover, M having an oriented Riemannian structure, there exists a natural form volume, denoted by vg , such that vg (X1 , X2 , X3 ) = 1 for every orthonormal frame (X1 , X2 , X3 ) according to the orientation. We will take μ = vg . To simplify notation, we no longer underline the odd forms. Evolution equation of μt . Let φt,s be a time-dependent flow (in M) generated by a vector field Xxt = X(x, t) of class C r , r ≥ 1, with φt = φt,0 an oriented diffeomorphism. We still denote by (φ˜t ) the evolution group in M˜ = M × R such

3.2 Thermodynamic Modelling

367

that φ˜ t (x, s) = (φt,s (x), t + s). Let Dt = φt,0 D, xt = φt,0 x ∈ Dt , and let ω be the 3-form on Dt given in a chart with coordinates (y j ) so that ωy = dy 1 ∧ dy 2 ∧ dy 3 . Thus at the point (xt ), we have ωxt = dxt1 ∧ dxt2 ∧ dxt3 . The pullback of ω by φt is the 3-form ωt on D defined by ωt = φt∗ ω = J (φt )ω0 ,

(3.81)

with ω0 = dx 1 ∧ dx 2 ∧ dx 3 and Jx (φt ) the determinant of Dφt (x) = (φt )∗ . Let μ = |ω| be the positive measure (associated with Euler coordinates) on Dt corresponding to the 3-form ω, and let μt = |ωt | be the positive measure (associated with Lagrange coordinates) on D0 corresponding to the 3-form ωt : ωt ∈ ∧3 D ↓

φt∗

←−

ω|Dt ∈ ∧3 Dt ↓

φt

μt = |ωt | ∈ M+ (D) −→ μ = |ω|Dt | ∈ M+ (Dt ) The image under φt of the measure μt = |J (φt )|.μ0 (with μ0 = |ω0 |) is the measure μ. In the following we do not distinguish μ from ω, nor μt from ωt , so that we write μt = φt∗ (μ) = |J (φt )|μ0 .

(3.82)

By time derivation of (3.82), we obtain with X = v, v = (  dμt  = LX μ = (divX) μ. dt t =0

j

∂xt ∂t

), (3.83)

Let f be a differentiable function f : M˜ = M × R → R. By integration on the domain Dt , we can write, since φ˜ t (D × {0}) = φt (D) × {t},     ft μ = f (φ˜ t ) μt , (3.84) φ˜ t∗ (f0 )φt∗ μ = φ˜ t∗ (f0 μ) = φt (D)

D

D

D

∗ φ˜ ∗ = φ ∗ with i the canonical injection of D in M. ˜ Let X˜ be since we have iD D t t   ∂f ˜ ) = the vector field on M˜ such that X(f ∂t + X(f )t =0 , also denoted by v˜ =  ∂ ∂ ∂ d j v ∂x i , or by dt . By time derivation of (3.84), we obtain, with ∂t + v.∇ = ∂t + the Lie derivative Lv˜ ,        d d ∗ φ˜ t (f μ) f μ = = Lv˜ (f μ). dt φt (D) D dt D t =0 t =0

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3 Fluid Mechanics Modelling

Thus d dt

 φt (D)

  f μ





= t =0

(v(f ˜ ) + f div v) μ = D

( D

∂f + div(f v)) μ. ∂t

(3.85)

We will essentially use the formula (3.85) of time derivation in the following form, with D =  a domain in E3 (with boundary = ∂D), μ = vg the Lebesgue measure (also denoted by dx or dx) on D, and a function f = uρ on : d dt

 φt ()

  u ρ μ

 = t =0



∂(uρ) μ+ ∂t

 (uρ) v.n d .

(3.86)



3.2.3 Mass Conservation In the modelling of a fluid evolution in a fixed domain M of E3 , we first have to write the mass conservation of the system, using  the measure (or odd 3-forms) = uρv . The total mass is expressed by m = u g M ρvg . We assume that there is no chemical reaction or electromagnetic field in the system. The mass conservation along the time (for a closed system) is coupled with a positive measure (or an odd 3-form) 1 , invariant under the evolution, i.e., such that 1,t

∗ = φt,0 (

1 ).

(3.87)

With the hypothesis that M has a Euclidean structure at t = 0, we take μ = vg the Lebesgue measure. The measure 1 that represents at time t = 0 the specific mass of the fluid is such that 1 = ρ0 μ. Then the corresponding measure 1,t at time t must be 1,t = ρt μt with ρt the (positive) specific mass at time t. Then (3.87) is written ∗ (ρ0 μ) = ρ0 μ. ρt μt = φt,0

(3.88)

Since μt = J (φt,0 ) μ, we have ρt J (φt,0 ) = ρ0 .

(3.89)

Let φt = φt,0 . Applying (3.84) with ft = ρt , we obtain the integral formulation of the mass conservation (for every compact piece D of M): 

 ρt μ = φt (D)

ρ0 μ. D

(3.90)

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The condition ρt ∈ L1loc (M), ∀t, gives a sense to (3.90) (and to (3.88) in the sense of measures). In formula (3.90), we assumed that for every x ∈ D, the fluid particle at x has a lifetime greater than t. More generally, we will assume that this lifetime is infinite (i.e., Xt = vt is complete). Local formulation of the mass conservation. If we take the derivative of (3.90) ˜ with respect to the time at t = 0, we obtain, with the immersion iM0 of M0 in M, d ∗ ∗ ∂ρ i φ˜ (ρ μ)|t =0 = Lv˜ (ρμ) = ( + v.∇ρ) μ + ρ (div v) μ = 0. dt M0 t ∂t A local expression of mass conservation is (see (1.366), (1.365) above) dρ + ρ div v = 0, dt with

dρ dt

= v(ρ) ˜ =

∂ρ ∂t

or

∂ρ + div (ρv) = 0, ∂t

(3.91)

+ v.∇ρ.

Remark 6 On the uniqueness of the invariant 3-forms by evolution. Let 11 and 2 1 be two positive measures satisfying (3.87). Then there exists a positive function ζ on M˜ = M × R constant along the trajectories (i.e., with Ix the lifetime interval, ζ (φt (x), t) = ζ (x, 0), ∀t ∈ Ix ) such that 11 = ζ 12 . Also we have for (3.91) that if ρ1 and ρ2 are two (positive) volume masses satisfying (3.91), then ρ1 = ζρ2 . ¶ Remark 7 We assume that there exists a solution of (3.91) in M, for every time, such that ρ ∈ C 0 (R, L1 (M)) (with the boundary condition v.n = 0). Then the quantity m = M ρ(x, t) μ is independent of t. Thus the total mass is conserved over time, i.e., with a probabilistic interpretation that the density of the death of particles is almost surely equal to the density of the birth of particles at every time t. ¶ Remark 8 On the metric associated with the density. The choice of the Euclidean metric on M is associated with the choice that the specific mass is at first a thermodynamic variable. Another choice may be the following. The momentum p of the velocity is defined by p = ρv or even by p = ρvμ. The momentum being usually related to the metric by p = Gv, it seems natural to define the metric on M by g(v, w) = kρv.w, with k a constant so that kρ is a density. In the case that ρ depends on x ∈ M and not on the time, we thus define a Riemannian metric on M that is adapted to the physical nature of the fluid. But if ρ depends on the time also, we would have to use a foliation of the tangent space, and thus of the phase space. ¶ From this point we will replace the notation μ by dx for the usual notation of the Lebesgue measure dx in fluid mechanics.

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3 Fluid Mechanics Modelling

3.2.4 Differential Structure: Foliated Fibre Bundles Let M˜ = M × R be the spacetime, T M˜ its tangent space, FM˜ = M˜ × F the (trivial) ˜ ˜ and T M˜ × ˜ F ˜ the fibre product bundle over M. thermodynamic bundle over M, M M Basic Differential Form and Foliation First, we have the foliation M˜ π of M˜ by the time, with the submersion π, π(x) ˜ = t. ˜ onto T R = R × R such ˜ The map T (π) = π∗ from T (M) (i) Foliation of T (M). ˜ that π∗ (x, ˜ v) ˜ = (t, λ) for x˜ = (x, t), v˜ = (v, λ) is also a submersion on T (M), ˜ ˜ ˜ and thus defines a foliation T (M)π∗ of T (M), with the leaves (T (M))t,λ . Note ˜ M˜ π ) is contained in (T (M)) ˜ π∗ : if (x, ˜ M˜ π ), that the space T (M, ˜ v) ˜ ∈ T (M, ∂ ˜ ˜ then π∗ (x, ˜ v) ˜ = (t, 0). Recall that ∂t ∈ / T (M, Mπ ). ˜ onto M˜ is a morphism from Note that the natural projection πM from T (M) ˜ π∗ of T (M) ˜ onto the foliation M˜ π of M, ˜ that is, it transforms the foliation T (M) ˜ π∗ into the leaves of M˜ π . Let v be a time-dependent section the leaves of T (M) ˜ and it is a morphism from the of T M. Then v˜ = (v, 1) is a section of T (M), ˜ π∗ . foliation M˜ π into the foliation T (M) We deduce the foliation of the tangent vectors to M˜ on the foliated boundary ˜ = ∂ M˜ π into T (∂ M˜ π ) ⊕ T (∂ M˜ π )⊥ with T (∂ M˜ π )⊥ the normal bundle of ∂ M˜ π . (ii) Foliation of F ˜ . Taking the state equations of the fluid into account, we M recall the foliation of F by the maximal integral manifolds (α ) of the basic differential forms θ e or θ s : θ e = de + P dτ − T ds,

θ s = ds −

P 1 de − dτ. T T

(3.92)

In the following we often simply denote F by F, and we consider only the case in which the “transverse” variable to the leaves is the variable s0 of θ s irreversible evolution (as for perfect polytropic gases). This means that the state equations are such that θ s is an exact differential form, thus such that θ s = dS 0 . Let F be the foliation of F by the “irreversible” entropy s0 , so S0 that S 0 (j ) = s0 , j ∈ F . This is an oriented foliation by the transverse variable s0 . The product (π, S 0 ) of the submersions π and S 0 , (π, S 0 )(x, ˜ j ) = (t, s0 ), is a submersion from the trivial product bundle FM˜ over M˜ on R × R. Thus we have a new foliation (FM˜ )(π,S 0 ) of FM˜ . ˜ π∗ × ˜ (F ˜ )(π,S 0 ) is an oriented foliation (iii) Then the fibre product bundle (T (M)) M M ˜ of the bundle T (M) ×M˜ FM˜ , with the transverse variables (t, λ, s0 ).

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371

We define the bundle T M˜ ×M˜ FM˜ as an associated bundle of the principal ˜ M˜ π ) over M˜ with group G = GL(3+1, 3); let R 4 ×F be the bundle P˜ = L(M, standard fibre, with the action of G on R 4 ×F given by a(ξ, ˜ j ) = (a˜ ξ˜ , j ) (if we assume that the group G does not operate on F, that is, on the thermodynamic variables). Then the fibre bundle over M˜ associated with P˜ , with standard fibre ˜ R 4 × F, G, P˜ ). R 4 × F, is denoted (see [Kob-Nom, ch. I.5, p. 55]) by E(M, 4 ˜ Let u˜ ∈ P , (ξ, j ) ∈ R × F. Then u.(ξ, ˜ j ) ∈ E, and it is identified with ((u.ξ ˜ ), j ) ∈ T M˜ ×M˜ FM˜ . ˜ π∗ × ˜ (F ˜ )(π,S 0 ) of the We could also define a foliation Eπ∗ ,S0 = (T (M)) M M bundle E as an associated bundle, where we have to replace the standard fibre by R 3+1 × FS0 .

˜ ט F˜ Some Sections of the Bundle T M M M We define the following sections of this bundle: Let v˜ be a section of T M˜ (a time-dependent field) such that v( ˜ x) ˜ = (v(x), ˜ 1), ˜ At first, v˜ does not represent the velocity of the with the notation x˜ = (x, t) ∈ M. fluid. We will indicate when we take v = dx dt to be the velocity. Let ζF be a section of FM˜ giving the thermodynamic state of the fluid at every x˜ by ζF (x) ˜ = j (x) ˜ = (e, τ, P , T , s)x˜ . Then ζ = (v, ˜ ζF ) is a section of T M˜ ×M˜ FM˜ . If ζ is of class C 2 , we can define the following tensors of type (1, 1) (a priori on T M): the strain tensor ˆv = (ji ), with ji = 12 (∂i v j + ∂j v i ), the stress viscous tensor τˆv = (τji ), and the stress tensor σˆ v = (σji ) by σˆ v = τˆv − Ph I, thus σji = τji − Ph δji ,

τji = λδji tr ˆv + 2μji ,

(3.93)

in the case of a Newtonian fluid, with λ, μ viscous coefficients, and with Ph a term of pressure (which may be different from the thermodynamic pressure, as we will see below). ˜ ט Then we recall some definitions related to the section ζ = (v, ˜ ζF ) of T (M) M FM˜ . θ -Admissible, θ -Reversible Evolutions Following the second law, in order that an evolution t ∈ R → x(t) ˜ ∈ M˜ be θ ˜ be nondecreasing in time. We can write admissible, it is necessary that S 0 (ζF x(t)) this condition in many ways. For instance, let π be the submersion from T M˜ ×M˜ FM˜ into R such that π(v, ˜ j ) = s0 ; then the section ζ from M˜ into T M˜ ×M˜ FM˜ must be such that π ◦ ζ (x(t)) ˜ = s0 is a nondecreasing function of time, that is (under smoothness condition),

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3 Fluid Mechanics Modelling

d ds0 (π ◦ ζ )|t =0 = Lv˜ (π ◦ ζ ) = < v, ≥ 0. ˜ d(π ◦ ζ ) > = dt dt

(3.94)

The nondecreasing condition on S 0 implies that the image (ζF )∗ v˜ of the “velocity” v˜ under ζF is θ -admissible: ˜ θ s > = < v, ˜ ζF∗ θ s > ≥ 0. < (ζF )∗ v,

(3.95)

The θ -reversibility of the evolution implies that the image of v˜ under the lift ζF is ˜ tangent to a leaf (F )s0 , that is, for a time evolution τˆ (t) in M, < (ζF )∗ v, ˜ θ s > = < v, ˜ ζF∗ θ s > = 0,

thus

ds0 =0: dt

(3.96)

ζF M˜ −→ F π ↓↑ xˆ S0 ↓ t ∈ R −→ s0 ∈ R

Thermodynamic Framework Following the fluid particle in its motion, the pullback by ζF of the differential form of heat θQ may be obtained by < v, ˜ ζF∗ (de) > = < v, ˜ ζF∗ (θW + θQ ) > .

(3.97)

The pullback of the differential form of work θW on F by ζF is, with (3.91), < v, ˜ ζF∗ θW > = − < v, ˜ P dτ > = −τ P div v.

(3.98)

˜ 3.2.5 Work and Heat on M The usual fluid equations, that is, the ENS (Euler–Navier–Stokes) equation, is obtained from a differential form of work θˆW , and then for the internal energy equation, see below (3.125), (3.126), and (3.130) (see [D-L19, ch. 1]), we will define a differential form of heat θˆQ . The differential form of work θˆW for the fluid (which is different from ζF∗ θW ) is defined by θˆW =



fj dx j =



∂i σji dx j ,   = ∂i τji dx j − ∂i Ph dx i ,

(3.99)

3.2 Thermodynamic Modelling

373

with the definitions (3.93) of σji and τji . Yet if v represents the velocity vector of the fluid particle, the power corresponding to the work is < v, ˜ θˆW > = −tr (σˆ .ˆ ) + div (σˆ .v) = −tr (τˆ .ˆ ) + div (τˆ .v)− < v, dPh > . (3.100) Of course if there are external forces f e , we have to add the work of those forces. The equivalence between the following different balances may be seen using the invariance of ρdx and the relation d df ∂f (f dx) = Lv˜ (f dx) == dx +f (div v) dx = ( +div (f v)) dx. (3.101) dt dt ∂t  Local power balance of the kinetic energy. Let f e = fje dx j be an external force. The kinetic energy 12 ρv 2 of a fluid particle is related to the differential forms θˆW of work and to the external forces f e , according to the principle of virtual powers, 1 < v, ˜ ρd( v 2 ) > = < v, ˜ θˆW > + < v, ˜ f e > = < v, θˆW > + < v, f e >, 2 (3.102) and we deduce ρ

d 1 2 ( v ) = −tr (τˆ .ˆ ) + div (τˆ v)− < v, dPh − f e > . dt 2

¶

(3.103)

Balance of heat and of internal energy. We assume that the differential form of heat on M is given by < v, ˜ θˆQ > = < v, ˜ ρζF∗ θQ > = tr (τˆ ˆ ) − div qF ,

(3.104)

which is the relation between the heat differential form and the heat flux qF . This term qF is linked with the “gradient” of the temperature T of the medium (in the absence of electric current) by the Fourier law20 qF = −k grad T ,

that is (qF )i = −

 j

j

ki

∂T , ∂x j

(3.105)

j

with a matrix operator k = (ki ) that depends on the medium and its thermodynamic state, notably its temperature. Moreover, k must be a positive definite operator, which corresponds to the fact that locally “heat passes from warm to cold.” In the isotropic fluid case, k is reduced to a positive number (which can depend on the temperature) and is called the thermal diffusion coefficient.

20 The physical existence of such a field seems natural from a finer modelling, microscopic or (and) random, with a diffusion process, for instance.

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3 Fluid Mechanics Modelling

From the first law, we have, with the thermodynamic work θW = −P dτ, according to (3.98), < v, ˜ ζF∗ de > = < v, ˜ ζF∗ (θW + θQ ) > = − < v, ˜ P dτ > + < v, ˜ ζF∗ θQ > . (3.106) Thus if we multiply by ρ and use the mass conservation (3.91), we have21 ρ

d eˆ + P div v = < v, ˜ ρζF∗ θQ > . dt

(3.107)

Then the evolution equation of the internal energy eˆ is ρ

d eˆ − tr (σˆ ˆ ) + div qF = 0, dt

(3.108)

with σˆ = τˆ − P I . If we assume that there is an internal heat source r0 by unit volume,22 then the evolution of the internal energy is given (as in [D-L19, ch. 1]) by ρ

d eˆ = tr (σˆ ˆ ) + (r0 − div qF ). dt

(3.109)

Entropy evolution. With the first law and the usual state equations, we recall θ = ds −

P 1 1 1 1 de− dτ = ds − (θW +θQ )+ θW = ds − θQ = ds0 . T T T T T

(3.110)

Thus θQ = T (ds − ds0 ), and the evolution of the entropy must satisfy d sˆ d sˆ0 − < v, ˜ ξF∗ θQ > = ρT ≥ 0, dt dt

(3.111)

1 d sˆ d sˆ0 + div ( qF ) − ρ η˙ = ρ ≥ 0, dt T dt

(3.112)

ρT which we can write23 ρ with

1 ρT η˙ = tr (τˆ ˆ ) − qF .( ∇T ), T

21 The internal energy eˆ is identified with (the pullback of) the thermodynamic internal energy e by the section ξF . 22 Which may be a model of a Joule effect induced by an electromagnetic field. 23 In a similar way to the Clausius–Duhem form; see [D-L19, chap. 1]).

3.2 Thermodynamic Modelling

375

which is positive with the Fourier relation, so that we can write ρ

1 d sˆ + div ( qF ) ≥ ρ η˙ ≥ 0. dt T

(3.113)

If the evolution of the system is θ -irreversible, then the evolution of the entropy s is governed by the relation (3.112), whereas for a θ -reversible evolution of the system, the evolution of the entropy s is governed by an equation. Note that if the evolution is adiabatic, that is, θQ = 0, then ddtsˆ = ddtsˆ0 , and we must have tr (τˆ ˆ ) − div qF = 0. Of course, the state equations of the fluid must complete these evolution relations. For instance, in the case of perfect gases, the entropy is given by s = (cv log e + r log τ ) + s0 , so that 1 dτ d sˆ0 d log τ d sˆ0 1 d eˆ d log(e) ˆ d sˆ = cv +r + = cv +r + . dt e dt τ dt dt dt dt dt

¶

The balance of the total energy (kinetic and internal energy) is given with (3.103), (3.108), by ρ

d 1 2 ( v + e) = div (τˆ v − qF )− < v, dPh − f e > −P div v. dt 2

(3.114)

Remark 9 The previous balances of kinetic energy and internal energy are made such that they are ready to be integrated on the current lines of the fluid. Let γtt0 ⊂ M × J be such a line. Then the balance of internal energy is  e(x(t), t) − e(x(t0), t0 ) = 

γtt0

ζF∗ de

 =

t t0

< v, ˜ ζF∗ de > dt

t

(3.115)

dτ + < v, ˜ ζF∗ θQ >]dt. [−P = dt t0 The use of the Lie derivative is made in order that the balances of kinetic energy24 and internal energy can be integrated on pieces D of R 3 , giving new balances of work and heat: 1 Lv˜ ( v 2 ρ dx ∧ dt) = < v, ˜ θˆW + f e > dx ∧ dt, 2 Lv˜ (eρ dx ∧ dt) = < v, ˜ ζF∗ (θˆW + θˆQ ) > ρdx ∧ dt.

24 Beware

(3.116)

of the simple notation v 2 ; we recall that this means g(v, v), with g the pullback of the initial metric by the inverse flow.

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3 Fluid Mechanics Modelling

Then 1 Lv˜ (( v 2 +e)ρdx∧dt) = < v, ˜ (θˆW +ρζF∗ θˆW )+ρζF∗ θˆQ +f e > dx∧dt. 2

(3.117)

Finally, we obtain 1 Lv˜ (( v 2 + e)ρ dx ∧ dt) 2

(3.118)

= [div (τˆ .v − qF )+ < v, f e − ∇Ph > −P div v] dx ∧ dt, with the (hydro) pressure Ph and the thermodynamic pressure P .

¶

Remark 10 Observe the following: (i) the unit of θˆW and of θˆQ is L−1 T −2 , that is, the unit of [ρde]; (ii) θˆW is a differential form on the foliation M˜ π (on the leaves (Mt )).

¶

Expressions of work and heat on pieces. The expressions on a piece D ⊂ M that represent the work (or the power) and the heat at a time t are deduced from the local formulas by integration. The “natural” functional spaces will be given according to the following relations. Let WD be the model of the work (in fact the power) due to the internal forces of the fluid in a domain Dt = D of M, expressed with the Stokes formula by     1 2 d ˆ v ρdx = WD = < v, ˜ θW > dx = (∂i σji ) v j dx dt D 2 D D    = Ph div v dx − tr (τˆ .ˆ ) dx + (σˆ v).n d∂D D

D

(3.119)

∂D

˜ ˜ with tr (τˆ .ˆ ) = ψ(v) density of the “resistance function” ψ(v) of the fluid, and  the i j with (σˆ .v).n = σj v ni = (τˆ .v).n − Ph v.n. The last formula of WD may be viewed as a weak formulation of the work. Let QD be the model of the heat that is (algebraically) received by the fluid in the domain D, expressed by 

< v, ˜ θˆQ > dx =

QD = 

D

=

 tr (τˆ ˆ ) dx −

D

 (tr (τˆ ˆ ) − div qF ) dx D

(3.120)

qF .nd∂D. ∂D

The last formula for QD may be viewed as a weak formulation of the heat. Expression of power and energy on pieces. If we assume a volume source of heat r0 and an external force f e , then the expression of the internal energy on a piece D ⊂ M in M at time t is deduced from the local formulas by integration:

3.2 Thermodynamic Modelling

d ED (t) = dt

 ρ 

D

=

377

d eˆ dx dt



(3.121)

[tr (τˆ .ˆ ) − P div v + r0 + v.f ] dx − e

D

qF .n d∂D. ∂D

By time integration on (t0 , t1 ), we see that the variation of the internal energy in the domain D is due to the following: (i) the work of viscosity forces in the domain D, (ii) the heat r0 produced in the domain D, (iii) the flux of heat of qF incoming on the surface ∂D. Remark 11 Flux, heat flux. The formulas (3.100) and (3.104) of balance of work and heat are to be compared with the formulas (1.368) and (1.369) of Chapter 1. The formulas of balance of internal energy (3.109) and of entropy (3.112) are to be compared with the formulas of balance such as ρ de dt + div Je = σe , see (1.370), and the Clausius–Duhem formula, see (1.374), which exhibit the fluxes of quantities such as the heat flux. In the expression (3.121) for the internal energy on a domain D, there appear a purely internal part and an integral on the  boundary that is the sum of the work (in fact the power) and a heat flux Qi∂D = ∂D qF .n d∂D. Note that this flux implied by qF is a transverse vector field to the tangent space T ∂D (so that its projection onto the normal space T (∂D)⊥ is not null). ¶

Evolution of the Temperature, or Heat Equation We consider a fluid with a state equation e = cv T and with the Fourier law; see (3.105). Then the evolution of the temperature must be such that with (3.109), cv ρ

dT + div qF = r0 + tr (σˆ ˆ ). dt

(3.122)

Then with the Fourier law, the equation for the temperature (called the heat equation) is cv ρ

dT − div k grad T = f, dt

with f = r0 + tr (σˆ ˆ ).

(3.123)

3.2.6 Some Remarks and Extensions Remark 12 The case of a fluid with a chemical reaction. Consider a system with several constituents (but in a model with a single fluid) with the possibility of chemical reactions. The differential form θe here is given by (see Chapter 1)

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3 Fluid Mechanics Modelling

θe = de − T ds + P dτ −



(3.124)

Aj dξj ,

with ξj the rate of the j th chemical reaction, and Aj its affinity (see Chapter 1). In a model with one velocity, by keeping the same formal expression of conservation of internal energy, we obtain the balance of energy in the form (3.109), with r0 = rTs , rs = −Aj

dξj dt

. ∂ξ

Then the evolution equations of the reaction rates are needed, ∂tj = Jj , with Jj given by the chemical kinetics as a function of the concentration or of the activity of each constituent (see, for instance, [Pri-Kon, ch. II.9]). We refer to [deG-Maz] for these equations with chemical reactions. Remark 13 The tiredness and aging notions of a material may be linked with irreversible thermodynamic evolution, through the “constants” (notably the Lamé coefficients) that intervene in the constitutive relations of the material (see Chapter 4); these “constants” depend on the thermodynamic state of the material (specially on the temperature), thus on the leaf Fs0 specifying the level of “irreversible” entropy s0 . ¶ Remark 14 It is necessary to distinguish clearly the notion of irreversible thermodynamic evolution from that of mechanics: the mechanical evolution may be irreversible in the sense that the system of equations does not admit a solution for the negative times, whereas the thermodynamic evolution may be reversible or not (in the sense of the first definition of Chapter 1). If the thermodynamic evolution is reversible, and if initially ζF (x, 0) is in the 2-dimensional leaf (F ˜ )s0 , then the section ζF has its image in this leaf (F ˜ )s0 . If M M the projection q from F into R 2 such that q(j ) = (e, τ ) is an isomorphism from (F ˜ )s0 onto its image in R 2 , we must be able to determine ζF by the map q ◦ ζF M from M˜ into R 2 from a section v˜ of T M˜ and from the evolution equations of (e, τ ). ¶ Remark 15 In the framework of the evolution of the density of a family of particles in a phase space (that is, at a macroscopic level finer than that of fluid mechanics), where the evolution is ruled by the Boltzmann equation, we are led to statistical thermodynamics. An entropy is defined that is increasing only along the time (“H theorem”), which corresponds to the irreversibility of the thermodynamic evolution due to the shocks of the particles, which is taken into account by the nonlinear term of the Boltzmann equation. We refer also to the “mathematical” entropy definition in fluid mechanics, following Lax (see, for instance, [God-Rav]). ¶ Remark 16 How to obtain the state equations of a medium: the relation between global thermodynamics and quantum physics. In order to obtain the state equations of a given medium at a given point, with the hypothesis that in a neighborhood of the point, the medium is in local thermodynamic equilibrium (i.e., that the representative points of the thermodynamic situation are on the same leave), we are led to zoom in on this neighborhood and to use a statistical quantum model

3.3 Mechanics Modelling

379

of the medium with a number N of particles in a domain of volume V . The use of global thermodynamics allows us to obtain the desired state equation. For such developments, see, for instance, [Lan-Lif1, ch. IV.42]. ¶

3.3 Mechanics Modelling Here we give the mechanical evolution equation of a fluid, called the Navier– Stokes equation, and the Euler equation when the fluid viscosity is neglected. Very generally, we will use the terminology of the ENS equation. We will sum up the main equations and balances of the fluid mechanics in several forms. We shall not return to the regularity hypotheses of existence of a local flow. The study of energy balance allows us to specify, for a Newtonian fluid, a “natural” (or “reasonable”) global functional framework with the weak formulation of the work, then of the heat in a subsequent section, so that the kinetic and internal energies are finite. In the case of the Navier–Stokes equation, this framework a priori eliminates the jumps of velocity across surfaces.

3.3.1 Euler–Navier–Stokes (ENS) Covariant Equations Let M be the domain of the considered fluid in the Euclidean space E3 , equipped with the Euclidean metric g. Let ∇t,v v = ∂v ∂t + ∇v v be the corresponding covariant derivative. Then the evolution equation of the fluid is expressed in a covariant way, similar to the fundamental equation of mechanics, by ρ∇t,v (Gv) = θˆW + f e ,

(3.125)

  j j ∂j σk dx k = ∂j τk dx k − dx Ph , and with f e the with (see (3.107)) θˆW = differential form of “external” forces. This equation may be still obtained by the principle of virtual powers (see (3.102)).

“Usual” ENS Equations Consider a Cartesian Euler coordinate system (x j ), with the metrics given by G = I , thus G( ∂x∂ j ) = dx j , or even gij = gij = δij . In this coordinate system, the d covariant derivative ∇t,v will be denoted by dt . The mechanics equation of the fluid is written in components in the following form (see [Germ, D-L19, Piro8, Cher]): ρ

 dv i = ∂j σ j,i + (f e )i , dt

with σˆ = (σ j k ) = τˆ − Ph I (see (3.93)).

i = 1, 2, 3,

(3.126)

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3 Fluid Mechanics Modelling

In fluid mechanics, it is usual to denote “divergence of a tensor field,” by25 (div σ˜ )i =



∂j σ ij



j

∂j σi , the ith component of the

and div σ˜ = div τ˜ − grad Ph .

Then (3.126) is written in the form ρ

dv = div τ˜ − grad Ph + f e . dt

(3.127) j

Recall that for a Newtonian fluid, τˆ is linked with ˆ (with i = 12 (∂i v j + ∂j v i )) by the behavior law (see (3.93)), which, using the viscosity coefficients λ, μ of the fluid, can be written as τˆ = λ I tr ˆ + 2μ . ˆ

(3.128)

Of course, from the ENS equation, we could write other evolution equations such as those of div v, curl v, and v .

Review of Other Equations (i) Mass equation (continuity law) dρ + ρ div v = 0. dt

(3.129)

(ii) Internal energy equation (see (3.109)) ρ

de = tr (σˆ P .ˆ ) + (r0 − div qF ), dt

(3.130)

with σˆ P = τˆ − P I ; qF is the field that represents the heat flux. (iii) Kinetic energy equation. The kinetic energy equation may also be deduced from equation (3.126) by an inner product on v (see (3.103)): ρ

d 1 2 ( v ) = −tr (τˆ .ˆ ) + div (τˆ v)− < v, dPh > +v.f e . dt 2

(3.131)

We will give a contravariant writing of the equations (3.126) and (3.127) in the case of a Newtonian fluid with constant μ using the tensor fields (of viscous constraints) of type (1, 1).

25 Its

intrinsic character is given in (3.69).

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381

Vector ENS Equations Let ˆ be the tensor field of type (1, 1) defined by (see (3.34)) ˆv =

1 −1 1 G Lv g = − (Av + A∗v ), 2 2

that defines the map  such that v = ˆv . Then in an intrinsic way, the “divergence” of ˆv is expressed by the differential operator ∗d (the adjoint in the sense of distributions of ), also denoted by ∗ in this chapter: div ˆv = ∗ v.

(3.132)

If v is null at the boundary, we also have ∗ v = grad div v −

1 curl curl v, 2

which is easily verified in a Cartesian coordinate system. Consider a Newtonian fluid with a constant viscosity coefficient μ (with respect to x). The stress viscous tensor τˆ being given by τˆ = An ˆ = λI tr ˆ + 2μˆ (see (3.128), (3.93)), then for every (smooth) w, v null at the boundary, τˆv = An ˆv , we have  < τˆv , w > = tr (An (v)(w)) dx = < ∗ τˆv , w > . M

The “divergence” of τˆv is obtained by Pv = div τv = ∗ An v = grad ((λ + 2μ) div v) − curl μ curl v.

(3.133)

(If the domain M is simply connected, and in the case of the Dirichlet condition, this expression corresponds to a Hodge decomposition of Pv into grad and curl, and we know that this decomposition is unique, so that from a condition like Pv ∈ L2 (M)3 , we can deduce regularity properties of v. We refer only to [D-L2, ch. IXA.1.4, Prop. 6].) Thus we can give a pure vector formulation of the ENS equations (3.125) by ρ∇t,v v = Pv − grad Ph + f e .

(3.134)

Remark 17 We can transform this contravariant equation (3.134) into a tensor equation of type (1, 3) (with unit MLT −2 , thus “with unit of a force”) by taking the product with μ = dx. Then ∇t,v v(ρdx) = ∇t,v (v ρdx) = (Pv)dx + (−grad Ph + f e ) dx,

(3.135)

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3 Fluid Mechanics Modelling

since ρdx, that is, ρ(xt , t)dxt (with xt = φt (x)) is time-invariant.

¶

Remark 18 On the kinetic energy equation. We can give different expressions for kinetic energy from the formula (3.103). The kinetic energy equation is written as the inner product of (3.134) on v: d 1 2 ( v ) = −(λ + 2μ)(div v)2 − μ(curl v)2 + (−grad Ph + f e ).v + div J, dt 2 (3.136) with J given by ρ

J = (λ + 2μ) v div v + μ grad (v 2 ) − 2μ ˆv .v,

(3.137)

or J = (λ + 2μ) v div v − 2μ ∇v v.

¶

(3.138)

3.3.2 Divergence Equations It is useful to write the equations of fluid mechanics in a “conservative form,” with only derivative operators on the left, so that they operate in the sense of distributions. This allows us to treat singularities such as shock waves. These relations are easily expressed with the Lie derivative, using the formula for a differentiable function f : Lv˜ (f dx ∧ dt) = (

∂f + div (f v)) dx ∧ dt. ∂t

(3.139)

Then (with suitable regularity conditions), we can apply the Stokes formula d dt



 M×J

f dx ∧ ds|t =0 =

M×J

Lv˜ (f dx ∧ ds),

which leads to balance laws. We can also apply the Lie derivative on dx only, so that we have    d ∂f + div (f v)) dx. f dx = Lv˜ (f dx) = ( dt M M M ∂t Thus with f = ρ, the mass conservation equation (see (3.91)) reads ∂ρ + div (ρ v) = 0. ∂t

(3.140)

With f = ρ( 12 v 2 + e), we have the balance of total energy given by (3.118):

3.4 Modelling Hypotheses

383

1 ∂ 1 (ρ( v 2 + e)) + div [vρ( v 2 + e) ∂t 2 2

(3.141)

= [div (τˆ .v − qF )+ < v, f e − ∇Ph > −P div v]. With f = ρe, we can write (3.109) in the form ∂(ρe) + div (ρe v) = tr (σ.) + r0 − div qF . ∂t

(3.142)

For the ENS equation, thanks to the relation ρ

∂v i ∂(ρv i ) dv i = ρ( + v j ∂j v i ) = + ∂j (ρv j v i ), dt ∂t ∂t

we obtain the following divergence expression, with σˆ h = τˆh − Ph : ∂(ρv i ) j,i + ∂j (ρv j v i − σh ) = (f e )i , ∂t

i = 1, 2, 3.

(3.143)

Then (without the entropy) we have five equations (3.140), (3.142), (3.143) with a priori two state equations for the seven unknowns v 1 , v 2 , v 3 , ρ, e, P , T . But information is lacking for specifying the (hydro) pressure Ph . It can be the gradient part of the external force f e in its Hodge decomposition, as in the case of incompressible fluids, or it can be a Lagrange multiplier in convex analysis. Of course we have to specify the mathematical sense of equations (3.129), (3.125), (3.130). Here we say only that from the nonlinearities, the mere framework of distributions does not allow one to give sense to these equations: we have to define products, and then we have to use the “natural” spaces (with finite or locally finite energy for instance). Besides, keep in mind that these equations are written with regularity hypotheses (existence of a unique flow, which allows using Lagrange coordinates as Euler coordinates), and it is not obvious that these equations contain all physical information if we admit discontinuities in the fluid. Writing conservation laws with balance of mass and energy leads to a usual mathematical framework, as we shall see in the next sections.

3.4 Modelling Hypotheses Writing the various conservation equations (specifically the kinetic energy equation) leads to a natural functional framework for various regular problems of fluid mechanics (the word “regular” is used in the sense that we want to follow the fluid particle in its motion). Very generally, the physical system to be studied is only a part of a system in the whole space, which is impossible to model completely, and one difficulty of the mathematical modelling is to represent the action of the

384

3 Fluid Mechanics Modelling

external system on the studied system through limit conditions from the mechanics and thermodynamics points of view. These conditions must be obtained (in the regular case) from the transmission conditions between two parts of the system, with a priori information on the external system. These conditions have to satisfy the following: (i) represent as well as possible the physical properties between the media (these exchanges are listed under the names of wall, “external” forces, mass exchange, heat exchange, . . . ), (ii) be in agreement with the operators with partial derivatives that intervene in the domain, (iii) be in agreement with the proposed functional framework, thanks to trace theorems, (iv) be independent, and if possible, lead to well-posed problems (existence and uniqueness of the solution, continuously depending on inputs). Of course there is a large variety of situations in fluid mechanics. The various systems are characterized as follows (see [Germ]) with respect to exchange of matter. Closed system: if M is the fixed domain of the fluid, there is neither incoming nor outgoing fluid from M, the “quantity of matter” (mass) of the system is constant. The wall is said to be impervious. Open system: it is possible to have an incoming (or outgoing) flux of fluid on a part of the boundary of the domain. For an incoming flux, given on a part 0 of the boundary, we will have the condition v.n| 0 = g < 0 (with n the external normal). A flow for a bounded domain corresponds to a situation with incoming flux of fluid on a part 0 of the boundary, and of outgoing flux of fluid on another part. A system in an infinite domain may be reduced to a flow in a bounded cylindrical domain, taking as side boundary a surface generated by streamlines. Many various situations and usual hypotheses on the physical (and chemical) nature of the fluid with respect to its mechanical behavior will be given.

3.4.1 Conditions on the Fluid Domain Here we consider a category of problems under the following assumptions: Hypothesis 1 The system is closed, the fluid domain M is fixed (time-independent), it is a bounded piece in E3 , and it is equipped with the natural Euclidean metric. We thus model a liquid or a gas contained in a motionless container. Of course, the case in which the fluid domain is time-dependent is also very interesting in applications.

3.4 Modelling Hypotheses

385

3.4.2 Hypotheses on the Fluid Behavior We successively make the hypotheses of thermodynamic fluid behavior, then of mechanical behavior, finally of behavior on the thermodynamics–mechanics fluid relations. Hypothesis 2 Thermodynamic behavior. The thermodynamic evolution of the fluid particle is without phase change, and without chemical reactions. The interesting cases with phase change or with chemical reactions may be very difficult to treat. Moreover, we consider the following situation (see Chapter 1): Hypothesis 3 Thermodynamic behavior. The fluid is described at each point as a system with two degrees of freedom; it satisfies the state equations PT = f (ρ) and e = g(T ), with given functions f, g, and the internal energy is a convex function of 1/ρ. Notice that from these state equations, we can deduce some boundedness properties on P and e, for instance if f, g are positive, then P and e are positive. Moreover, for the mechanical behavior, following the fundamental principle (see Chapter 4), we admit the following condition. Hypothesis 4 Mechanical behavior. There exists a resistance function φ on E = S 2 T ∗ M, invariant under the rotation group, that is convex and homogeneous, so that the stress viscous tensor τˆ and the rate strain tensor ˆ are related26 by τˆ ∈ ∂φ(ˆ ) and ˆ ∈ ∂φ ∗ (τˆ ). Moreover, we assume the following. Hypothesis 5 Mechanical behavior. The fluid is Newtonian, i.e., the resistance function φ(ˆ ) is a quadratic positive function, so that φ(ˆ ) = φλ,μ (ˆ ) =

λ (tr ) ˆ 2 + μ tr (ˆ ◦ ˆ ), 2

(3.144)

with λ, μ coefficients of fluid viscosity such that K = λ + 23 μ ≥ 0, μ ≥ 0. Two types of fairly different situations are generally considered. The first leads to the Navier–Stokes equations: Hypothesis 6 The viscosity coefficients are such that α = inf(3λ + 2μ, 2μ) > 0, i.e., φ(ˆ ) satisfies the (strong) ellipticity condition φ(ˆ ) ≥ α tr (ˆ ◦ ˆ ),

∀ˆ ∈ S 2 T ∗ M.

(3.145)

The second situation leads to the Euler equation and corresponds to the limiting case μ = 0. 26 See

Chapter 4 for this notation.

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3 Fluid Mechanics Modelling

Relations between thermodynamics and mechanical behavior. Thermodynamics acts on the mechanical behavior (on the divergence of the velocity) as a constraint; pressure intervenes as a Lagrange multiplier in the framework of convex analysis (see Chapter 4), but the nature of the pressure is to be specified. The mechanical– thermodynamic interaction is often modeled according to the following (see (3.105)): Hypothesis 7 The Fourier law. The field qF that represents the heat flux and the gradient of temperature are linked, with k > 0 (which can depend on T ), k being a scalar or a positive definite matrix, by qF = −k grad T .

(3.146)

Hypothesis 8 Kinetic condition. There exists a pseudogroup (φt,s ) on M, or a group (φ˜ t ) on M˜ = M × R, of evolution of the fluid at every time. But proving that this is a diffeomorphism group is generally out of reach.

3.4.3 Mechanical Conditions on the Boundary In the framework of Hypotheses 1 and 8, we have φt,s (∂M) = ∂M, and then the velocity vector on the boundary of M must be tangent to ∂M; this gives naturally the condition v.n = 0 on = ∂M. Many boundary conditions are possible, as indicated in Chapter 4. Recall two types of boundary conditions on the velocity. In the case of a rough wall, with a viscosity coefficient μ = 0, the condition of nonslipping at the boundary is implied by the following hypothesis. Hypothesis 9 For a rough surface, the fluid velocity on the boundary ∂M of M is equal to that of the wall (for a moving or fixed domain). Thus the fluid velocity is known on ∂M if the velocity of the wall is given. Likewise, at the entrance of a flow, the fluid velocity v is generally assumed to be known. In the case that the wall is a smooth surface, the fluid can slip along the wall, and the velocity at the boundary must satisfy a condition of Neumann type. More generally, the mechanical action of the outside of the domain on the system may be modeled by known surface forces (but then the (tangent) velocity at the boundary is unknown), which corresponds to the following conditions: Hypothesis 10 The stress constraint σˆ v = τˆv − Ph I at the boundary must be such that σˆ v .n = f e given on = ∂M. In the case of a free boundary (that is time-dependent t = ∂Mt ), the slipping condition of the fluid along the wall is f et = 0. Notice that the surface forces are in fact rarely known despite it being important to determine them.

3.4 Modelling Hypotheses

387

3.4.4 Thermodynamic Boundary Conditions (a) Conditions on thermodynamic equilibrium. By assuming the thermodynamic equilibrium of the system with the outside at each point of the boundary of the domain M, we often write the equality of outside–inside temperatures and also of pressures (that is, of intensive variables; see Chapter 1). If the outside of the domain is occupied by a rigid medium whose thermodynamic freedom is of one degree, for instance temperature, we write the equality of temperatures at the boundary between the fluid and the rigid medium. If the temperature of the rigid medium (called a reservoir) is known, that leads to the condition that the temperature of the fluid is given on the boundary. In order to have the possibility of imposing such conditions on the pressure and on the temperature, it is of course necessary that the traces on the boundary for these functions make sense from the point of view of functional analysis. The space L2 (M) is not smooth enough to allow a trace notion. Other boundary conditions on the pressure are obtained by writing the equality of outside–inside surface forces, and this seems to be the more “natural” condition to impose on P , with the positivity for a gas. (In a transmission problem, with two nonmixing fluids on each part of a surface S, we can have a jump across S of the surface forces, which leads to a surface tension (see Chapter 4).) (b) A given heat flux. In the case of a given flux on the boundary , we have (see, for instance, [D-L19, ch. IA.1.7]) the following hypothesis. Hypothesis 11 Let be a given density of heat flux on the boundary ; the balance of heat flux on is given by − qF .n =

− f e .v, on .

(3.147)

With the Fourier law, the boundary condition is k

∂T | = ∂n

− f e .v.

(3.148)

Note that the condition qF .n = 0 on implies that the heat flux across is null (we say that the wall is adiabatic) and also implies, through the Fourier law, the condition nk ∂T ∂n | = 0. Other thermodynamic conditions may be written in the case of open systems (see, for instance, [Piro8, ch. I.3]), and this leads to problems that may be well posed. Then it is necessary to specify the specific mass on the part of the boundary where the fluid is incoming.

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3.5 “Natural” Functional Framework In a sense we leave here differential geometry for functional analysis, which may be viewed as a weak formulation of differential geometry, that is, we abandon the framework of smooth functions for the framework of Sobolev spaces, with their usual trace properties at the boundary. Yet the difficulty in modelling fluid equations is to choose the framework that agrees with the physicist’s model, and it will be especially difficult to specify a “natural” framework for the pressure.

3.5.1 Global Conditions on the Specific Mass A priori, the specific mass, the (absolute) temperature, and, for gases, the pressure and internal energy are positive quantities. Thus their representations by distributions must be positive measures on M. Moreover, they have to give sense to the state equations of the fluid and to the differential expressions of thermodynamics (if the usual regularity in differential geometry is missing, we can hope to give sense to these equations using convex analysis). We first begin with the natural hypothesis of modelling the specific mass of a fluid. Along with Hypothesis 1, we have the following.  Hypothesis 12 The total mass is finite: m0 = M ρ0 dx < ∞ : ρ0 ∈ L1+ (M). Thanks to mass conservation, this implies  ρt dx = m0 < ∞, thus ρt ∈ L1+ (M), ∀t ∈ J = [0, t0 ], and M

(3.149)

ρ ∈ L∞ (J, L1+ (M)), i.e. sup &ρt &L1 (M) < ∞. t ∈J

The existence of the flow requires that the Jacobian determinant D(φt ) satisfy 0 < c0 ≤ D(φt ) ≤ c1 (with constants c0 and c1 ), then that div v be bounded (see Theorem 1), whence with (3.149), we must have dρ = −ρdiv v ∈ L∞ (R, L1+ (M)) dt and then ρ ∈ C([0, t0 ], L1+ (M)). Moreover, we usually assume the following hypothesis: Hypothesis 13 The specific mass is bounded, ρt ∈ L∞ + (M), ∀t ∈ R, that is, there exists ρ1t > 0, ρt (x) ≤ ρ1t , ∀x ∈ M.

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Furthermore, in order to eliminate domains in M without fluid, and thus to give sense to the thermodynamic variable τ (and to the thermodynamic section ξF ), we make the following hypothesis. Hypothesis 14 There exists a real number ρ 0 > 0 such that ˜ i.e., τ = 1/ρ ∈ L∞ (M). ˜ ρ 0 ≤ ρt (x), ∀(x, t) ∈ M,

(3.150)

The problems with ρ = 0 at the boundary are called “degenerate,” and they are more difficult to deal with. They don’t always require boundary conditions.

3.5.2 Framework for the Physical Model We first specify, thanks to the balance of the main variables of fluid mechanics, a natural functional framework linked to a variational point of view. Then we give various hypotheses in order that this framework can be realized. The natural framework for the velocity field is that of L2 functions, whereas the natural framework for energy and internal energy is the space of L1 functions, with the specific mass satisfying Hypothesis 14. Thus there is a natural mixing of three functional frameworks L1 , L2 , and L∞ . The first requirement is that the total work WM , the total heat QM , and the work of the thermodynamic pressure be finite, to give sense to (3.119), (3.120), thus that < v, ˜ θˆW >,

ρ < v, ˜ θˆQ >

and ρ < v, ˜ ζF∗ (θW ) > ∈ L1 (M × J ).

(3.151)

Smooth Natural Framework for the Velocity Field v˜ The evolution equation of kinetic energy is in a first formal writing27 d dt



 1 2 |vt | ρt dx + < τˆ , ˆ > dx M 2 M    = (σˆ .v).n d + Ph .div vt dx + f e .vt dx,

M

(3.152)

M

with = ∂M. By integrating over time (on an interval J = [0, t]), we obtain the balance of global kinetic energy:

27 With

the time-dependent notation on v and ρ.

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3 Fluid Mechanics Modelling

 K= M



1 |vt |2 ρt dx + 2

+





1 |v0 |2 ρ0 dx 2 M  Ph .div vs dxds + f e .vs dx ds.

< τˆ , ˆ > dxds = M×J



(σˆ .v).n d ds + ×J

M×J

M×J

(3.153) A natural functional framework28 leads us to assume that the left-hand side of relation (3.153) is finite, that is,  |vt |2 ρt dx < ∞, ∀t, (3.154) M

and 

 < τˆ , ˆ > dxdt < ∞, thus M×J

< ˆ , ˆ > dxdt < ∞,

(3.155)

M×J

which implies (in the case of a Newtonian fluid, which is all that we consider) the following: (i) v ∈ L2 (M × J, ρt dx dt)3 , and with Hypothesis 14, that v ∈ L2 (M × J )3 . (ii) the strain tensor is in L2 (M × J )9 , and thus div v ∈ L2 (M × J ), j

i ∈ L2 (M × J ).

(3.156)

Recall that moreover, we must have, at least for the flow, tr ˆ = div v ∈ L∞ (M × J ).

(3.157)

Let H 1 (M) be the usual Sobolev space. By the Korn inequality (see below), we deduce that the section v˜ of the bundle T M˜ over M˜ must be such that v˜ ∈ L2 (J ; H 1(M)3 )∩L∞ (J ; L2 (M)3 ), with div v ∈ L∞ (M ×J ).

(3.158)

Remark 19 The conditions ij ∈ L2 (M × J ), with curl v ∈ L2 (M × J )3 , directly imply that vt is in the space L2 (J, H 1 (M)3 ). Indeed, for every i, j we have ∂vj ∂vj ∂vi 1 ∂vi 1 ∂vi = ( + )+ ( − ), ∂xj 2 ∂xj ∂xi 2 ∂xj ∂xi

(3.159)

thus ∂vi /∂xj ∈ L2 (M × J ), ∀i, j , giving the (natural) regularity result (3.158).

28 At least for a Newtonian fluid and if the velocity is null on the boundary of M (that is, the Dirichlet condition).

3.5 “Natural” Functional Framework

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Note that this functional framework of v for a Newtonian fluid does not allow a discontinuity of v across a surface t in M. ¶ Some “Reasonable” Hypotheses for Obtaining the “Natural” Framework Global initial conditions. A first obvious hypothesis for (3.153) is this one. Hypothesis 15 The global initial kinetic energy in a bounded domain is finite:  |v0 |2 ρ0 dx < ∞, i.e., v0 ∈ L2 (M, ρ0 dx)3 . M

Global conditions on the pressure Ph and the “external” forces. We have first to specify the notion of pressure when there is an external force. Let us consider a motionless fluid with a constant gravitational force as “external” force. Then the ENS equation is reduced to f e − grad Ph = 0,

with f e = −ρge3 ,

with g the gravitational constant, and e3 a unit vector. If ρ is a constant (for an incompressible fluid), then the pressure Ph is given by Ph = P (0) − ρgx 3 . If we assume that the fluid satisfies the van der Waals equation as state equation, that is, (P + aτ −2 )(τ − b) = RT , then does Ph satisfy this equation? We have Ph τ = P (0)τ − gx 3 , and then, with T0 a constant, the van der Waals equation is reduced to −gx 3 (1 − bτ −1 ) = R(T − T0 ), so that the temperature will be increasing with the pressure and with the depth.29 On the other hand, P (0) may be a thermodynamic pressure, so that it satisfies the state equation with τ0 , T0 . Such a situation allows us to visit the Archimedes law. Consider a container full of water with a rigid body in it. Let  , respectively , be the domain of the water, respectively of the body, with boundary  , respectively ,  = ∪ 0 with 0 the boundary of the domain 0 of the container. Then the sum of the forces on the domain is transformed into pressure at the boundary  according to 

 F =



f e dx =

 

grad Ph dx =



Ph n.e3 d  .

29 Of course, the use of the van der Waals equation may be criticized, but the same result will be obtained with any reasonable state equation.

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3 Fluid Mechanics Modelling

Moreover,  









Ph n.e3 d =

Ph n.e3 d 0 − 0 





f dx = −ρge3 | |, e

Ph n.e3 d ,

f e dx = −ρge3 |0 |, 0

with | |, |0 |, the volume of the water and the container respectively. Thus the gravitational force of the water on the rigid body is given by the Archimedes law (so that the pressure Ph has a trace on the boundary)  F =

Ph n.e3 d = −ρg||e3 .

From the ENS equation (3.127), we simply notice that if f e is decomposed (according to the Hodge decomposition) into f e = grad φ + curl w, then we can choose Ph = φ if there is no thermodynamic state equation for Ph , and this is the case of incompressible fluids. Then we have only to consider external forces f e such that div f e = 0. But in the general case, the physical modelling of the pressure Ph in fluid mechanics seems to be an open question. On “external” forces. The main point is the (Hodge) decomposition of the “external” force f e into f e = grad φ + curl w. This Hodge decomposition of f e is well known if f e satisfies the following hypothesis. Hypothesis 16 f e ∈ L2 (M × J )3 . This hypothesis on every finite time interval J implies f e ∈ L2loc (R, L2 (M)3 ). In fact, Hypothesis 16 may be broadened, a main point being that the integral of f e .v or more generally that < v, f e > is finite, so that from a mathematical point of view, the space of the “external” forces will be the dual space of the velocities (see [Lio1] for incompressible fluids). Of course the L2 norms of the external force and of the pressure have no obvious physical sense, but it is a useful framework and often sufficient. If there are “external” surface forces, the “natural” hypotheses on these forces are not obvious.30 On the Pressures Ph and P . The regularity condition on Ph is linked with the smoothness of f e : if f e ∈ L2 (M × J )3 , then we can take Ph ∈ L2 (J, H 1 (M)). A simple hypothesis from a mathematical point of view that leads to a “regular” framework for the thermodynamic pressure P is the following. Hypothesis 17 The pressure P is such that P ∈ L2 (M × J ).

30 Their

frameworks may be specified thanks to the trace theorems.

3.5 “Natural” Functional Framework

393

 With the hypothesis div v ∈ L∞ (M × J ), in order that M×J P .div vs dxds be finite, and in agreement with the internal energy,31 we are led to the following hypothesis. Hypothesis 18 The pressure is such that P ∈ L1 (M × J ). In the case of gases, we recall that the thermodynamic pressure P must be positive, as implied by the state equation. But this condition does not imply that the pressure has a sense at the boundary. In order to give sense to the trace of P , we have to assume a regularity condition such as the following. Hypothesis 19 grad P ∈ L1 (J, L1 (M)3 ), or even P ∈ L2 (J, H 1 (M)). But Hypothesis 19 is not compatible with a jump of P (t) across a surface t in M.

3.5.3 A Priori Estimates from the Previous Hypotheses We assume that the fluid is Newtonian, following Hypothesis 5. With the previous hypotheses on v0 , f e , Ph , v , with32 v = 0, see Hypotheses 15, 16, 9, respectively, we will prove that the “natural” framework for (3.153) is satisfied in the “regular” case, with f e and Ph ∈ L2loc (R, L2 (M)). Notice that the hypothesis grad Ph ∈ L2loc (R, L2 (M)3 ) would be simpler for proving this result. Proposition 5 With the previous hypotheses with Ph and f e ∈ L2loc (R, L2 (M)), with the Dirichlet condition, every field v ∈ L2 (M × J )3 that satisfies (3.153) is in L2 (J, H01 (M)3 ), even in L∞ (J, H01 (M)3 ), for every J = (0, t). Proof Let  φ(t) = M

1 |vt |2 ρt dx, 2

 C0 = M

1 |v0 |2 ρ0 dx, 2

 &u&2 =

|u|2 dxdt. M×J

With the usual inequality ab ≤ 12 (ηa 2 + η1 b2) for all η > 0 and Hypothesis 6, the relation (3.153) gives  φ(t)+α M×J

1 η &Ph &2 + &div v &2 + < ˆ , ˆ > dxds ≤ 2η 2

 f e .vs dx ds+C0 . M×J

(3.160) Now we have the following lemma, which is easily proved by diagonalizing: Lemma 2 We have the following inequality for every symmetric matrix v on R 3 : &div v&2 = &tr v &2 ≤ 3 &tr (v .v & = 3 &v &2 .

(3.161)

instance with the state equation P = γρe. order to simplify; without this hypothesis, we would have v ∈ L2 (J, H 1/2 ( )), and similar results.

31 For 32 In

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3 Fluid Mechanics Modelling

Then from (3.160), we have φ(t) + (α − With the choice α −

1 3η &Ph &2 + ) &v &2 ≤ 2 2η

3η 2

 f e .vs dx ds + C0 . M×J

> 0 (for instance 3η = α) we obtain 

φ(t) ≤

f e .vs dx ds + M×J

1 &Ph &2 + C0 . 2η

Moreover, for every η1 > 0, 

1 2η1

f e .vs dx ds ≤≤ M×J

 M×J

1 η1 |f e |2 dx ds + ρ 2

 |v|2 ρdx ds. M×J

Thus  φ(t) ≤ η1

φ(s)ds + C, with C = J

1 2η1

 M×J

1 1 &Ph &2 + C0 . |f e |2 dx ds + ρ 2η

From the Gronwall lemma (see [D-L19, ch. XVIII.5.2.2]) we obtain φ(t) ≤ C exp(η1 t). Then from (3.153), we deduce that From (3.152), we have

 M×J

1 d φ(t) ≤ η1 φ(t) + dt 2η1



< ˆ , ˆ > dxds ≤ ∞.

1 1 |f e |2 dx + ρ 2η M

 |Ph |2 dx. M

Then from (3.152), we obtain 1 3η ) &v &2 ≤ (α − 2 2η



 |Ph | dx +

|f e .v|dx + |

2

M

M

d φ(t)|, dt

and thus with 3η = α, we obtain, for every t ∈ J , α 3 &v &2 ≤ 2 α

 |Ph |2 dx + M

1 η1



1 |f e |2 dx + η1 φ(t). ρ M

Thus for every t ∈ J , we have &v &2 ≤ ∞. Since &vt & ≤ ∞, both v and v are in L2 (J, L2 (M)); as a consequence of the Korn inequality (see [D-L19, ch. VII.2, Thm. 2]) with a smooth boundary, for instance C 2 by pieces, we deduce the proposition. ¶ From these formulas, we see that the L2 norm of  on M × J and the norm of vt on M are controlled by the sum of the L2 norm on M × J of f e , Ph , and the

3.5 “Natural” Functional Framework

395

L2 norm on M of v0 . Likewise, we control the norm of v in L2 (J, H01 (M)3 ) and as well the L2 norm on M × J of div v and curl v. Furthermore, we have to specify the sense of the derivative dv dt (and of the boundary conditions if v = 0). With the Dirichlet condition, from v ∈ L2 (J, (H01 (M))3 ), we deduce by the Sobolev inclusion that v ∈ L2 (J, (Lp (M))3 ), 2 ≤ p ≤ 6, so that we obtain v.∇v ∈ L1 (J, (Lq (M))3 ), 1 ≤ q ≤ 32 , and thus is in L1 (J, H −1 (M)3 ). Then we have to give sense to the initial condition v0 . With the condition Ph ∈ L2 (M × J ), the “divergence” of σˆ is in L2 (J ; H −1(M)3 ) (with H −1 (M) the dual 1 −1 3 space of H01 (M)). Then from the ENS equation, we have ∂v ∂t ∈ L (J ; H (M) ); this implies v ∈ C([0, t0 ], H −1 (M)3 ). This gives sense to the initial condition on the velocity, v0 ∈ H −1 (M)3 , so that the “natural” condition v0 ∈ L2 (M)3 is allowed.

3.5.4 “Natural” Framework for the Thermodynamic Field A natural hypothesis is that the global heat (see (3.104), (3.151)) is finite: < v, ˜ θˆQ > ∈ L1 (M × J ).

(3.162)

Thus with the “natural” hypotheses on the strain tensor, this is equivalent to the total heat flux is finite, that corresponds to: div qF = −div k grad T ∈ L1 (M × J ).

(3.163)

This implies ρ ddteˆ ∈ L1 (M × J ) (see (3.109)). Since the measure ρdx is invariant d under the flow, we have dt (eρdx) ˆ = ddteˆ ρdx, and thus by integrating over time, 1 we also have ρ eˆ ∈ L (M × J ). With the properties of ρ, we obtain the following “natural” thermodynamic hypotheses: eˆ ∈ L1 (M × J ),

d eˆ ∈ L1 (M × J ). dt

(3.164)

Thus eˆ ∈ C([0, t0 ], L1 (M)), so that the initial condition e(0) ˆ ∈ L1 (M) is allowed.  It is natural to admit that the flux qF is such that M×J |qF | dxds is finite, and thus qF ∈ L1 (M × J )3 , so that with (3.163), qF has trace n.qF on the boundary of M in the space L1 (J, W ) with W the dual of the Lipschitz functions on ∂M.

3.5.5 Consequences of This Thermodynamic Framework These conditions (3.164) give a sense to the following balances of internal power, and of internal energy by integrating (see (3.121))

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3 Fluid Mechanics Modelling

d dt



 eˆt ρt dx = M

[< σˆ , ˆ > −div qF ] dx,

(3.165)

M

and 





eρ ˆ dx − M

eˆ0 ρ0 dx = M

(< σˆ , ˆ > −div qF ) dx ds,

(3.166)

M×J

 with the “natural” hypothesis that M eˆ0 ρ0 dx < ∞, that is, the initial internal energy is finite on M, and with the previous hypotheses on the stress and the strain tensor fields. With the state equation e = cv T , with cv a constant, the temperature T must be in the same framework as the internal energy, which must be positive in this case. Then with given stress and strain tensors, thus with given < σˆ , ˆ >, the internal energy or the temperature must satisfy the heat equation (3.123), which is a well-known33 parabolic equation in the framework (3.164). The trace of the temperature on the boundary ∂M × J makes sense, since T and qF (thus grad T ) are in L1 (M × J ). Moreover, T ∈ C([0, t0], L1 (M)), and thus T (0) ∈ L1 (M). In order to apply the maximum parabolic principle to prove that the temperature is always positive, we have to require (i) regularity results both on the given velocity field (at least continuous), and on the temperature (usually C 2,1 in space and time); (ii) r0 + tr (σ ) ≥ 0. With r0 = 0, this is equivalent to tr (τ ) − P tr  ≥ 0, and this condition is satisfied if P div v ≤ 3λ(tr )2 , whence |P | ≤ 3λ|div v|, which implies P ∈ L∞ (M × J ).

3.5.6 Evolution with a Surface Discontinuity Wave Shock Problem in a Model of a Newtonian Fluid The functional framework L2 (J ; H 1(M)3 ) (or L2 (J ; H01 (M)3 ) relative to the velocity field is incompatible with the possibility of discontinuity of this field across a surface  in M (or in M × J ). This is related to the parabolic nature of the Navier–Stokes equation (with respect to the velocities). Recall that the choice of this functional framework is directly linked to the behavioral laws of the fluid (notably, for a Newtonian fluid, this choice is a consequence of the Korn inequality). (We can remark that if v admits a discontinuity across a surface  (with limits on each part of ), then there appears, by writing the Navier–Stokes equation in the sense of distributions, a distribution of order 1, concentrated on , due to the “divergence of the discontinuity of the stress tensor” across , i.e.,

33 Especially

in stochastic theory (see, for instance, [Par]).

3.5 “Natural” Functional Framework

397

∂j (λ[n.v] δ δij + μ[nj vi + ni vj ] δ ), and this term cannot be eliminated by any other term.) In contrast, for an ideal fluid for which the viscosity coefficient μ is null, the evolution equation of velocities is a nonlinear hyperbolic equation, and it is possible to have a discontinuous velocity field, which leads us to abandon the framework of finite total kinetic energy of Hypothesis 15. Then we have the possibility of modelling a shock wave; this leads to working with more general functional spaces, such as spaces of functions of bounded variation.

General Framework We want to study the evolution of a fluid without viscosity with discontinuities of ˜ for instance due to a shock wave in some variables across a given surface  in M, the fluid. We specify some notions in a general setting. Consider a system of equations in the divergence form ∂u  ∂ j + f (u) = 0, or g given, ∂t ∂x j ∂uk  ∂ j + f (u) = 0, or gk , k = 1 · · · d, ∂t ∂x j k

(3.167)

with the unknown u(x, t) ∈ R N , x ∈ R d ,34 with smooth functions f j from R N to R N (and g(x, t) ∈ R N ). We assume that u and f j (u), j = 1, . . . , d, are such that we can write the system in the sense of distributions in R d × R, and that we can give a sense to the different traces afterward. We assume that u is discontinuous across a surface  ⊂ R d × R with limits on each side of the surface.35 We use essentially the formula that expresses the divergence of a vector field w in the sense of distributions, denoted by divx,t w, as a function of its classical expression denoted by (divx,t w), and with the jump [w.ν] of w.ν across , where ν is a normal to : divx,t w = (divx,t w) − [w.ν] δ . We assume that the surface  is (globally) given by the equation  = {(x, t), ϕ(x, t) = 0},

34 We

could also consider a domain  in R d , bounded or not. that this is possible notably with u ∈ BV (R d )N the space of bounded variations.

35 Note

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3 Fluid Mechanics Modelling

with the function ϕ (of class C 1 ) such that dϕ(x, t) = 0, ∀(x, t).36 Let t ⊂ R d be such that t = {x ∈ R d , ϕ(x, t) = 0}.37 Every tangent vector to , w˜ = (w, w0 ) = ((wj ), w0 ) is such that < w, ˜ dϕ > =



wj

∂ϕ ∂ϕ = 0. + w0 ∂x j ∂t

,

n=

Thus ker dϕ = T . Let σ =

∂ϕ ∂t

|gradx ϕ|

gradx ϕ . |gradx ϕ|

(3.168)

The vector v˜d = (vd , 1) = (−σ n, 1) is tangent to , whereas (n, σ ) is orthogonal to ; the vector vd is transverse to t . Let (x(t)), t ∈ J, be such that ϕ(x(t), t) = 0. The corresponding velocity vector dx dt satisfies  dx j ∂ϕ d ∂ϕ ϕ(x(t), t) = = 0, + dt dt ∂x j ∂t

(3.169)

and there exists λ ∈ R such that λ

 ∂ϕ ∂ϕ ( j )2 + = 0, ∂x ∂t

which gives λ dx dt = −σ n = vd . Thus the equation dx (t) = vd (x(t), t), dt

with x(s) = xs

for t ≥ s defines a curve xs ∈ s → x(t) ∈ t , t ≥ s, giving the trajectory of the discontinuities, and vd is the velocity vector of propagation of discontinuities. The existence of a lifetime interval for this equation indicates the possibility that discontinuities appear or disappear. We denote by [u] (respectively [f j (u)]) the jump of u (respectively f j (u)) across . The system of equations (taken in the sense of distributions) implies the system of relations [u]

36 Note

∂ϕ ∂ϕ  j + [f (u)] j = 0. ∂t ∂x

(3.170)

that this condition implies that the surface  is orientable. that sometimes we assume that ϕ(x, t) = 0 is of the form ψ(x) + t = 0, which implies that t ∩ s is empty if t = s. 37 Note

3.5 “Natural” Functional Framework

399

These relations may be considered equations of propagation of discontinuities with unknowns ϕ or , and the jumps [uα ] of uα across . At first we deduce relations between jumps.

Jump Relations for a Perfect Fluid Consider a nonviscous fluid (thus τˆ = 0), with k = 0, thus without heat flux: qF = 0, with r0 = 0. The fluid mechanics equations (3.91), (3.109), (3.143), (3.142) may be written in divergence form (a priori in R 3 ), summing on repeated indices: (i)

∂ρ + ∂j (ρ v j ) = 0, ∂t

∂(ρv i ) + ∂j (ρv j v i ) + δ ik ∂k P = (f e )i , i = 1, 2, 3, ∂t ∂ (iii) (ρe) + ∂j (ρv j e) + ∂j (P v j ) = f e .v. ∂t (ii)

(3.171)

Note that in these equations, we have identified the pressures P and Ph , and in the third equation, we have written f e .v = (∇P ).v. d (ρu) + ρ div u = ρ du Proof Indeed, we have dt dt with u = v for (ii) and u = e for (iii). Moreover, we have div (σ v) = − div (P v), since the tensor τˆ is null. ¶

We assume that the fluid evolution (both for the mechanical and thermodynamic evolution) is smooth on each side of the surface , corresponding to a shock wave that flows across the fluid, so that the sections v˜ and ξF of the bundle E = T M˜ ×M˜ FM have a jump across . In (3.167), let u0 = ρ, uj = ρv j , j = 1, 2, 3, u4 = ρe. Then f j,0 = ρv j , f j,k = ρv j v k + P δ j k , f j 4 = (ρe + P )v j , j, k = 1, 2, 3. Thus we obtain38 ∂ϕ  j ∂ϕ + [ρv ] j = 0, ∂t ∂x  ∂ϕ ∂ϕ [ρv i ] + [ρv i v j + P δ ij ] j = 0, ∂t ∂x ∂ϕ  ∂ϕ [ρe] + [(e + P τ )ρv j ] j = 0. ∂t ∂x [ρ]

(3.172)

grad P = (grad P )−[P ]nδ , with grad P taken in the sense of distributions, (grad P ) in the classical sense, and n the unit normal to .

38 We use the formula

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3 Fluid Mechanics Modelling

∂ϕ It is a system of five equations, linear with respect to ∂ϕ ∂t , ∂x j , j = 1, 2, 3. So that there are nontrivial solutions, we have to write the compatibility conditions. Usually we remark that there are no tangential discontinuities of the velocity field. Let v be the velocity of the fluid particle, n the normal to t , following (3.168), then vn = v.n their inner product. Then (3.172) is equivalent to

(i) σ [ρ] + [ρvn ] = 0, (ii) σ [ρvn ] + [ρvn2 + P ] = 0,

(3.173)

(iii) σ [ρe] + [(e + P τ )ρvn ] = 0. Let vd be the velocity of propagation of the fluid discontinuities. Then let v˜n = vn + σ = (v − vd ).n. Thus v˜n is the normal relative velocity of the fluid with respect to the velocity of propagation of . Let w˜ = e + P τ be the enthalpy. Then (3.173) becomes (i) [ρ v˜n ] = 0, (ii) [ρ v˜n2 + P ] = 0,

(3.174)

1 (iii) [w˜ + v˜n2 ] = 0, if ρ v˜n = 0. 2 Proof of (3.174)(ii) We have [ρvn2 ] + σ [ρvn ] = [ρvn v˜n ] = [ρ v˜n2 ] with (3.174)(i). Proof of iii) We observe that (3.173)(iii) is [(e + P τ )ρ v˜n ] − σ [P ] = 0, and thus with (3.173)(i), [ e + P τ + 12 (v˜n − σ )2 ρ v˜n ] − σ [P ] = 0. We then deduce 1 1 [((w˜ + v˜n2 ) − σ v˜n + σ 2 )ρ v˜n ] − σ [P ] = 0. 2 2 But σ [ρ v˜n2 + P ] = 0, σ 2 [ρ v˜n ] = 0 by (i) and (ii), and thus (iii) if ρ v˜n = 0.

¶

The relations (3.174) appear as conditions on the possible jumps. We can have several discontinuity types, first a discontinuity of specific mass ρ, but not of velocity vn (this is the model of the evolution of two nonmixing fluids). We have vn = −σ = vd through the relation (3.173)(i) (the velocity of the fluids at  is that of the discontinuity), and (ii) gives [P ] = 0, i.e., the pressure has no jump across . Finally, condition (iii) is trivially satisfied. The more interesting case is that of the velocity vn , corresponding to a shock wave. The relations (3.173) imply the discontinuities of thermodynamic variables, notably of the specific mass ρ.

3.5 “Natural” Functional Framework

401

From the relations (3.174), we can eliminate the fluid velocity, and we obtain a compatibility relation, the Hugoniot relation, where only thermodynamic variables intervene.

Hugoniot Relation We assign the subscript 1 or 2 to the variables on each part of . Let M = ρ1 v˜n1 = ρ2 v˜n2 ,

(3.175)

which corresponds to a fluid mass flux; thus v˜n1 and v˜n2 have the same sign. Then v˜n1 = Mτ1 ,

v˜n2 = Mτ2 ,

hence M[τ ] = [v˜n ].

(3.176)

hence M 2 [τ ] = −[P ].

(3.177)

Then (3.174)(ii) implies τ1 M 2 + P1 = τ2 M 2 + P2 , We deduce 2 2 − v˜n1 = M 2 (τ22 − τ12 ) = M 2 (τ2 − τ1 )(τ2 + τ1 ) = −[P ](τ2 + τ1 ). [v˜n2 ] = v˜n2

Let τm = 12 (τ2 +τ1 ), Pm = 12 (P2 +P1 ). Thus we have [ 12 v˜n2 ] = −[P ]τm . Moreover, through the relation [P τ ] = [P ]τm + Pm [τ ], (3.174)(ii) implies 1 [e + P τ + v˜n2 ] = [e] + [P τ ] − [P ]τm = [e] + Pm [τ ] = 0. 2 Thus we have obtained the Hugoniot relation, which links the jump of internal energy to the jump of the inverse of the specific mass, 1 (e2 − e1 ) + (P2 + P1 )(τ2 − τ1 ) = 0, 2

(3.178)

which may be a way to write the relation [ξF∗ (θQ )] = [ξF∗ (de + P dτ )] = 0. Inequalities Across

We choose to orient the surface  t according to the vector vd (if vd = 0). Let D2 be the domain that contains vd (in a neighborhood of (x, t)), D1 its complement domain. The surface  t and thus  are divided into three independent parts according to the sign of the product vd .v˜n :

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3 Fluid Mechanics Modelling

 t,− = {x ∈  t , vd .v˜n < 0},

 t,+ = {x ∈  t , vd .v˜n > 0},

 t,0 = {x ∈  t , vd .v˜n = 0}.

(3.179)

On  t,+ the normal velocity of the fluid is greater than that of the shock wave. If x ∈  t,+ , we have M > 0, whereas if x ∈  t,− , we have M < 0. We denote by the subscript 2 (respectively 1) the variables on side D2 (respectively D1 ) of  t . If M > 0, then if x ∈  t,+ , we have, by (3.176) and (3.177), the equivalences v˜n2 < v˜n1 ⇔ τ2 < τ1 ⇔ ρ2 > ρ1 ⇔ P2 > P1 .

(3.180)

For the fluid particle, the evolution before and after the shock is adiabatic and occurs with constant entropy; the crossing of the shock wave must be related to a jump of (irreversible) entropy. Following the second law, we must have s0,2 ≥ s0,1 if x ∈  t,+ , and s0,2 ≤ sO,1 if x ∈  t,− . In the case of an ideal gas, we have P2 > P1 ⇔ s2 > s1 . Proof We substitute the internal energy using the state equation e = Hugoniot relation. We obtain 2 (P2 τ2 − P1 τ1 ) = (P1 + P2 )(τ1 − τ2 ). γa − 1

(3.181) 1 γa −1 P τ

in the

(3.182)

Let Pm = 12 (P1 + P2 ), τm = 12 (τ1 + τ2 ); we have 1 (Pm [τ ] + [P ]τm ) + Pm [τ ] = 0, γa − 1 thus γa Pm [τ ] + [P ]τm = 0, or γa Let Pr =

P2 P1 ,

τr =

τ2 τ1 .

[τ ] [P ] + = 0. τm Pm

(3.183)

We divide (3.183) by P1 τ1 , and we obtain γa

τr − 1 Pr − 1 + = 0. τr + 1 Pr + 1

(3.184)

3.5 “Natural” Functional Framework

403

Thus τr =

Pr + μ μPr + 1

with μ =

γa + 1 . γa − 1

(3.185)

Moreover, the reversible evolution of entropy is expressed by srev = log e + (γa − 1) log τ = − log(γa − 1) + log(P τ γa ). cv We apply this to each side of  t,+ . Taking the difference, we obtain γ

1 P2 τ2 a def 1 1/γ [srev ] = (srev,2 − srev,1 ) = log( ) = γa log(τr Pr a ). cv cv P1 τ1γa Let ψ(Pr ) =

Pr +μ 1/γa . μPr +1 Pr

(3.186)

Then we have [srev ] = cv γa log ψ(Pr ).

(3.187)

Furthermore, ψ  (Pr ) ≥ 0, for Pr > 0, and thus ψ is an increasing function such that ψ(1) = 1, whence [srev ] is positive when Pr ≥ 1 and negative when Pr ≤ 1. We then deduce the jump of entropy across  t,+ , [s] = s2 − s1 = (srev,2 + s02 ) − (srev,1 + s01 ) = [srev ] + (s02 − s01 ) > 0, since we must have (s02 − s01 ) ≥ 0. Thus we have proved that if x ∈  t,+ , then s2 > s1 , P2 > P1 , τ2 < τ1 thus ρ2 > ρ1 , e2 < e1 , and v˜n2 < v˜n1 .

¶

(3.188)

Furthermore, with (3.176) and (3.176), we have ' [v˜n ] = M[τ ] =

( ( P2 − P1 (τ2 − τ1 ) = − (P2 − P1 )(τ1 − τ2 ) = − −[P ][τ ]. τ1 − τ2

But observe that we did not prove that the motion of the fluid particle before and after the crossing of the shock wave is on different integral manifolds such that s02 − s01 = 0. Now we prove this. With vˆ = ζ∗F v, ˜ we have < v, ˆ ρθ s > = < v, ˆ ρds > + < v, ˆ

ρ (de + P dτ ) > . T

But by (3.109), we have < v, ˆ ρ(de + P dτ ) > = 0, and thus < v, ˆ ρds > = ρ

ds d dρ d = (ρs) − s = (ρs) + ρs div v. dt dt dt dt

404

3 Fluid Mechanics Modelling

Therefore, < v, ˆ ρθ s > =< v, ˆ ρds > =

∂ (ρs) + div (ρs v). ∂t

The reversibility condition < v, ˆ ρθ s > = 0 gives the jump condition [sρ vˆn ] = 0.  0, a contradiction with (3.187), which But [ρ vˆn ] = 0, which implies [s] = 0 if M = thus proves the θ -irreversibility of shocks. ¶ We study the relation of the shock with the velocity of the sound, still for an ideal gas. Let us prove that for x ∈  t,+ , we have the inequalities ∂P P2 − P1 ∂P (τ2 , s2 ) < (τ1 , s1 ). < ∂τ τ2 − τ1 ∂τ

(3.189)

Through the relation (3.185) we have P2 μ − τr , = Pr = P1 μτr − 1 and

∂P ∂τ

gives dPr 1 − μ2 = . dτr (μτr − 1)2

For τr = 1, we have

dPr dτr (1)

= − μ+1 μ−1 . Thus the relation (3.189) gives

μ+1 1 − μ2 Pr − 1 μτr − 1 > 0, μ1 < τr < 1. Let ci be the velocity of sound in the domain of the subscript i; then we deduce 2 2 ρ12 c12 < M 2 = ρ12 v˜n1 = ρ22 v˜n2 < ρ22 c22 ,

3.5 “Natural” Functional Framework

405

and thus finally c1 < |vn1 − vd |,

|vn2 − vd | < c2 with x ∈  t,− .

(3.192)

Therefore, with x ∈  t,+ , the relative velocity of the fluid with respect to the shock wave is supersonic for the forward front (for D2 ), and subsonic for the backward front (for D1 ). This is generalized from the case of ideal gases with convexity conditions (see, for instance, [God-Rav]).

Chapter 4

Behavior Laws

Behavior laws of fluid mechanics. Force–velocity relations. The “mechanics and thermodynamics behavioral laws” of a considered fluid will complete the equations of fluid mechanics, in order to determine: the mechanical state of the fluid, through the velocity field v and the rate strain tensor  (giving the fluid evolution by the pseudogroup φt,s ), the specific mass ρ and the pressure p, the thermodynamic state of the fluid, given by (τ, s, e, P , T ) (with τ = 1/ρ). A question is to determine whether the thermodynamic pressure P is identical to the pressure p. These equations show the coupling between the mechanical bundle (the tangent bundle T M, or the phase space T ∗ M, or the bundle of symmetric covariant tensors of degree 2, S 2 M = S 2 T ∗ M, for ) and the thermodynamic bundle. They express the following ideas, under the fundamental hypothesis that the velocity field generates a global flow (which is necessary for the use of Lagrange variables): • Conservation (of the mass) of the fluid, essential link between mechanics and thermodynamics; • for mechanics, balance of the kinetic energy and balance of momenta (or conservation equations of kinetic energy and of momenta); • for thermodynamics, balance of internal energy. The balance of several variables will be successively given, with the various behavioral laws of fluids. In rigid-medium mechanics, the (mechanical) behavioral law of a medium is a relation between the stress tensor and the strain tensor, which is specific to the considered medium. In the framework of fluid mechanics, the behavioral law is a relation between the stress tensor and the rate of strain tensor, which is specific to the considered fluid. There are many ways to introduce such laws. Here we will do this, following [Mor1] in a fairly general case, from the framework and the notion of resistance, thanks to convex analysis properties. The very powerful method leads © Springer International Publishing AG, part of Springer Nature 2018 M. Cessenat, Mathematical Modelling of Physical Systems, https://doi.org/10.1007/978-3-319-94758-7_4

407

408

4 Behavior Laws

to a direct reliance on forces and velocity from a behavioral law. For these relations that are both local and global, it is convenient to specify the boundary conditions. Numerous examples will be given in the framework of Newtonian fluids. The force and velocity relations then allow us to write the fluid mechanics equations directly. We first briefly recall several properties of convex analysis, and give the Fenchel theorem (in a generalized form, following [Aub]), which is the foundation of all optimization problems that we consider here, and notably the problems with constraint, the thermodynamic variables acting as constraints with respect to the fluid mechanics.

4.1 Some Review of Convex Analysis 4.1.1 Basic Definitions Let E and E ∗ be locally convex vector spaces that are in duality by a bilinear form, denoted by < ., . >, and let φ be a function from E into R ∪ {+∞}. With the applications in view, E will be a Hilbert space, which will not necessarily be identified with its dual. Definition 1 Gâteaux differential. We call the limit (if its exists) φ  (u; v) = lim

λ→0+

φ(u + λv) − φ(u) λ

the directional derivative of φ at u in the direction v. Furthermore, if there exists u∗ ∈ E ∗ such that φ  (u; v) = < v, u∗ >, ∀v ∈ E then we say that φ is Gâteaux differentiable at u, and u∗ is the Gâteaux differential of φ at u, and it is denoted by Dφ(u) (we also say that Dφ(u) is the gradient of φ at u). Definition 2 Subgradient and subdifferential. We say that f ∈ E ∗ is a subgradient of φ at v ∈ E if φ(v) is finite and if < u − v, f > +φ(v) ≤ φ(u),

∀u ∈ E.

(4.1)

The set of subgradients of φ at v is called the subdifferential of φ at v, and denoted by ∂φ(v). Note that this set may be empty. The subdifferential notion is a generalization of the differential notion for convex functions: a convex function φ that is Gâteaux differentiable at u admits a subdifferential at u such that ∂φ(u) = {Dφ(u)}. Recall the following notions.

4.1 Some Review of Convex Analysis

409

Definition 3 A function φ from a metric space E into R¯ = R ∪ {+∞, −∞} is called lower semicontinuous (l.s.c.) at x0 if φ(x0 ) ≤ lim inf φ(x), with lim inf φ(x) = sup x→x0

x→x0

inf

η>0 x∈B(x0 ,η)

φ(x),

where B(x0 , η) is the ball with center x0 and radius η. A function φ from E into R ∪ {+∞} is called strict (or proper) if there exists v in E such that φ(v) is finite (its domain is nonempty). Definition 4 Let φ be a strict function from E to R ∪ {+∞} . Then the Fenchel conjugate function φ ∗ of φ is the function from E ∗ into R ∪ {+∞} defined by φ ∗ (f ) = sup (< u, f > −φ(u)) ,

(4.2)

u∈E

and its biconjugate is the function φ ∗∗ from E into R ∪ {+∞} defined by φ ∗∗ (u) = sup

f ∈E ∗

  < u, f > −φ ∗ (f ) .

(4.3)

Let us assume the following hypothesis. Hypothesis 1 The function φ is strict and convex l.s.c. Then we can inverse the relation f ∈ ∂φ(v)

(4.4)

thanks to the conjugate function φ ∗ ; the relation (4.4) is also equivalent to v ∈ ∂φ ∗ (f ), and also to

φ(v) + φ ∗ (f )− < v, f > = 0.

There are many equivalent ways to denote the same thing. Let φf (v) = φ(v)− < f, v > . Then (4.4) and (4.5) may be given by 0 ∈ ∂φf (v) ⇔ v ∈ ∂φf∗ (0) ⇔ φf (v) + φf∗ (0) = 0. Indeed, we have φf∗ (p) = φ ∗ (p + f ), and φf (v) + φf∗ (0) = φ(v)− < f, v > +φ ∗ (f ) = 0.

(4.5)

410

4 Behavior Laws

Notice also the equivalence 0 ∈ ∂φf (v) ⇔ φf (v) ≤ φf (w), ∀w ∈ E ⇔ φf (v) = inf(φ(w)− < f, w >). w

With the supplementary hypotheses on φ, Hypothesis 2 φ(0) = 0,

0 ∈ ∂φ(0),

we have φ ∗ (0) = supv∈E (−φ(v)) = −φ(0) = 0, and thus φ ∗ and φ are positive functions. Indicator functions They are special functions that are of great importance in convex analysis. The indicator function χK of the set K is defined by χK (v) = 0, if x ∈ K, χK (v) = +∞ otherwise.

(4.6)

We call the conjugate function of χK the support function of K, denoted by σK = (χK )∗ , and thus σK (p) = supv∈K < p, v > . If K is a convex set, we have   ∂χK (v) = p ∈ E ∗ , < p, v > = σK (p) .

(4.7)

Let v ∈ K. Then ∂χK (v) = NK (v) is called the normal cone to K at v. If K is a closed convex cone, then   ∂χK (v) = p ∈ E ∗ , < p, w − v > ≤ 0, ∀w ∈ K ,

(4.8)

  K − = p ∈ E ∗ , < p, v > ≤ 0, ∀v ∈ K

(4.9)

and the set

is called the negative polar cone of K.

¶

Example 11 Some examples of special conjugate functions (i) Constant function: φ(v) = C, then φ ∗ (f ) = +∞, ∀f = 0, φ ∗ (0) = −C, and thus ∂φ ∗ (0) = E. (ii) Linear function: φ(v) = < p, v > . We have φ ∗ (f ) = 0 if f = p, and then φ ∗ (f ) = +∞ if f = p, and thus φ ∗ = χ{p} , the indicator function of the set reduced to p. (iii) a convex nondifferentiable function: φ(u) = |u| , u ∈ R. With the change of notation f = p and u = x, calculation of φ ∗ (p) gives φ ∗ (p) = sup (x.p − |x|) = 0 if p ∈ [−1, +1] , φ ∗ (p) = +∞ otherwise, x∈R

whence φ ∗ (p) = χ[−1,+1] (p), the indicator function of the interval [−1, +1]. Then ∂φ(x) and ∂φ ∗ (p) are expressed by

4.1 Some Review of Convex Analysis

411

(i) ∂φ(x) = sign0 x, that is: ∂φ(x) = {−1} if x < 0, ∂φ(x) = {+1} if x > 0, ∂φ(x) = [−1, +1] if x = 0, (ii) ∂φ ∗ (p) = {0} if p ∈ ]−1, +1[ , ∂φ ∗ (p) = ]−∞, 0] if p = −1, ∂φ ∗ (p) = [0, +∞[ if p = +1. We verify that φ(x) + φ ∗ (p) = p.x, since |x| = (sign0 x).x.

¶

The case in which the function φ is a positive quadratic form boils down to the following example. Example 12 Standard example. Let E be a real Hilbert space with scalar product (, ) and norm &.&, and let φ be the function from E given by φ(u) = &u&2 /2; here E ∗ is identified with the space E. We have φ ∗ (f ) = sup ((u, f ) − u∈E

1 1 1 &u&2 ) = &f &2 − inf (&u − f &2 ), 2 2 2 u∈E

and then the supremum is reached with u = f , and we get φ ∗ (f ) = &f &2 /2 = (u, f ) − φ(u).

(4.10)

We deduce the case φ(u) = (Au, u)/2 with a symmetric continuous coercive operator A on E, i.e., there exist positive constants c0 , c1 such that c0 (u, u) ≤ (Au, u) ≤ c1 (u, u), ∀u ∈ E. We have φ ∗ (f ) = (A−1 f, f )/2; then f ∈ ∂φ(u), u ∈ ∂φ ∗ (f ), that is, f = Au, and u = A−1 f. ¶ Example 13 Let V ⊂ H ⊂ V ∗ be a variational framework, that is, V , H, V ∗ are Hilbert spaces, with V ∗ the dual space of V , with &u&V ≥ &u&H , ∀u ∈ V , with V dense in H . Thus the natural injection of V in H (and then of H in V ∗ ) is continuous. Let φ(u) = 12 (u, u)H = 12 (u, u). Then we have, for p ∈ V ∗ , / H. φ ∗ (p) = sup (< u, p > −φ(u)) = φ(p) if p ∈ H, and φ ∗ (p) = +∞ if p ∈ u∈V

Proof Since V is dense in H , the first part is obvious. Now let p ∈ / H . Let us assume that φ ∗ (p) is finite. Then ∃k ∈ R such that < u, p > −φ(u) ≤ k, ∀u ∈ V . Let λ ∈ R, so that u = λu0 , &u0 &H = 1. Thus λ < p, u0 > − 12 λ2 ≤ k for all λ ≥ 0. The supremum is reached at λ = < p, u0 >, so that we obtain 12 (< p, u0 >)2 ≤ k, and thus | < p, u0 > | ≤ (2k)1/2 &u0 & , so that p is continuous on H , contrary to the assumption p ∈ / H.

412

4 Behavior Laws

We can obtain such a framework from an unbounded operator A in H that is closed, injective, with dense domain D(A) in H . Let V = D(A), and then &u&V = (&u&2H +&Au&2H )1/2 . Then we have φ ∗ (p) = φ(p) if p ∈ H, and φ ∗ (p) = ∗ (p) = sup +∞ if p ∈ / H. But if we define φA u∈D(A) (< u, p > −φ(Au)), we ∗ ∗ obtain φA (p) = φ(Av) for p = A Av, v ∈ D(A). ¶ First we indicate some notation. Let E = X and Y be two real Hilbert spaces, and A a linear continuous map from E into Y . We denote by Im A its range (Im A = A(E)), by ker A its kernel (ker A = A−1 ({0})), and by A∗ the transpose of A, which is the linear continuous map from Y ∗ into E ∗ such that < A∗ p, x > = < p, Ax >,

∀x ∈ X, p ∈ Y ∗ .

Let f be a function from X into R ∪ {+∞}. The domain of f is the set, denoted by Dom f , of elements such that f (x) is finite: Dom f = {x ∈ X, f (x) ∈ R}. Let X be a topological vector space, and let N, N1 , N2 be subsets of X. We denote by Int (N) the interior of N, by N1 + N2 (respectively N1 − N2 ) the set of elements n = n1 + n2 (respectively n1 − n2 ), ni ∈ Ni , i = 1, 2. We remark that X + N = X and X − N = X, ∀N ⊆ X. Recall the following properties of convex analysis (see [Aub], [Eke-Tem], with various extensions). Let E = X and Y be Hilbert spaces, and let f be a strict function from X into R ∪ {+∞}: (i) Let A be an isomorphism from E onto Y ; we have (f ◦ A)∗ = f ∗ ◦ (A−1 )∗ .

(4.11)

(ii) Let f be a strict l.s.c. convex function, and let 0 ∈ Int (Im A - Dom f ) (see [Aub, ch. I.4.5]). Then ∂(f ◦ A)(x) = A∗ ∂f (Ax).

(4.12)

4.1.2 Generalized Fenchel Theorem Here we recall a fundamental theorem (see [Aub, ch. I.5.2, p. 69]). The framework of the theorem is the following. Let X and Y be two Hilbert spaces; let f and g be two strict l.s.c. convex functions, respectively from X and Y into R∪{+∞}, and let A be a linear continuous operator from X into Y . Then we consider the following optimization problems, where y ∈ Y, p ∈ X∗ occur as parameters,

4.1 Some Review of Convex Analysis

413

(i) h(y) = inf (f (x)− < p, x > +g(Ax + y)), x∈X

(ii) e∗ (p) = inf∗ (f ∗ (p − A∗ q) + g ∗ (q)− < q, y >).

(4.13)

q∈Y

Theorem 1 The Fenchel theorem. With the previous hypotheses, if p ∈ Int (Dom f ∗ +A∗ Dom g ∗ ), then there exists a solution x¯ of the problem h(y), and we have h(y) + e∗ (p) = 0. If, moreover, we assume that y ∈ Int (Dom g − A Dom f ), then there exists a solution q¯ of the problem e∗ (p), and we have the equivalences ¯ + A∗ ∂g(Ax¯ + y). x¯ solution of h(y) ⇔ x¯ ∈ ∂e∗ (p) ⇔ x¯ solution of p ∈ ∂f (x) Likewise, q¯ solution of e∗ (p) ⇔ q¯ ∈ ∂h(y) ⇔ q¯ solution of y ∈ ∂g ∗ (q) ¯ − A∂f ∗ (p − A∗ q). ¯ Moreover, x¯ and q, ¯ respectively solutions of the problems h(y) and e∗ (p), are related by 

p ∈ ∂f (x) ¯ + A∗ (q) ¯ ∗ y ∈ −Ax¯ + ∂g (q). ¯

(4.14)

(The convex functions h and e∗ are called marginal functions. They give the variation of optimal values as a function of the parameters y and p. A minimal solution x˜ of h(y) gives the variation rate of the marginal function e∗ with the variation of p. There exist many variants of this theorem; see [Aub, ch. V].) Application. Minimization problem with constraints. Let K be a closed convex subset of Y . The primal minimization problem h(y) =

inf

(Ax+y)∈K

(f (x)− < p, x >)

(4.15)

and its associated dual problem e∗ (p) = inf∗ (f ∗ (p − A∗ q) + σK (q)− < q, y >) q∈Y

(4.16)

enter in the previous framework with g(x) = χK (x), the indicator function of K, ∗ . Let Int D be the interior of the domain D. Then the conditions on p and σK = χK and y are p ∈ Int (Dom f ∗ + A∗ Dom σK ), and for K = {0},

−y ∈ Int (K − A Dom f ),

(4.17)

414

4 Behavior Laws

p ∈ Int (Dom f ∗ + Im A∗ ),

−y ∈ Int (A Dom f ).

(4.18)

If K is a closed convex cone, then x¯ and q¯ are such that p ∈ ∂f (x) ¯ + A∗ NK (Ax¯ + y) ¯ + NK − (q), ¯ y ∈ −A∂f ∗ (p − A∗ q)

(4.19)

and they are linked by ¯ p ∈ ∂f (x) ¯ + A∗ q,

Ax¯ + y ≥ 0, q¯ ≤ 0, < q, ¯ Ax¯ + y > = 0.

(4.20)

4.2 Force–Velocity Relations in Convex Analysis From the fundamental principle of mechanics [Mor1], the admissible velocity field v of a system and the corresponding admissible force field f (which belong respectively to the spaces E and E ∗ ) are in duality, and are related through a function φ from E, called a resistance function of the force law (for the considered fluid), satisfying Hypotheses 1 and 2, by one of the following equivalent relations: − f ∈ ∂ψ(v), or v ∈ ∂ψ ∗ (−f ), or still ψ(v) + ψ ∗ (−f )+ < v, f > = 0. (4.21) The dispersed power during the motion of velocity v, when the resistance function is quadratic, is the positive quantity given by − < v, f > = 2ψ(v).

(4.22)

Hypothesis 2 implies the positivity of (4.22), and the possibility of having null both f and v. With respect to the “usual” mechanics of systems, the duality between force and velocity leads us to consider the force as an element of the phase space T ∗ M, where M is the configuration space of the fluid particle (generally an open set of R 3 ), which allows us to associate a work to each force.

4.3 Stress–Strain Relations in Mechanics In the case of a continuum medium, the resistance function generally depends on the “thermodynamic state” of the fluid, at every point of the space. Passing from a point where the medium is in a solid state to a point where the medium is in a liquid state leads, for example, to passing from a Hooke constitutive law (between the stress tensor σ and the strain tensor u at temperature T ) such as

4.3 Stress–Strain Relations in Mechanics

415

σ = λe I tr u + 2μe u − 3Ke αT I, (with the Lamé elasticity coefficients λe , μe and Ke = λe + (2/3)μe ; 3α is a volume dilatation coefficient, and I the unit matrix),1 to a constitutive law between the stress tensor σ and the rate strain tensor v at pressure Ph , such as σ = λI tr v + 2μv − Ph I, with λ, μ the viscosity coefficients of an isotropic Newtonian fluid. These coefficients may also depend on the thermodynamic state of the fluid. Below we will consider only the case of fluids. We will apply the previous principle, which is (4.21), to the fluid mechanics, at first locally, with the duality between the rate strain tensor and the viscous stress tensor, instead of the velocity field and the force field. Then we will return to a global point of view, with the velocity field and the force field, essentially in the case of Newtonian fluids. Let M be a bounded connected open set of R 3 . Let S 2 T ∗ M (also denoted by 2 S M) be the space of symmetric covariant (of degree (0, 2)) tensors , and let 1 (M) be the space of symmetric mixed (of degree (1, 1)) tensors , T1S ˆ that is (with the metric g), (X, Y ) = (Y, X) = g(ˆ X, Y ) = g(X, ˆ Y ),

∀X, Y ∈ Tx M.

1 (M). The space We identify these spaces thanks to the map G−1 , ˆ = G−1  ∈ T1S 2 ∗ E = S T M is identified with its dual by

< ˆ , ˆ  > = tr (ˆ ◦ ˆ  ). The orthogonal group O(3) acts in E by u(ˆ ) = t u ◦ ˆ ◦ u = u−1 ◦ ˆ ◦ u,

1 u ∈ O(3), ˆ ∈ T1S (M).

A function φ on E is called invariant under the group O(3) if it satisfies φ(u(ˆ )) = φ(ˆ ),

1 ∀u ∈ O(3), ˆ ∈ T1S (M).

A resistance function φ on the space E is associated with a given homogeneous and isotropic fluid; this function satisfies the following properties: It is (i) invariant under the group O(3), (ii) convex in , (iii) homogeneous. The viscous stress tensor is a symmetric tensor, here denoted by τ (of degree (2, 0)) or τˆ (of degree (1, 1)), such that τ ∈ ∂φ() (or τˆ ∈ ∂φ(ˆ )), where  is the rate strain tensor. This relation is called the behavioral law of the fluid.

1 See,

for instance, [Germ], [D-L19, ch. IA.2.6].

416

4 Behavior Laws

In the following, in order to simplify writing, we denote ˆ and τˆ simply by  and τ , and the components of ˆ by ij (instead of ˆji ), and likewise for τ . Condition (i) implies that φ is a function φ() = ψ(I , I I , I I I ) of the three basic invariants (with respect to the group O(3)): I = φ1 () = tr , I I = φ2 () = tr ( ◦ ), I I I = φ3 () = tr ( ◦  ◦ ). We verify that the partial derivatives of these invariants are such that ∂I I ∂I I I ∂I = δij , = 2 ij , = 3 ( ◦ )ij . ∂ij ∂ij ∂ij Then the behavioral law is τij =

∂φ ∂ψ ∂ψ ∂ψ ∂ψ ∂ψ ∂ψ = δij + 2ij + 3(◦)ij , τ = I+ 2+ 3◦. ∂ij ∂I ∂I I ∂I I I ∂I ∂I I ∂I I I

In the case of a differentiable resistance function φ, the relations between τ and  may be written “in a differential form” dφ = < τ, d > = tr(τ ◦ d).

4.3.1 Stress–Strain Relations for Newtonian Fluids For numerous fluids, we furthermore assume that the resistance function is homogeneous, of degree 2 in . Let def

tr  =



ii ,

def

tr ( ◦ ) =



j k kj .

A Newtonian fluid has a resistance function φ on E (which is invariant under O(3)) given by 2φ() = 2φn () = λ (tr )2 + 2μ tr ( ◦ ), = (, An ),

with An  = λI tr  + 2μ .

(4.23)

The coefficients λ and μ are called viscous coefficients of the fluid, and are such that φ() is positive for every symmetric tensor  (see, for example, [Germ]). We have the decomposition of every symmetric tensor  into  =  S +  D , with  S = (q/3)I, q = tr  and tr  D = 0,

(4.24)

where  D is called a deviator. This decomposition is orthogonal, since tr ( S ,  D ) = 0. Then we obtain with a coefficient K = (λ + 23 μ), called the “bulk modulus,”

4.3 Stress–Strain Relations in Mechanics

φ() = φ(q,  D ) =

417

K 2 q + μ tr ( D . D ) = φ0 (q) + φD ( D ). 2

(4.25)

The resistance function splits into φ = φ0 + φD with the decomposition of . The positivity condition of φ() for every symmetric  is equivalent to the following: Hypothesis 3 K ≥ 0 and μ ≥ 0. The case μ = 0 (respectively μ = 0) leads to the Euler equation (respectively Navier–Stokes equation). Note that Hypothesis 3 implies that φ satisfies Hypothesis 1 (moreover, φ is a C ∞ function of ). We will make the following hypothesis (stronger than Hypothesis 3, since fluids without viscosity are excluded): Hypothesis 4 α0 = inf(K, μ) > 0. This hypothesis implies that 2φ() is the square of a Hilbert norm on S 2 M, and thus φ() = &&2n /2, the scalar product being defined by (1 , 2 )n = λ(tr 1 ) (tr 2 ) + 2μ tr (1 ◦ 2 ) = (An (1 ), 2 ).

(4.26)

The conjugate function of φ is φ ∗ (τ ) = sup (< τ, 1 > −φ(1 ). 1 ∈S 2 M

By the standard example, see (4.10), since φ is a quadratic function of , the supremum is reached by 1 = A−1 n (τ )). The inverse matrix of An is A−1 n (τ ) =

λ 1 (τ − I tr τ ), 2μ 3K

2 K = λ + μ. 3

Thus the conjugate function of φ is such that φ ∗ (τ ) = φ(A−1 n (τ )).

(4.27)

We simply obtain the viscous stress tensor by taking the derivative of φ, so that the behavioral law of the Newtonian fluid is given by τ = Dφ() = An (), that is, also in the general case, by the relations τ ∈ ∂φ() and  ∈ ∂φ ∗ (τ ). Then the power dispersed by viscosity is given by < τ,  > = tr (τ ◦ ) = 2φ().

(4.28)

418

4 Behavior Laws

Remark 1 The fluids for which the viscous stress tensor is an affine function of the rate strain tensor are still called Newtonian fluids. This is a priori a more general behavioral law than (4.26). But if, moreover, we impose that this law is independent of the coordinate system, and isotropic, then the behavioral law is necessarily of type (4.26) (see, for instance, [Ciar]). The hypothesis of a Newtonian fluid is not necessarily satisfied, as for the Bingham fluids with a nonlinear law (see below and, for example, [Duv-JLL]), which is deduced from a convex resistance function. ¶

4.3.2 Stress–Strain Relations for Bingham Fluids We call an incompressible fluid behaving as a rigid medium in some domains and as a viscoplastic medium in other domains a Bingham fluid; in the latter case, such a fluid is then called a “rigid viscoplastic incompressible fluid.” As usual, let σ be the stress tensor, and let  be the strain tensor, with σ = −P I + σ D ,

def

tr σ D = 0, and let σI I =

1 tr (σ D .σ D ). 2

The behavioral law of a Bingham fluid is expressed thanks to two positive constants, g (plasticity threshold) and μ (fluid viscosity), by 1/2

 = 0,

σI I < g,

1/2

σI I ≥ g,

=

g 1 (1 − 1/2 )σ D . 2μ σI I

Here we express the strain tensor in terms of the viscous stress tensor. By this law, the stresses below a threshold do not induce a strain. Let us show that this situation may be modeled in the framework of convex analysis with a resistance function. To simplify notation, let √ ˜ = (2μ/g 2) ,

√ √ τ˜ = (1/g 2) τ = (1/g 2) σ D , and &τ˜ & = (tr (τ˜ .τ˜ ))1/2 .

We have thus defined a norm on the space of symmetric matrices. Furthermore, we 1/2 have σI I = g &τ˜ &. Hence in these new variables, the behavioral law is &τ˜ & < 1,

˜ = 0,

&τ˜ & ≥ 1,

˜ = (1 − 1/ &τ˜ &) τ˜ .

Then we define the function φ ∗ by2 φ ∗ (τ˜ ) =

1 (&τ˜ & − 1)2 if &τ˜ & ≥ 1, 2

φ ∗ (τ˜ ) = 0 otherwise.

notation anticipates the fact that φ ∗ , being a strict l.s.c. convex function, is equal to the conjugate function of φ.

2 This

4.3 Stress–Strain Relations in Mechanics

419

This function is continuous and convex. Its conjugate function is (φ ∗ )∗ = φ φ() = sup (< , τ˜ > − φ ∗ (τ˜ )) = sup(I1 , I2 ), τ˜

with I1 = sup < , τ˜ > = && , &τ˜ & − (&τ˜ & − 1)2 ) = sup (λ && − (|λ| − 1)2 ) 2 2 &τ˜ &≥1 τ˜ =λ/&&, |λ|≥1 &&2 1 = && (1 + && /2). = sup(λ && − (λ − 1)2 ) = && + 2 2 λ≥1 Hence we have φ() = && (1 + && /2). Note that this is a convex continuous and positive function satisfying Hypothesis 2. The relations ˜ ∈ ∂φ ∗ (τ˜ ), τ˜ ∈ ∂φ(˜ ), imply φ ∗ (τ˜ ) + φ(˜ ) = < ˜ , τ˜ > . We thus obtain an expression for τ˜ as a function of : ˜  τ˜ =

(1 + &&)˜ ˜  / &˜ & if &˜ & = 0, unknown, but &τ˜ & < 1, for &˜ & = 0.

If &τ˜ & > 1, then we have &˜ & = &τ˜ & − 1 , with ˜ / &˜ & = τ˜ / &τ˜ & , which is a straightforward consequence of the given behavioral law. Now we return to the 2 τ initial variables. Letting ψ ∗ (τ ) = gμ φ ∗ ( √ ), we obtain that its conjugate function g 2 is given by ψ ∗∗ () = ψ() =

√ g2 2μ 1/2 1/2 φ( √ ) = && (g 2 + μ &&) = 2I I (g + μI I ), μ g 2

√ 1/2 with I I = && / 2, I I = (tr .)/2. The function ψ ∗ is expressed by ψ ∗ (τ ) =

1 1/2 1/2 (τ − g) if τI I > g, 2μ I I

ψ ∗ (τ ) = 0 otherwise.

420

4 Behavior Laws

Thus the relation between  and τ is not univalued for a Bingham fluid, in contrast to the case of a Newtonian fluid. For g = 0, note that the fluid is simply a Newtonian fluid. For more general behavioral laws for viscoplastic media, see [Duv-JLL, Bou-Suq].

4.3.3 Stress–Strain Relations for Tensor Fields Here we limit ourself to Newtonian fluids. Let  be a symmetric tensor field in the Hilbert space3 E = L2 (M, S 2 M) (i.e., a square integrable section of the bundle S 2 M). Note that this choice of space may be considered a regular framework for the fluid mechanics. We identify E with its dual by the scalar product  (1 , 2 ) =

tr(1 (x) ◦ 2 (x)) dx. M

Remark 2 The homogeneous fluids are such that λ and μ are x-independent; but λ and μ very generally depend on the “thermodynamic state” of the fluid (especially on the specific mass and on the temperature). Thus we will make the hypothesis even stronger than Hypothesis 4 (where the infimum is relative to x and to the thermodynamic conditions): Hypothesis 5 α = inf (K, μ) > 0, with sup (K, μ) < +∞. Then K and μ with their inverses are bounded. We will be induced to assume some smoothness of these coefficients for the boundary condition of M. ¶ From the resistance function (4.23) of a Newtonian fluid, we define a new resistance function on the space E by ˜ φ() = φ˜ n () =

 (4.29)

φn ((x)) dx. M

With Hypothesis 5, we define (using (4.26)) a scalar product and a norm that is equivalent to the natural norm on L2 (M, S 2 M) by  (1 , 2 )n =

(1 (x), 2 (x))λ,μ dx,

1/2

&&n = (, )n .

(4.30)

M

From this resistance function, by (4.26), we define a viscous stress tensor τ , which is a square integrable symmetric tensor field of degree (2, 0) or (1, 1), the relation between τ and  being

that the vector bundle S 2 T ∗ M is naturally equipped with a fibre metric, since M is a Riemannian (even Euclidean) manifold, so that L2 (M, S 2 M) is a Hilbert space.

3 Note

4.4 Relations Between Force and Velocity Fields

˜ τ ∈ ∂ φ(), and  ∈ ∂ φ˜ ∗ (τ ),

421

thus τ = An  and  = A−1 n τ.

(4.31)

The dispersed power by viscosity is such that 

 < τ (x), (x) > dx = 2

φn ((x)) dx.

M

(4.32)

M

By Hypothesis 5, there exists α > 0 such that for every symmetric matrix , 2φn () ≥ α tr ( ◦ ), whence 

 φn () dx ≥ α

2 M

tr( ◦ ) dx,

(4.33)

M

for every “admissible” strain tensor field  in L2 (M, S 2 M).

4.4 Relations Between Force and Velocity Fields 4.4.1 Introduction Here we consider essentially Newtonian fluids. Let  be the rate–strain tensor field linked to the velocity field as in Chapter 3; the resistance function on the tensor fields induces a resistance function on the velocity field, and then on a space of sections of the tangent bundle T M (still for a Newtonian fluid, with φn given by (4.29), with (4.23)) by ψ(v) = ψn (v) = φ˜ λ,μ (v ) =

 φn (v ) dx, M

1 ∂v i ∂v j with v = ((v )ij ) = ( j + ). 2 ∂x ∂x i

(4.34)

The application of the mechanics principle in the viscosity framework requires us to specify a functional space E that is related to global properties for the velocity fields, and notably with boundary conditions. A natural framework E for the domain of definition of ψ is the Sobolev space H 1 (M)3 , a subspace, or a quotient space. Then we will be led to use the dual space of H 1 (M)3 , which we will specify in a next section. We also use the basic map :

v ∈ H 1 (M)3 → (v) = v ∈ L2 (M, S 2 M).

(4.35)

422

4 Behavior Laws

We underline the importance of the kernel and the image of this map. In the Dirichlet case, that is, with the boundary condition v = 0 on , the kernel is reduced to 0, which simplifies the method. With An defined by (4.26), we have ψ(v) =

1 2

 tr (An (v) ◦ (v)) dx = M

1 &(v)&2λ,μ . 2

(4.36)

With (4.26), the viscous stress tensor τ = τv is such that τ = An v = λI tr v + 2μ v ,

(4.37)

˜ v ) = 2ψ(v) ≥ 0. so that < τv , v > = 2φ( Then ψ is a convex l.s.c. function on H 1 (M)3 , and moreover, ψ is Gâteaux differentiable, that is, ∂ψ(v) = {Dψ(v)} has one element for every v in H 1 (M)3 . Also notice that v ∈ ker  implies ψ(v) = 0, whence Dom ψ ∗ = {f ∈ (H 1 (M)3 )∗ , ψ ∗ (v) < +∞} = (ker )⊥ . The range of the map  is a closed subspace Y of L2 (M, S 2 M), and the conjugate function of  is the map ∗ from Y ∗ into E ∗ . From the resistance function (4.34), ˜ with (4.31) and (4.12), since τ ∈ ∂ φ(v), the viscous relations between force and velocity are expressed by ˜ − f = ∗ τ ∈ ∗ ∂ φ(v) = ∂(φ˜ ◦ )(v) = ∂ψ(v), v ∈∂ψ ∗ (−f ) = ∂(φ˜ ◦ )∗ (−f ) = ∂(φ˜ ∗ ◦ ∗−1 )(−f )   = (∗−1 )∗ ∂ φ˜ ∗ (−∗−1 f ) = −1  .

(4.38)

In (4.38), φ˜ is in fact the restriction φ˜ Y to Y of the map φ˜ defined by (4.29) on ⊥ 2 2 L2 (M, S 2 M).  Let Y be the orthogonal of Y in L (M, S M)2for the2scalar product (1 , 2 )n = M (1 (x), 2 (x))λ,μ dx, with (4.26). Every  ∈ L (M, S M) splits into ˜ ˜ 0 +  1 ) = φ( ˜ 0 ) + φ( ˜ 1 ), and = φ(  =  0 +  1 with  0 ∈ Y,  1 ∈ Y ⊥ , and φ() ∗ then for every τ ∈ Y the dual of Y , the conjugate function of φ˜ is given by ˜ ˜ 0 +  1 )) φ˜ ∗ (τ ) = sup (< τ,  > −φ()) = sup (< τ,  0 > −φ( 

 O , 1

˜ 0 )), = sup (< τ,  0 > −φ(  0 ∈Y

and thus we deduce that φ˜ ∗ (τ ) = φ˜ Y∗ (τ ), ∀τ ∈ Y ∗ . Let us specify the previous framework, according to the boundary conditions, knowing that  is an isomorphism from V = H01 (M)3 onto (H01 (M)3 ) = YD in the Dirichlet case, and from E = H 1 (M)3 / ker  onto YN = (H 1 (M)3 ) in

4.4 Relations Between Force and Velocity Fields

423

the general case. The injection H 1 (M, T M) → L2 (M, T M) being compact (with smooth bounded M), the Peetre lemma implies that YN is a closed subspace of  1/2 is a norm on E equivalent to L2 (M, S 2 M). Moreover, M tr (ˆv (x).ˆv (x) dx) the quotient norm of H 1 (M, T M) by ker . Then the map  is an isomorphism from E onto YN .

4.4.2 Relations f, v with Homogeneous Dirichlet Conditions Here we consider the modeling of a situation in which the fluid fills up the whole (bounded) domain M of a container and cannot slip along the wall of this container. The velocity field must be null on the boundary of the domain M; the velocity field space E is the Sobolev space E = H01 (M, T M) identified with E = H01 (M)3 . The dual space E ∗ is the Sobolev space H −1 (M)3 . The conjugate function of ψ is defined (for a force field f in E ∗ ) by ψ ∗ (−f ) = sup (< −f, w > −ψ(w)) = sup (< −f, w > −φ˜ n (w )). w∈V

w∈V

The following propositions are equivalent: (i) v ∈ E is a solution of the variational problem  (v , w )λ,μ dx = − < f, w >, M

∀w ∈ H01 (M)3 ;

(ii) v ∈ E is such that −f ∈ ∂ψ(v), and v ∈ ∂ψ ∗ (−f ). Starting from (i), we prove the existence and uniqueness of the solution of the variational problem (with given f in E ∗ ), by the Lax–Milgram lemma with Hypothesis 4. Then we obtain, with the standard Example 12, ψ ∗ (−f ) =

1 &v &2n = ψ(v) = < −f, v > −ψ(v), 2

(4.39)

which implies the relations (ii). Starting from (ii), the uniqueness of the solution v of (ii) (with given f in the same conditions) is a consequence of the fact that the functional ψ is coercive (i.e., that lim ψ(v) = +∞ when &v& → ∞) and strictly convex (see [Eke-Tem, ch. II.1]). ¶

424

4 Behavior Laws

The relations between f and v are made explicit by4 (An v, w) = − < f, w >,

∀w ∈ E,

(4.40)

Pv = − ∗ An v = grad (λ divv) − ∗ (2μv) = f, and if μ is constant, this relation is Pv = grad ((λ + 2μ) div v) − curl(μ curl v) = f.

(4.41)

If we define the viscous stress tensor field τ = τv by  < τv , w > = (v , w )n =

(v (x), w (x))λ,μ dx,

∀w ∈ E,

M

then we have −f = ∗ τv , which we (formally) write f = div τv , or with components fi = ∂j τij . Indeed, we have 



(v , τ ) =

(∂j vi ).τij dx = − M

vi (∂j τij ) dx = < v, ∗ τ > .

M

Thus ∗ is the continuous map from L2 (M, S 2 M) into H −1 (M)3 defined by (∗ τ )i = −∂j τij , i = 1, 2, 3. We also set ∗ τ = −div τ. Then the dispersed power by viscosity is given by (if μ is constant)  − < fv , v > = 2ψ(v) =

tr (τv ◦ v ) dx M

=

 

 (λ + 2μ) |div v|2 + μ |curl v|2 dx.

M

Let us specify the image space YD = (H01 (M)3 ), which is a closed subspace of L2 (M, S 2 M), by its orthogonal space YD⊥ in L2 (M, S 2 M) for the usual scalar  product. Let us denote ∂j τij , i = 1, 2, 3 by div τ . The space orthogonal to YD is   YD⊥ = τ ∈ L2 (M, S 2 M), div τ = 0 . Then we can sum up the previous results as follows. Consider the following functional framework: the set of admissible velocities (respectively viscous forces) is the Sobolev space E = H01 (M)3 (respectively E ∗ =

4 The

viscous coefficients can be x-dependent.

4.4 Relations Between Force and Velocity Fields

425

H −1 (M)3 ), and the resistance function ψ of the viscous fluid is given by (4.34) in the case of a Newtonian fluid. Proposition 1 The relation (4.21) between velocity and force, given by the mechanic principle, with homogeneous Dirichlet boundary condition on the velocity, is given by equation (4.40), and this correspondence is an isomorphism under Hypothesis 5. Remark 3 Hypothesis 3 is a simple “algebraic” condition that does not take the possibilities of application of the Green formula into account. In the present case, in which the velocity field is null on the boundary of the domain,  we obtain, with the hypothesis of μ constant, the positivity of the expression M 2φλ,μ (v (x)) dx under the conditions λ + 2μ ≥ 0 and μ ≥ 0, weaker than Hypothesis 3. Indeed, M 2φλ,μ (v ) dx is reduced to  

 (λ + 2μ) |div v|2 + μ |curl v|2 dx

M

if μ is constant. Hypothesis  3 seems to imply necessarily, for every boundary condition, the positivity of M 2φλ,μ (v ) dx. ¶ Remark 4 In the case f ∈ L2 (M)3 , the corresponding velocity v is in the domain D(P) of the unbounded operator P in L2 (M)3 defined thanks to the bilinear form a(v, w) = (An v, w) on E = H01 (M)3 by D(P) = {v ∈ E, such that the map w → a(v, w) is continuous on L2 (M)3 }, so that ∀v ∈ D(P), we have a(v, w) = (Pv, w), ∀w ∈ E. Thus if f ∈ L2 (M)3 , the velocity v, a solution of (4.41), is in D(P). ¶

4.4.3 Dual Space of H 1 (M)3 In the case where the velocity space is V = H 1 (M)3 , the general framework of forces is given by the dual space V ∗ which is not a distribution space. Dual Space of H 1 (M)3 with the Scalar Product (., .) (i) Duality of the “scalar” space H 1 (M). We first specify the dual space of the “scalar” space E = H 1 (M) with the scalar product:  (grad u.grad v + u.v)dx.

(u, v) = M

(4.42)

426

4 Behavior Laws

Let φ(u) = &u&2 /2. Let fM ∈ L2 (M), f ∈ H −1/2 ( ). Then f = (fM , f ) defines a continuous map f on E (thus f ∈ E ∗ ) by  fM .w dx + < f , w| >,

< f, w > =

∀w ∈ H 1 (M).

(4.43)

M

Since E is a Hilbert space, there exists u ∈ E such that (u, w) = − < f, w > for every w ∈ E. Then we verify the equivalence f ∈ L2 (M) × H −1/2 ( ) ⇔ u ∈ H 1 (, M),

(4.44)

  with H 1 (, M) = u ∈ H 1 (M), u ∈ L2 (M) , thanks to the isomorphism A defined by  ∂u  ) = f = (fM , f ). Au = ((− + I )u, ∂n 

(4.45)

With the subdifferential of φ, we write u ∈ ∂φ ∗ (−f ), and −f ∈ ∂φ(u). (ii) Duality of the “vector” space V = H 1 (M)3 . The space V is decomposed into V = H01 (M)3 ⊕ W with  W = {w = (w ) ∈ V , j

( M

 ∂wj ∂uj  + wj uj )dx = 0, ∀u ∈ H01 (M)3 }. ∂x i ∂x i

Thus W is defined by W = {w = (wj ) ∈ V , −wj + wj = 0}. The problem: find w ∈ V , a solution of the problem (i) − w + w = 0,

that is − wj + wj = 0, j = 1, 2, 3,

(ii) w| = w ∈ H 1/2 ( )3 being given, has a unique solution w ∈ W , so that we can identify W with H 1/2( )3 . Thus the dual space of V = H01 (M)3 ⊕ W is identified with V ∗ = H −1 (M)3 ⊕ H −1/2 ( )3 , so that with the decompositions v = (u, w) ∈ H01 (M)3 ⊕ W,

and f = (fM , f ) ∈ H −1 (M)3 ⊕ H −1/2 ( )3 ,

we have < v, f > = < u, fM > + < w, f > .

4.4 Relations Between Force and Velocity Fields

427

Dual Space of H 1 (M)3 for (., .)n Here we specify the dual space V ∗ , thanks to the decomposition of V into V = V0 ⊕ W,

with V0 = H01 (M)3 ,

(4.46)

the elements w of W being defined with the following differential operator: ∗d : τ ∈ L2 (M, S 2 M) → (−∂j τi ) ∈ H −1 (M)3 (also denoted by −div τ ), by j

w ∈ W ⇔ {w ∈ V , with d∗ An w = 0},

(4.47)

since the space D(M)3 of functions C ∞ with compact support is dense in V0 . Thus W is orthogonal to V0 for the scalar product (v , w )n :  W = w ∈ V , (v, w)n = (v, ∗d An w) = 0,

 ∀v ∈ V0 .

(4.48)

Then (4.47) is equivalent to def

Pw = −∗d (An w) = grad (λ div w) − ∗ (2μ w) = 0, with w ∈ V , or, if μ is constant, to Pw = grad ((λ + 2μ)div w) − curl (μ curl w) = 0, with w ∈ V . Since the trace γ is a continuous map from V onto H 1/2( )3 , with kernel V0 , γ is an isomorphism from W onto H 1/2( )3 , so that we can identify W with H 1/2 ( )3 . We remark that ker  ⊂ W, and thus we can identify the space E = V / ker  with E = V0 × V1 with V1 = H 1/2( )3 / ker ,

(4.49)

where we identify ker  with its space of traces on (which we also denote by ker ). The decomposition of every v ∈ V is given by v = v0 + w,

w = R1 (v| ) ∈ V solution of

Pw = 0, w| = v| ,

v0 = v − w ∈ V0 .

(4.50)

Then to this decomposition of V = H 1 (M)3 corresponds the decomposition of the dual V ∗ into V ∗ = H −1 (M)3 ⊕ H −1/2 ( )3 ,

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4 Behavior Laws

and the dual space E ∗ of E admits the following decomposition: E ∗ = V0∗ ⊕ V1∗ , with V0∗ = H −1 (M)3 , V1∗ dual of V1 ,

(4.51)

that is, V1∗ = {f ∈ H −1/2( )3 , f |ker  = 0}, and thus < f , a + b × x > = 0,

∀a, b ∈ R 3 ,

def

that is, with < u > = < u, 1 >, ∀u, < f > = 0,

< f × x > = 0.

Then the decomposition of an element f ∈ E ∗ is given by f = (fM , f ) with fM ∈ H −1 (M)3 , f ∈ V1∗ ,

(4.52)

so that for all w = (w0 , w1 ) ∈ E = V0 × V1 and f ∈ E ∗ , we have < f, w > = < fM , w0 > + < f , w1 > .

(4.53)

The Calderón operator (also called capacity operator, or DtN). We apply the Green formula to elements u, v ∈ W :     j j ∗ ∗ 0 = ((d An u).v − u.(d An v))dx = ∂j τk (u)v k − uk τk (v) dx M

M



j

=

j

(nj τk (u)v k − uk nj τk (v)) d .

Then we define the Calderón operator,5 which is the map C from H 1/2( )3 into H −1/2( )3 , for every u = u| ∈ H 1/2( )3 , u ∈ W by j

(C(u ))k = (nj τk (u)| ),

∀u ∈ W.

(4.54)

5 This operator is similar to the capacity operator (often called Dirichlet-to-Neumann operator) in potential theory (see [D-L19]), or to the Calderón operator (or impedance operator) in electromagnetism (see [Ces]).

4.4 Relations Between Force and Velocity Fields

429

We obtain the property of symmetry for all6 u , v ∈ H 1/2( )3 , < C(u ), v > = < u , C(v ) > .

(4.55)

We also have for all v, w ∈ W  0 = ((∗d An u).v + u.(∗d An v)) dx M





=

∂j M

j τk (u)v k

j + uk τk (v)





j

dx − M

j

[τk (u)jk (v) + jk (u)τk (v)] dx,

and thus  < C(u ), v > + < u , C(v ) > =

[τ (u).(v) + (u).τ (v)] dx,

(4.56)

M

that is, with (4.55), < C(u ), v > = ((u).(v))n = ((u), (v))n .

(4.57)

Thus we see that the Calderón operator C is such that ker C = ker ; its range is R(C) = V1∗ . It is an isomorphism between V1 and V1∗ . Moreover, we can identify C with a positive definite operator on V1 , by (u , u ) = < u , Cu > . We have the following basic property with respect to the resistance function ψ (see (4.36)) in the decomposition (4.50) of V : ψ(v) = ψ0 (v0 ) + ψ (v ),

with ψ (v ) =

1 (v , Cv ). 2

(4.58)

With the decomposition (4.52) of E ∗ and (4.53), we obtain the decomposition of the conjugate ψ ∗ to the resistance function ψ, on E ∗ : ψ ∗ (f ) = sup (< f, v0 > −ψ(v0 )+ < f, v > −ψ(v )) = ψ0∗ (fM ) + ψ ∗ (f ). (v0 ,v )

Hence ψ ∗ = ψ0∗ + ψ ∗ ,

(4.59)

of confusing the notation u with u (x), x ∈ for the velocities: u is not necessarily tangent to , but is in Tx M = Tx ( ) ⊕ Tx ( )⊥ . But we keep the notation f for the forces in Tx∗ M. But in some cases, the more natural notation u is used instead of u .

6 Beware

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4 Behavior Laws

which gives the decompositions of the corresponding subdifferentials ∂ψ(v) = ∂ψ0 (v0 ) + ∂ψ (v ) ∗

∂ψ (f ) = ∂ψ0∗ (fM ) + ∂ψ ∗ (f ).

(4.60)

Remark 5 We can use these results for the following problem: let f ∈ E ∗ , find v ∈ H 1 (M)3 such that (v , w )n = − < f, w >,

∀w ∈ H 1 (M)3 .

(4.61)

Let f0 (respectively f1 ) be the restriction of f to V0 = H01 (M)3 (respectively V1 ). An element v solution of this equation, is characterized in two steps: (i) Let w be in the space D(M)3 (dense in V0 ), in (4.61). We then have (v, w)n = (An v, w) = < d∗ An v, w > = − < f, w > = − < f0 , w >, and hence in the sense of distributions − Pv = d∗ An v = −f0 .

(4.62)

(ii) Let w be in V1 , in (4.61); we have (v, w)n = (v, An w) = − < f, w > = − < f1 , w > .

(4.63)

Then with the Green formula if Pv ∈ L2 (M)3 , we obtain (An v, w) = −(Pv, w)+ < Bv, w > ,

(4.64)

with Bv ∈ H −1/2( )3 , such that (Bv)i = λni div v + 2μ nj (v )ij = (n.τv )i ,

i = 1, 2, 3.

(4.65)

But in (4.64), Pv is neither in the sense of distributions, nor orthogonal to V1 . Let Pv = f0 + fP with fP ∈ V1∗ be its decomposition (An v, w) =< Bv − fP , w >

(4.66)

and fP = Bv 0 with v0 ∈ H01 (M)3 solution of Pv0 = f0 , and then < Bv − fP , w1 > = − < f1 , w1 >,

∀w1 ∈ V1 .

(4.67)

Thus Bv = fP − f1 .

(4.68)

4.4 Relations Between Force and Velocity Fields

431

Note that the Calderón operator C is related to B by C(v| ) = Cγ v = Bv − Bv0 ,

∀v ∈ E, Pv ∈ L2 (M)3 .

(4.69)

The main point is that the elements f of the pivot space L2 (M)3 must be decomposed according to the decomposition (4.51) of E ∗ . ¶ In the case of a Newtonian fluid, we can make the Calderón operator C explicit in the following form (4.70), by C(v) = λn div v + μC1 (v), that is in matrix form, with div v given by (3.80),   n (λ + 2μ) ∂v vn + 2λRm vn + λ div v ∂n = C . v μ (grad vn + ∂v ∂n )

(4.70)

This gives the force (fn , f ) = C(0, v ) at the boundary in the special case vn = 0.

4.4.4 Relations f, v with Boundary Conditions Consider the following functional framework: the set of admissible velocities (respectively of admissible viscous forces) is the Sobolev space E = H 1 (M)3 / ker  (respectively E ∗ ), and the resistance function is given by (4.34). Proposition 2 (General framework for a Newtonian fluid) The mechanic principle expressed by (4.21) in this framework leads to the following “viscous” relations (bijective modulo the kernel of , with Hypothesis 5) between velocity v and force f , globally expressed by (4.61), or by f = −∗ An v (with the previous decompositions of f into f0 and f1 ) by −f0 ∈ ∂ψ0 (v0 ), v0 ∈ ∂ψ0∗ (−f0 ), − < v0 , f0 >= ψ0 (v0 ) + ψ0∗ (−f0 ) = 2ψ0 (v0 ), −f ∈ ∂ψ (v ), v ∈ ∂ψ ∗ (−f ), − < v , f >= ψ (v ) + ψ ∗ (−f ) = 2ψ (v ), that is, also Pv0 = f0 ,

Cv = −f .

(4.71)

The relation between v (respectively (v0 , v )) and f (respectively (f0 , f )) is still an isomorphism. In the present framework of convex analysis, for given forces f (respectively (f0 , f )) in a Newtonian fluid, the proof of the existence of v (or of (v0 , v )) is a consequence of the Fenchel theorem, Theorem 1 (see [Aub, ch. I.3.3; 3.2]).

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4 Behavior Laws

Uniqueness is due to the coercivity of ψ(v) (up to rigid motions). This implies that the problems (4.61) and (4.71) with given f (or (f0 , f )) in E ∗ admit one and only one solution v in E that solves a “Neumann” problem, since (f )k = j (nj τk (u)), f = n.τ. Relations f, v with Inhomogeneous Dirichlet Conditions Let us return to the previous situation of a Newtonian fluid in a bounded domain M, but in the case that the fluid velocity is given on the boundary of M, and a force field f is given on M. Finding the velocity field v with given velocity on the boundary of M, v ∈ H 1/2( )3 , and with a given force field f = f0 ∈ V0∗ = H −1 (M)3 , is obtained by means of the primal minimization problem Problem 1 h(v ) =

inf

v∈V ,γ (v)=v

(ψ(v)+ < v, f >),

where γ is the usual trace mapping, v ∈ V → γ (v) = v| ∈ H 1/2( )3 . Let γ ∗ be the transposed map of γ , from H −1/2( )3 into V ∗ . Then the dual problem is as follows. Problem 2 e∗ (−f ) =

inf

q∈H −1/2 ( )3

(ψ ∗ (−f − γ ∗ q)+ < q, v >).

In this problem, q¯ is a surface force f on the boundary of the fluid, which appears as a Lagrange multiplier for Problem 1, for which the stress is the boundary condition on γ (v) = v . The Fenchel theorem, Theorem 1, implies the existence of the solutions v¯ (still denoted by v) of Problem 1 and q¯ of Problem 2, under the conditions − f ∈ Int (Dom ψ ∗ + Im γ ∗ ),

v ∈ Int (γ Dom ψ).

(4.72)

We have Dom ψ ∗ = (ker )⊥ , and Im γ ∗ = (ker γ )⊥ ; thus Dom ψ ∗ + Im γ ∗ = (ker )⊥ + (ker γ )⊥ = (ker  ∩ ker γ )⊥ = (H 1 (M))∗ . Thus the first condition is satisfied. The second condition is trivially satisfied. Then the solutions v¯ and q¯ are related by ¯ − f ∈ ∂ψ(v) ¯ + γ ∗ q.

(4.73)

˜ Using (4.12) with (see (4.34)) ψ(v) = φ(v), with (4.40), we obtain, for a Newtonian fluid,

4.4 Relations Between Force and Velocity Fields

433

− f = ∗ An v¯ + γ ∗ q. ¯

(4.74)

This corresponds to the formula, here with B = Cγ (this notation may be confusing) ∗ An v = −Pv + γ ∗ Bv.

(4.75)

With v = v, ¯ we have P v¯ = f ∈ H −1 (M)3 , and thus using (4.74) and (4.75), we ∗ obtain γ (B v¯ + q) ¯ = 0, which finally gives P v¯ = f,

B v¯ + q¯ = 0.

(4.76)

As a consequence, with V = H 1 (M)3 , we have defined the map for the Dirichlet problem up to rigid motions (f, v ) ∈ H −1 (M)3 × H 1/2( )3 → (v, f ) ∈ V × H −1/2( )3 , which is to be compared with the map of the problem (fM , f ) ∈ H −1 (M)3 × H −1/2( )3 , → v ∈ H 1 (M)3 / ker .

4.4.5 Mixed Type Condition Here we consider a situation in which the domain M occupied by a fluid is bounded by a wall, such that on a part 0 , the fluid cannot slip, on the other part, we have a free boundary condition. We assume that the part 0 of the boundary (of nonnull measure) is fixed and then that the velocity field is null. Let   E = H 10 (M)3 = v ∈ H 1 (M)3 , v| 0 = 0 .

(4.77)

If 1 = \ 0 is a regular part of with nonnull measure, then the set of traces v| with v ∈ E is identifiable with the space   1/2 V 1 = H00 ( 1 )3 = v ∈ H 1/2( )3 , v| 0 = 0 .

(4.78)

The dual space is V ∗1 (also denoted by (H00 ( 1 )3 ) ).7 The kernel ker  of the map  : v ∈ H 10 (M)3 → v ∈ L2 (M, S 2 M) is reduced to 0. The image of this map is a closed subspace Y 0 of L2 (M, S 2 M) (thanks to the Korn inequality). The dual space of E is not a distribution space (if 1 = \ 0 is of nonnull measure). 1/2

7 For

the properties of these spaces, see [Lio-Mag].

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4 Behavior Laws

Let W 0 be the subspace of H 1 (M)3 defined by   W 0 = w ∈ H 1 (M)3 , Pw = 0, w| 0 = 0 ⊂ W.

(4.79)

We can identify W 0 with the space V 1 thanks to the lift R1 defined by v = R1 v 1 ⇔ v ∈ H 10 (M)3 , Pv = 0, v| 1 = v 1 .

(4.80)

We split the space E into the orthogonal spaces E = H 10 (M)3 = H01 (M)3 ⊕ W 0 ⊂ H01 (M)3 ⊕ W = H 1 (M)3 .

(4.81)

Then E is identified with the product H01 (M)3 × V 1 . By duality, we obtain E ∗ = (H 10 (M)3 )∗ = H −1 (M)3 ⊕ W ∗0 ,

(4.82)

identified with H −1 (M)3 × V ∗1 . An element f = f0 + f1 ∈ E ∗ , with f0 ∈ H −1 (M)3 , f1 ∈ W ∗0 , is characterized by the fact that there exists v in E such that (v , w ) = − < f, w >,

∀w ∈ E.

(4.83)

This is equivalent to (as in (4.62), (4.67)) Pv = f0 ,

< Bv, w > 1 = − < f1 , w >, ∀w ∈ W 0 ,

(4.84)

where B = C1 γ is a continuous boundary operator W 0 → V ∗1 , so that Bv = −f1 , with the Calderon operator C1 defined from V 1 into V ∗1 . With the lift R1 (defined from V 1 onto W 0 ), the operators C1 and B are also related by: C1 (v 1 ) = BR1 (v 1 ). As previously, the resistance function ψ splits into ψ(v) = ψ(v0 ) + ψ(v1 ) = ψ0 (v0 ) + ψ 1 (v 1 ), with ψ 1 (v 1 ) =

1 < v 1 , C1 v 1 > 1 ; 2

C1 is (identifiable with) a symmetric positive operator on V 1 .

(4.85)

4.4 Relations Between Force and Velocity Fields

435

L2 -framework of admissible forces. If the space of regular admissible force fields f = (fM , f 1 ) is Z = L2 (M)3 × V ∗1 , then the space of admissible velocity fields is defined, with μ constant, by   E0 = H 10 ,A (M)3 = v ∈ H 10 (M)3 , Pv ∈ L2 (M)3 with Pv = grad ((λ + 2μ)div v) − curl (μ curl v). The relation between force and velocity is still expressed with E = H 10 by inf (ψ(v) + (fM , v)+ < f 1 , v 1 >),

v∈E

which is equivalent to −fM − γ1∗ f 1 = ∗ An v and Pv = fM ,

C1 (v| 1 ) = −f 1 .

¶

(4.86)

Then we have the following result. Proposition 3 (Mixed framework for a Newtonian fluid) The “viscous” relations between force and velocity due to the mechanic principle, applied with the mixed boundary conditions, are given by (4.83) (equivalent to (4.78) and v null on 0 ). These relations split according to (4.60), with the replacement of the Calderón operator C by C1 with (4.85). Then this relation between velocity and force is an isomorphism. Remark 6 Duality between velocity and viscous stress. Until now we have used “symmetric” dualities between strain tensor fields and viscous stress tensor fields, then between velocity and force vector fields. We can also use an “asymmetric” duality between the velocity vector fields and the viscous stress tensor fields, which we make explicit in the framework of mixed limit conditions, following a method explained in [Eke-Tem, ch. VII.4] about elasticity, transposable to the case of fluid mechanics. The velocity (respectively viscous stress) space is E = H 10 (M)3 (respectively Y 0 = Im  ⊂ L2 (M, S 2 M)). These spaces are in duality through the scalar product (v, τ ). Here we consider the minimization problem called primal with given f in E ∗ : Problem 3 ˜ < f, v >). inf (φ(v)+

v∈E

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4 Behavior Laws

We set Ff (v) = < f, v > . The dual problem (in the space of viscous stress tensor fields) is then defined through the conjugate functions φ˜ ∗ and Ff∗ of φ˜ and Ff by Problem 4 sup (−φ˜ ∗ (τ ) − Ff∗ (−∗ τ )) = − inf∗ (φ˜ ∗ (τ ) + Ff∗ (−∗ τ )). τ ∈Y

τ ∈Y ∗

0

0

Furthermore, Ff∗ (−∗ τ )) = 0 if −∗ τ = f, Ff∗ (−∗ τ )) = +∞ otherwise. Then Problem 4 becomes (with constraint) the problem −

inf

τ ∈Y ∗ ,−∗ τ =f

(φ˜ ∗ (τ )), or still: −

0

inf

τ ∈Y ∗ ,−∗ τ =f

˜ −1 (φ(A n τ ))

(4.87)

0

using (4.27). Naturally we have the relations between the fields v and τ giving the extrema of Problems 3 and 4: Ff (v) + Ff∗ (−∗ τ ) = − < ∗ τ, v >, ˜ φ(v) + φ˜ ∗ (τ ) = < τ, v >,

(4.88)

these relations being equivalent to ˜ − ∗ τ ∈ ∂Ff (v), (i.e., − ∗ τ = f ), and τ ∈ ∂ φ(v).

(4.89)

In this last relation, there occurs only the behavioral law between  and τ . ¶

4.4.6 Stream Here we consider a situation in which the fluid enters through − in the domain M with a given velocity and leaves M through + with an unknown velocity. The remaining boundary is that of a wall along which the fluid cannot slip. The whole bounded domain M is occupied by the fluid. This situation is typical of the stream problems, notably that M is a cylindrical pipe (we could also consider a situation with semipermeable walls, also with unknown v− ). Thus the boundary of the domain M has three complementary parts, l , − , + , on which we have the conditions (with the orientation of the normal n toward the outside of M) v| l = 0,

v| − = v− , v− given so that n.v− ≤ 0, and n. v| + ≥ 0.

(4.90)

Then we define the velocity domain   K(v− ) = v ∈ H 1 (M)3 , v| l = 0, v| − = v− , n. v| + ≥ 0 ,

(4.91)

4.4 Relations Between Force and Velocity Fields

437

which is a closed convex subset of H 1 (M)3 . In a first step, we will use a lift of the condition on − , null on + , so that we boil things down to the homogeneous condition v− = 0. This goes back to treating a stream problem without entrance of fluid, but with a new external force. Let 0 be the closure of l ∪ − , and let 1 = + ; we will use the set (which is a closed convex cone)   K = v ∈ H 1 (M)3 , v| 0 = 0, n. v| 1 ≥ 0 .

(4.92)

With E = H 10 (M)3 , and with a force field f ∈ E ∗ , or simply f ∈ L2 (M)3 , we consider the following problem. Problem 5 h(0) = inf (ψ(v)+ < f, v >). v∈K

(4.93)

A. Velocity–Force Duality ∗ be the support function of Let χK be the indicator function of K, and let σK = χK K. Since K is a cone, σK is also the indicator function of the negative polar cone8

  K − = q ∈ E ∗ , < q, v > ≤ 0, ∀v ∈ K . The dual problem of Problem 5 is (following (4.16)) the following. Problem 6 e∗ (−f ) = inf∗ (ψ ∗ (q − f ) + σK (−q)). q∈E

Then Problem 6 can be written e∗ (−f ) = inf ψ ∗ (−f + q). q∈K −

(4.94)

The solutions v¯ and q¯ of Problems 5 and 6 satisfy, through the Fenchel theorem, Theorem 1, with X = Y = E, A = I (identity),9  (i)

8 See 9 See

ψ(v) ¯ + ψ ∗ (q¯ − f ) = < q¯ − f, v¯ >, ¯ + σK (−q)+ ¯ < q, ¯ v¯ > = 0, (ii)χK (v)

(4.9), and [Aub, ch. I.3.5, p. 46]. [Eke-Tem, ch. VII.4], [Aub, ch. 1.5.2; 5.3].

(4.95)

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4 Behavior Laws

or v¯ and q¯ are also such that 

¯ (i) v¯ ∈ ∂ψ ∗ (q¯ − f ), or q¯ − f ∈ ∂ψ(v), ¯ = NK (v), ¯ (ii) − q¯ ∈ ∂χK (v)

(4.96)

with NK (v) = ∂χK (v) the normal cone (see (4.7), (4.8)), that is,   NK (v) = q ∈ E ∗ , < u − v, q > ≤ 0, ∀u ∈ K .

(4.97)

Thus v, ¯ q¯ satisfy the relations ¯ q¯ − f ∈ ∂ψ(v), ¯ thus q¯ − f = An v,

(4.98)

and from (4.95)(ii), we have v¯ ∈ K, q¯ ∈ K − , < q, ¯ v¯ > = 0. B. Velocity–Force Duality. A Variant Another way (which specifies the cones K, K − and NK (v)) is the following. Let 1/2

K1 = {μ ∈ Y, μ ≥ 0} .

Y = H00 ( 1 ),

Let γn be the map v ∈ E = H 10 (M)3 → v.n| 1 ∈ Y. Then K1 = γn (K). Problem 5 is inf

v∈E,γn (v)∈K1

(ψ(v)+ < f, v >).

(4.99)

Then we can consider the following dual problem. Problem 7 e∗ (−f ) = inf ∗ (ψ ∗ (−f − γn∗ q0 ) + σK1 (q0 )), q0 ∈Y

∗ = χ − , the negative polar cone to K with the support function of K1 , σK1 = χK 1 K 1

simply being K1− = {q ∈ Y ∗ , q ≤ 0}; see (4.9).

1

Thus this dual problem is (as in (4.95)) e∗ (−f ) = inf ψ ∗ (−f − γn∗ q0 ). q0 ∈K1−

Since 

(i) − f ∈ Int (Dom ψ ∗ + γn∗ Dom σK ) = Int (E ∗ + γn∗ K1− ) = E ∗ (ii) 0 ∈ Int (Dom χK1 − γn Dom ψ) = Int (K1 − γn E) = Y,

(4.100)

4.4 Relations Between Force and Velocity Fields

439

the conditions of the Fenchel theorem, Theorem 1, for (4.99) and (4.100) are 1/2 satisfied. Thus with X = E, Y = H00 ( 1 ), A = γn , the Fenchel theorem implies the existence of solutions of Problems 5 and 7. The solutions v¯ and q¯ of (4.99) and (4.100) satisfy 

(i) − f ∈ ∂ψ(v) ¯ + γn∗ (q¯0 ) ∗ (q¯ ). (ii) 0 ∈ −γn v¯ + ∂χK 0 1

¯ = This last relation is equivalent to q¯0 ∈ ∂χK1 (γn v) χK = χK1 ◦ γn , (4.96)(ii) gives, with (4.12),

(4.101) NK1 (γn v). ¯ Since

¯ = γn∗ ∂χK1 (γn v), ¯ −q¯0 ∈ ∂(χK1 ◦ γn )(v) whence we have the relation −q¯ = γn∗ (q0 ). We see the equivalence between (4.96) and (4.101). We also have NK (v) = Nγ −1 (K1 ) (v) = γn∗ (NK1 (γn v)). Furthermore, we have (for all q) n

∗ (q) = ∂χK − (q) = NK − (q) ¯ = {y ∈ K1 , < y, q¯0 > = 0} . ∂χK 1 1

1

Thus (4.101)(ii) implies γn v ≥ 0, and < γn v, q¯0 > = 0,

(4.102)

¯ since q¯0 ∈ K1− . Also note that which implies that q¯0 is null on the support of γn v,   Im γn∗ = γn∗ (Y ∗ ) = (ker γn )⊥ , with ker γn = v ∈ H 10 (M)3 , v.n| 1 = 0 .

C. Interpretation of the Solutions of Problems 5, 6, 7 From now on we don’t write an overline on the solutions v, q, and q0 . L2 -Forces. In the case f ∈ L2 (M)3 , we have a very easy interpretation of the solutions using the formula (4.75), with (4.101)(i). For a Newtonian fluid, if γ is the trace v ∈ H 10 (M)3 → v| 1 ∈ V 1 , we obtain −f = −Pv + γ ∗ Bv + γn∗ q0 , which implies Pv = f , and the conditions on 1 (Bv)t = 0,

and (Bv)n = n.Bv = −q0 ,

denoting by (Bv)t and (Bv)n the tangential and normal components of Bv. Thus q0 is a normal surface force to 1 such that q0 ∈ K1− , < q0 , vn > 1 = 0. For

440

4 Behavior Laws

f ∈ L2 (M)3 , the velocity field v ∈ H 1 (M)3 , with v| 0 = 0, vn | 1 ≥ 0, also satisfies (Bv)t | 1 = 0, (Bv)n | 1 ≥ 0, with < (Bv)n , vn > 1 = 0

Pv = f,

(thus the surface force on 1 , −(Bv)n is null on the support of vn ). Then the velocity field v is a solution of a problem with inequality constraint. General case for f in the dual of (H 10 M)3 . We use the decomposition of E in the product H01 (M)3 × V 1 . Through lifting R1 (see (4.80)), the space W 0 is 1/2 identified with the space V 1 = H00 ( 1 )3 . We also use the decomposition of its dual E ∗ according to (4.82). Thus we have, for all f ∈ E ∗ , f = (f0 , f 1 ), with f0 ∈ H −1 (M)3 , f 1 ∈ V ∗1 . We return to (4.96). Let f v ∈ E ∗ be such that −f v ∈ ∂ψ(v). Then f0v = Pv, f1v = −B ∗ (v 1 ), or with f v1 = R1∗ f1v ,

f v1 = −Cv 1 .

Then (4.96) gives f − f v = q ∈ −NK (v);

(4.103)

hence f0 = f0v = Pv,

f 1 − f v1 = f 1 + Cv 1 ∈ −R1∗ NK (v).

(4.104)

We will prove below that R1∗ NK (v) = Np−1 (K1 ) (v 1 ).

(4.105)

f 1 + Cv 1 ∈ −Npn−1 (K1 ) (v 1 ).

(4.106)

n

Then (4.104) is f0 = Pv,

We deduce that < f 1 − f v1 , v 1 > = 0, thus < −f 1 , v 1 > = < −f v1 , v 1 > = < Cv 1 , v 1 > ≥ 0. Let gτ and gn be the tangential and normal components of an element g ∈ V 1 ; we also obtain (f 1 + Cv 1 )τ = 0,

(f 1 + Cv 1 )n ∈ −K − , i.e., (f 1 + Cv 1 )n ≥ 0,

4.4 Relations Between Force and Velocity Fields

441

with the orthogonality (see (4.98)) < q, v > = < (f 1 − f v1 )n , (v 1 )n > 1 = < (f 1 + Cv 1 )n , (v 1 )n > 1 = 0. By (4.96), this implies the following relation showing that the dispersed energy by viscosity in the stream is independent of q since < q, v > = 0: < −f, v > = < q − f, v > = ψ(v) + ψ ∗ (q − f ) = 2ψ(v) ≥ 0. 1/2

1/2

Proof of (4.105) Let V 1 = H00 ( 1 )3 , Y = H00 ( 1 ), let pn be the map v 1 ∈ V 1 → n.v 1 ∈ Y , and let γ be the trace v ∈ H 10 (M)3 → v| 1 ∈ V 1 . Then we have γn = pn ◦ γ . This implies the following relations between the normal cones in Y ∗ , V ∗1 , E ∗ : Np−1 (K1 ) (v 1 ) = pn∗ NK1 (pn v 1 ), NK (v) = γ ∗ Np−1 (K1 ) (γ v) = γ ∗ Np−1 (K1 ) (v 1 ) n

n

n

(see [Aub, ch. I.4.6, p. 66]). Since γ ◦ R1 = I in V 1 we then deduce R1∗ NK (v) = R1∗ γ ∗ Npn−1 (K1 ) (v 1 ) = (γ ◦ R1 )∗ Npn−1 (K1 ) (v 1 ) = Npn−1 (K1 ) (v 1 ). Conclusion. In a stream of a Newtonian fluid, the relations between velocity and force are such that q = f − f v ∈ −∂χK (v) = −NK (v),

v ∈ ∂ψ ∗ (q − f ) with v ∈ K;

(4.107)

see (4.96), (4.97), with −f v ∈ ∂ψ(v), and (4.98).

D. Stream with Nonnull Entrance We directly treat the stream problem with a given nonnull entrance velocity of the fluid v| − , and with a given volume force f ∈ L2 (M)3 . The primal problem is Problem 8 inf

v∈K(v− )

(ψ(v) + (f, v)).

Let γ−,n be the map v ∈ H 1l (M)3 → (γ− v, γn ) = (v| − , n.v| + ) ∈ Y with 1/2

1/2

Y = H00 ( − )3 × H00 ( + ). The constraint of Problem 8 is written (γ−,n v + (−v− , 0)) ∈ {0} × K1 .

442

4 Behavior Laws

Then the dual problem is Problem 9 inf

q=(q− ,q1 )∈Y ∗

∗ (ψ ∗ (−f − γ−,n q) + σK1 (q1 )+ < q− , v− >).

∗ q = γ ∗ q + γ ∗ q . By the Fenchel theorem, the solutions v¯ and Observe that γ−,n − − n 1 q, ¯ still denoted by v, q, of these problems are such that ∗ −f ∈ ∂ψ(v) + γ−,n q, with q1 ∈ NK1 (γn v).

Using the formula (4.75), we obtain − f = −Pv + γ ∗ Bv + γ−∗ q− + γn∗ q1 ,

(4.108)

which gives, besides f = Pv, the boundary conditions γ−∗ (Bv + q− ) = 0, γn∗ (Bv + q1 ) = 0, γτ∗ (Bv) = 0. Therefore, every velocity field v that is a solution of Problem 8 satisfies the relations v ∈ H 1 (M)3 , v| l = 0, v| − = v− , n.v| + ≥ 0, and furthermore, Pv = f, with (Bv)τ | 1 = 0, n.Bv| 1 + q1 = 0, with q1 ≤ 0, and < q− , γn v > = 0. Remark 7 The passage between the two minimization methods, with or without lifting the incoming velocity condition, can be made in the following way: let v r be a lift of v− in El = H 1l (M), null on + = 1 . We remark that E = H 10 (M)3 ⊂ El . Then the elements of K(v− ) are v = v r +w, with w ∈ H 10 (M)3 , and thus we have, ∀v ∈ El and ∀q ∈ El∗ , K(v− ) = v r + K, χK(v− ) (v) = χK (v − v r ), σK(v− ) (q) = < q, v r > +σK (q), and also 

∂σK(v− ) (q) = v r + ∂σK (q) = v r + ∂χK − (q) = v r + NK − (q), ∂χK(v−) (v) = ∂χK (v − v r ) = NK (v − v r ).

Moreover, we verify ψ(v)+ < f, v > = (ψ(w)+ < f, w > +(v r , w)n ) + (ψ(v r )+ < f, v r >).

4.4 Relations Between Force and Velocity Fields

443

Let F = f + ∗ An v r = f − f r . We see that by changing (v, f ) into (w, F ), we obtain (4.107) for (w, F ), which is then written F − f w ∈ −NK (w). By returning to (v, f ), we obtain that (v, f ) satisfies f − f v ∈ −∂χK(v−) (v). ¶ E. Duality Between Velocity and Viscous Stresses Then the primal problem is (see (4.93)) Problem 10 ˜ inf (φ(v) + χK (v)+ < f, v >).

v∈E

Let ˜ ). Ff (v) = χK (v)+ < f, v >, G(σ ) = φ(σ Then we can write the dual problem (with respect to the viscous stress), with Y = (H 10 ) ⊂ L2 (M, S 2 M), as Problem 11 inf (φ˜ ∗ (τ ) + Ff∗ (−∗ τ )).

τ ∈Y ∗

The conjugate of Ff is given by ∗ (q − f ) = σK (q − f ) = χK − (q − f ), Ff∗ (q) = χK

q ∈ E∗,

and thus if −∗ τ − f ∈ K − , then Ff∗ (−∗ τ ) = χK − (−∗ τ − f ) = 0,

Ff∗ (−∗ τ ) = +∞ otherwise.

The dual problem is written as a minimization problem with constraint inf

τ ∈Y ∗ ,−∗ τ −f ∈K −

φ˜ ∗ (τ ).

The elements v and τ , solutions of Problems 10 and 11, satisfy the relations 

Ff (v) + Ff∗ (−∗ τ ) = − < ∗ τ, v >, ˜ φ(v) + φ˜ ∗ (τ ) = < τ, v >,

which is also equivalent to 

−∗ τ ∈ ∂Ff (v) = NK (v) + f, ˜ τ ∈ ∂ φ(v).

444

4 Behavior Laws

As in (4.38), we obtain the previous results (4.107), with q¯ = −∗ τ . Then let 1/2 f = (fM , f 1 ) ∈ E ∗ , and fM ∈ L2 (M)3 , f 1 ∈ (H00 ( 1 )3 )∗ . We have Pv =

−∗ An v = div τ = f0 ∈ L2 (M)3 and −Bv = n.τv | ∈ H −1/2 ( )3 according to (4.60). Application of the Green formula 





τij ∂i vj dx =

< −τ, v > = − M

∂i τij vj dx − M

ni τij vj d 1

in the calculus of Ff∗ (−∗ τ ) gives Ff∗ (−∗ τ )

 = sup (< −τ, v > − < f, v >) = sup (− v∈K

v∈K

(ni τij + fi )vj d 1 ). 1

Then we get Ff∗ (−∗ τ ) = +∞, except if (n.τ +f 1 )τ = 0, and (n.τ +f 1 )n ≥ 0, that is, (−Bv + f 1 )τ = 0, and (−Bv + f 1 )n ≥ 0, in which case we have Ff∗ (−∗ τ ) = 0. ¶

4.4.7 Slipping Condition on the Boundary Here we consider a situation in which a Newtonian fluid fills the whole bounded domain M of a container and slips on the boundary. The velocity space is     V = H 1 (M) = v ∈ H 1 (M)3 , n.v| = 0 = v ∈ H 1 (M)3 , v(x) ∈ Tx , x ∈ . We have to know whether the condition (a + b × x).nx = 0, ∀x ∈ implies a = b = 0, in order to specify the kernel of , ker  in V . For instance, taking a sphere with centre O, we see that the previous condition implies a = 0, but b is arbitrary. In the case of a plane with nx = e3 , we obtain a3 = 0, b1 = b2 = 0, but a1 , a2 , b3 are arbitrary. Thus ker  = {0} if there exists a displacement leaving the surface globally invariant. Thus except in the case in which M is a ball (for bounded M), and in very particular geometric situations, such as the half-space for unbounded domains, we have ker  = {0}. In the following, we assume for simplicity that this is satisfied. The velocity space V is a closed vector subspace of H 1 (M)3 , containing 1 H0 (M)3 . We have the decomposition V = H01 (M)3 ⊕ W with W , which we can identify with   1/2 H ( ) = v ∈ H 1/2( )3 , v (x) ∈ Tx , x ∈ ,

4.5 Relations σ,  with Trace Constraint

445

so that V is identified with the product E = H01 (M)3 × H ( ). Then the dual W ∗ 1/2 of W is identified with the dual of H ( ), which we can also identify with 1/2

−1/2

H

  ( ) = f ∈ H −1/2( )3 , f (x) ∈ Tx∗ , x ∈ , −1/2

(or n.f = 0), so that the dual E ∗ of E is E ∗ = H −1 (M)3 × H ( ), with the decomposition of f ∈ E ∗ into f = (f0 , f ). 1/2 Let J be the natural injection of H ( ) into H 1/2 ( )3 , and P = J ∗ its 1/2 −1/2 dual. Then we define the Calderón operator C from H ( ) into H ( ) (as in (4.54)) by j

(C(u ))k = (P nj τk (u)| ),

1/2

∀u ∈ W , u| = u ∈ H ( ).

(4.109)

Then the Calderón operator C is related to the Calderón operator C defined by (4.54) by C = P ◦ C ◦ J . Let ψ (v ) = 12 (v , C v ). The resistance function ψ splits according to ψ(v) = ψ0 (v0 ) + ψ (v ). Then the conjugate function ψ ∗ of ψ splits as ψ ∗ (f ) = ψ0∗ (f0 ) + ψ ∗ (f ). The “viscous” relation between force and velocity, expressed by (4.21) thanks to the resistance function ψ (see (4.34)) and its conjugate according to the mechanic principle, is given by Pv0 = f0 ,

C (v ) = −f ,

with C (v ) = P Cv , and v = v| , n.v| = 0,

(4.110)

and we have the relation − < f, v > = 2ψ(v).

4.5 Relations σ, with Trace Constraint We first consider a minimization problem with constraint on the trace of the rate strain tensor  ∈ S 2 T M, where the stress tensor σ ∈ S 2 T ∗ M intervenes as parameter: Problem 12 h(q) =

inf (φ())− < σ,  >).

, t r =q

446

4 Behavior Laws

Let A0 be the map from S 2 M into R such that A0  = tr . The conjugate A∗0 from R into S 2 M is defined by < A∗0 q,  > = < q, A0  > = < q, tr  > = tr (q) = < qI,  >, and thus A∗0 q = qI, with I the identity. We introduce the dual optimization problem of Problem 12, by taking A = −A0 in (4.14), with q ∈ R as a parameter. Problem 13 e∗ (σ ) = inf (φ ∗ (σ + pI ) − < p, q >). p∈R

In a sense, these Problems 12 and 13 are trivial, since the resistance function φ splits into φ = φ0 + φD in the decomposition (4.24) (see (4.25)). The application of the Fenchel theorem to this minimization problem with constraint gives the following results [Aub, ch. I.5.3, Cor. 5.2]. Proposition 4 Trace constraint for a Newtonian relation. There exist solutions ˆ and p, ˆ respectively of Problem 12 (i.e., h(q)) and Problem 13 (i.e., e∗ (σ )), and these are also solutions of ˆ ∈ ∂e∗ (σ ),

pˆ ∈ ∂h(q), with tr ˆ = q,

(4.111)

ˆ thus σ = An (ˆ ) − pI.

(4.112)

and they are related by σ ∈ ∂φ(ˆ ) − pI, ˆ

Furthermore, we have h(q) + e∗ (σ ) = 0. Proof We return to the decompositions of φ() (see (4.25)) and < σ,  > according to the decomposition of tensors into deviator and trace in (4.24): φ() = φ0 (A0 ) + φD ( D ), < σ,  > = < σ S ,  S > + < σ D ,  D > =

1 A0 σ.A0 + < σ D ,  D > . 3

The decomposition of the conjugate function φ ∗ of φ is 1 1 2 ∗ q . φ ∗ (σ ) = φ0∗ ( A0 σ ) + φD (σ D ), with φ0∗ (q) = 3 2K

(i) Expression of h(q). We obtain h(q) = φ0 (q) −

1 < A0 σ, q > + inf(φ1 ( D )− < σ D ,  D >), 3

4.5 Relations σ,  with Trace Constraint

447

and thus h(q) =

K 2 1 q − < A0 σ, q > −φ1∗ (σ D ). 2 3

We note that h(0) = −φ1∗ (σ D ). (ii) Expression of e∗ (σ ). The dual problem is given by 1 e∗ (σ ) = φ1∗ (σ D ) + inf φ0∗ ( A0 σ + p) − pq) p∈R 3 = φ1∗ (σ D ) + inf

p∈R

1 1 (( A0 σ + p)2 − pq). 2K 3

The infimum is reached by pˆ = −(qσ /3) + Kq, with qσ = A0 σ = tr σ, and thus 1 K −e∗ (σ ) = q(− qσ + q) − φ1∗ (σ D ) = h(q). 3 2

¶

From the stress tensor σ and from the trace q of the strain tensor , we have obtained the strain tensor ˆ and pˆ as solutions of the two dual optimization problems, Problems 12 and 13, by 1 D ˆ = A−1 n σ + I q, 3

1 pˆ = Kq − tr σ. 3

(4.113)

The condition p ≥ 0 implies the inequality Kq ≥ qσ /3, i.e., K tr  ≥ tr σ/3. When q = 0, we have pˆ = −tr σ/3. In this case of incompressible fluids, we can think of pˆ as the pressure Ph of Chapter 3. In the general case, a conjecture is that the decomposition (4.113) of pˆ into pˆ = Kq − 13 tr σ corresponds to the decomposition of the pressure pˆ into pˆ = Ph + P , with P = Kq the thermodynamic pressure. ¶ A variant, with Lagrange multiplier. We introduce the function (often called the “Lagrangian”) L(, p) = Lp () = φ() − p tr ,

p ∈ R,

(4.114)

with p a Lagrange multiplier. The conjugate function L∗p of Lp is given by L∗p (σ ) = sup (< σ,  > −Lp ()) ∈S 2 M

= sup (< σ + pI,  > −φ()) = φ ∗ (σ + pI ). ∈S 2 M

(4.115)

448

4 Behavior Laws

Notice that L∗p (σ ) is also expressed by L∗p (σ ) = sup sup (< σ,  > −φ() + pq) = sup (pq − h(q)) = h∗ (p). q∈R , tr =q q∈R (4.116) Let σ be the rate stress tensor and let  be the strain tensor defined by ¯ ∈ ∂L∗p (σ ) = ∂φ ∗ (σ + pI ),

σ¯ ∈ ∂Lp (),

(4.117)

so that (suppressing the overline) Lp () + L∗p (σ ) = < σ,  >, or φ() − p tr  + φ ∗ (σ + pI ) = < σ,  > . Let τ = σ + pI be the relationship between the stress and the viscous stress. We again obtain φ() + φ ∗ (τ ) = < τ,  > . We see that every solution of the dual problem intervenes as a Lagrange multiplier. This is a general fact (see [Aub, ch. I.5.6, p. 76]). Proximity map. We return to our previous study from a slightly different point of view. Let p be a positive number. Let pˆ = p/3K, with K = λ + (2/3)μ. Then we can write, with (4.26), ) )2 p tr  = (pI, ˆ )n , and )pI ˆ )n = p2 /K.

(4.118)

Let φ (p) be the resistance function (relative to p) defined by (p)

def

φ (p) () = φn () =

)2 1) ) − pI ˆ )n , 2

∀ ∈ S 2 M.

(4.119)

Thus using (4.118), we have def

φ (p) () =

1 1) ) 1) ) 2 2 &&2n − p tr  + )pI ˆ )n = L(, p) + )pI ˆ )n . 2 2 2

(4.120)

The conjugate function is given by φ (p) ∗(σ ) = sup (tr ((σˆ + pI ) ◦ ˆ ) − φ()) − ∈S 2 M

)2 1) )pI ˆ )n , 2

whence φ (p) ∗ (σ ) = Lp∗ (σ ) −

)2 1) 1) ) 2 )pI ˆ )n = φ ∗ (σ + pI )) − )pI ˆ )n . 2 2

(4.121)

4.6 Force–Velocity with Constraint on the Divergence

449

Thus the stress tensor σ and the rate strain tensor  are linked through the resistance function (4.119) by the fundamental relations σ ∈ ∂φ (p) (),

 ∈ ∂φ (p) ∗(σ ).

(4.122)

ˆ → ) (with (4.121)) is called a proximity map The map p →  ∈ ∂φ (p) ∗(σ ) (or pI (see [Eke-Tem, ch. II.2]). The “dispersed” power is given by < σ,  > = φ (p) () + φ (p)∗ (σ ) = Lp () + L∗p (σ ).

(4.123)

With (4.120) and (4.121) (and φ ∗ (τ ) = φ ∗ (σ + pI ) = φ()), we also have < σ,  > = 2φλ,μ () − ptr  = &&2n − (, pI ˆ )n ) )2 ) )2 ) − )(p/2)I ) . = ) − (p/2)I ˆ ˆ n n

(4.124)

Remark 8 Recall that the dispersed power by viscosity is a positive quantity: < τ,  > = < σ + pI,  > = 2φ() ≥ 0. If < σ,  > is a dispersed power, i.e., also a positive quantity (which corresponds to the name of the constraint of σ ), then we must have )2 )2 ) ) ) − (pI ˆ /2)n , ˆ /2))n ≥ )pI

(4.125)

) ) ˆ /2)n ), that is, and thus  must be outside the ball Bλ,μ (pI ˆ /2, )pI √ p∈ / Bλ,μ (pI /6K, p/(2 K)). Conversely, p must satisfy p tr  ≤ &&2n , whence if tr  > 0, then p ≤ &&2λ,μ / tr . ¶ Remark 9 The resistance function relative to p is such that φ (p) (0) =

)2 1) )pI ˆ )n = 0, and 0 ∈ / ∂φ (p) (0). 2

(We have 0 ∈ ∂φ (p)(pI ˆ ) and 0 ∈ ∂φ (p) ∗(pI ).) It does not satisfy Hypothesis 2.

¶

4.6 Force–Velocity with Constraint on the Divergence 4.6.1 General Case for a Newtonian Fluid Let V be a closed subspace of H 1 (M)3 , with H01 (M)3 ⊆ V ⊆ H 1 (M)3 . If V ∩ ker  = {0}, let E be the quotient space of V by the kernel of the map .

450

4 Behavior Laws

∗ We consider  the following minimization problems, where f ∈ E is a force field, with ψ(v) = M φ(v)dx (see (4.34)).

Problem 14 hf (q) = h(q) =

inf

v∈V , div v=q

(ψ(v)+ < f, v >).

Let A be the operator from V (or from E) into L2 (M) defined by Av = −div v.

(4.126)

In order to treat this problem, we consider the dual problem Problem 15 q

e∗ (−f ) = e∗ (−f ) =

inf

p∈L2 (M)

(ψ ∗ (−f − A∗ p) − < p, q >).

We also use the “Lagrangian” ˜ L(v, p) = L˜ p (v) =

 L(∇v , p) dx M



(4.127)

(φ(∇v ) − p tr ∇v )dx = ψ(v)+ < p, Av >,

= M

with p ∈ L2 (M) a function on M that intervenes as a Lagrange multiplier. Let ψ p be the resistance function on E relative to the function p (linked with the proximity map), defined by 

p φn (v)dx

ψ (v) = p

M



1 = 2

 M

&∇v − pI ˜ &2n dx =

1 &∇v − pI ˜ &2n 2

1 1 ˜ &2n = L˜ p (v) + &pI ˜ &2n . = ψ(v) − p tr ∇v dx + &pI 2 2 M

(4.128)

Usually we exchange Problem 14 with constraint for the following problem. Problem 16 inf (L˜ p (v)+ < f, v >), or inf (ψ p (v)+ < f, v >).

v∈E

v∈E

Then the conjugate function L˜ p∗ of L˜ p (or (ψ p )∗ of ψ p ) is L˜ p∗ (−f ) = sup (< −f, v > − < p, Av > −ψ(v)) v∈V

= sup (< −f, v > − < A∗ p, v > −ψ(v)) = ψ ∗ (−f − A∗ p). v∈V

(4.129)

4.6 Force–Velocity with Constraint on the Divergence

451

We also observe that inf (L˜ p (v)+ < f, v >) = − sup (< −f, v > −L˜ p (v)) = −L˜ ∗p (−f ).

v∈E

v∈E

˜ Let v, f be the velocity field and the force field defined from the Lagrangian L(v, p) by thus − < f, v > = L˜ p (v) + L˜ p∗ (−f ). (4.130) Of course we would have similar relations with the function ψ p . Let f p = A∗ p ∈ E ∗ . With (4.129), we verify the equivalences v ∈ ∂ L˜ p∗ (−f ),

−f ∈ ∂ L˜ p (v),

v ∈ ∂ L˜ ∗p (−f ) ⇔ v ∈ ∂ψ ∗ (−f − f p ) ⇔ −f − f p ∈ ∂ψ(v) ⇔ (−f ∈ ∂ L˜ p (v) ⇔ − < f + f p , v > = ψ(v) + ψ ∗ (−f − f p ).

(4.131)

Furthermore, in the present case in which ψ is quadratic, we have − < f + f p , v > = ψ(v) + ψ ∗ (−f − f p ) = 2ψ(v) ≥ 0.

(4.132)

We remark that f0 = f + f p is a viscosity force and that (4.132) corresponds to the dispersed power by viscosity. Notice that (4.132) may still be written − < f, v > + < p, div v > = 2ψ(v) ≥ 0.

(4.133)

We can deduce (4.130) from (4.117), in a similar way to (4.38). With (4.128), we have −f = ∗ σ ∈ ∗ ∂φ (p) (v) = ∂(φ (p) ◦ )(v) = ∂ψ (p) (v) = ∂ L˜ p (v). We can also prove (4.130) in the following way. From (4.39), we have ψ ∗ (−f − f p ) + ψ(v) = − < f + f p , v >,

(4.134)

ψ ∗ (−f − A∗ p) + ψ(v)+ < A∗ p, v > = − < f, v >,

(4.135)

ψ p∗ (−f ) + ψ p (v) = L˜ p∗ (−f ) + L˜ p (v) = − < f, v >,

(4.136)

and thus

whence

which gives (4.130). These relations (4.130) are still equivalent to that v ∈ E is obtained from f ∈ E ∗ by solving the “Stokes problem”

452

4 Behavior Laws



(v , w )λ,μ dx = − < f + A∗ p, w >,

(v , w )n =

∀w ∈ E.

(4.137)

M p

Let τv and σ = σv be the tensor fields defined by 

 tr (τv ◦ w )dx = M

(v , w )n dx,

∀w ∈ E,

(4.138)

M

thus with (4.26), τv = An v , and let σ be the stress tensor σ = σvp = −pI + τv = −pI + An v = (−p + tr v )I + 2μv .

(4.139)

We see that σ is solution of the problem (still called the Stokes problem)  tr (σ ◦ w )dx = − < f, w >,

∀w ∈ E.

(4.140)

M

Then using, for instance, the proximity map, we verify 1 ˜ w ) − 1 &pI ˜ &2n = (v , v )n − φ(v ) − &pI ˜ &2n , ψ p∗ (−f ) = sup (v , w )n − φ( 2 2 w∈E and then ψ p ∗ (−f ) =



tr (σ ◦ v )dx − φ˜ p (v ) = − < f, v > −ψ p (v), M

which is (4.130). Proposition 5 Let the resistance function be given by the function ψ p (4.128) or by the “Lagrangian” L˜ p (4.127). The mechanic principle (see (4.21)) applied to this function with the constraints of Problems 14 and 15 gives the “viscous relations” (4.130) between velocity and force, with given p. These relations are equivalent to the “Stokes problem” (4.137) (or (4.140) with (4.139)). Proposition 6 The relation between (v, p) ∈ E × X∗ and (−f, q) ∈ E ∗ × X is obtained by solving the two optimization problems, Problems 14 and 15, whose solutions v and p satisfy q

v ∈ ∂e∗ (−f ),

p ∈ ∂hf (q), with q = div v,

−f ∈ ∂ψ(v) + A∗ p,

(4.141)

and the relation is an isomorphism between the following spaces, respectively in the Dirichlet and the Neumann conditions:

4.6 Force–Velocity with Constraint on the Divergence

E=

X=

H01 (M)3 ,

∗

E =H

−1

L20 (M)

∗

453

   2 = q ∈ L (M), q dx = 0 , M

(4.142)

(M) , X = L (M)/R 3

2

(respectively E = H 1 (M)3 / ker , X = X∗ = L2 (M)). Proof of (4.141) from (4.130). Let v be a solution of (4.130) with given f and p and let q = div v. We have q ∈ div ∂ψ ∗ (−f − A∗ p) = div ∂ L˜ p∗ (−f ). Then (see Section A.1 or [Aub, ch. I.5.3, p. 73]) p ∈ ∂h(q), p is solution of e∗ (−f ); furthermore, the relation −f ∈ ∂ψ(v) + A∗ p implies v ∈ ∂e∗ (−f ). Thus (4.141).¶ Note that the case that q = 0 corresponds to an incompressible fluid, treated in [Eke-Tem, ch. IV.2.5] with μ = 1. We have equivalence between (4.141) and the following problem, which expresses the relations between velocity and force fields, divergence of the velocity, and the pressure p: 

(v , w )n + (p, div w) = − < f, w >, div v, ϕ) = (q, ϕ), ∀ϕ ∈ L2 (M).

∀w ∈ E,

(4.143)

The existence and uniqueness of the solution (v, p) of (4.143), in a space E × X∗ with given (f, q) ∈ E ∗ × X, results from the Fenchel theorem. We also find a proof in [Gir-Rav]. The functional framework and the Stokes problem, according to various boundary conditions, are given below.

4.6.2 Relation f, v with Constraint, Dirichlet Condition Let V = E = H01 (M)3 and A∗ p = grad p ∈ E ∗ = H −1 (M)3 . For f ∈ H −1 (M)3 , the conjugate function L˜ p∗ of L˜ p is given by L˜ p∗ (−f ) = ψ ∗ (−f − grad p). Here the viscosity force is f0 = f + grad p, and the dispersed power by viscosity is 2ψ(v) = − < f0 , v > = − < f + grad p, v > ≥ 0. The Stokes problem (4.137) is  (v , w )n =

(v , w )n dx = − < f + grad p, w >, M

∀w ∈ E.

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4 Behavior Laws

The functional framework is given by (4.142). The space X∗ is X∗ = L2 (M)/R, which is not identified with X in order that p can be positive (see below, Remark 11). Then we can assert the existence and uniqueness of the solution (v, p) of (4.143). Note that in the case f ∈ L2 (M)3 , the relation between force and velocity is also that given by solving the Stokes problem in the form − Pv + grad p = −f, in M (Bv − pn)| = 0.

Inhomogeneous Dirichlet Problem with Divergence Constraint The main change with respect to the homogeneous Dirichlet problem is that the conjugate A∗ of the operator A : w ∈ H 1 (M)3 → −div w ∈ L2 (M) is no longer a gradient. It is defined by ∗





< A p, w > = −

p div w dx = − M

tr (pI w) dx = < −pI, w >, M

∀w ∈ H 1 (M)3 . This proves that A∗ p = ∗ (−pI ) (for the usual duality of the trace). We remark that ker  ⊂ ker A, and A is also a map from E = H 1 (M)3 / ker  into L2 (M). We consider the “regular case” in which −Pv + grad p ∈ L2 (M)3 . Then we can apply the Green formula, ∀w ∈ H 1 (M)3 

 (An v − pI, w) dx = M

(−Pv + grad p).w dx+ < (Bv − pn), w > . M

(4.144) This formula implies that B p v = Bv − pn ∈ H −1/2( )3 . Notice that a priori, neither Pv nor grad p is in L2 (M)3 , and then neither Bv nor pn makes sense in H −1/2( )3 . We can still deduce from the Green formula the relation similar to (4.75) ∗ (An v − pI ) = (−Pv + grad p) + γ ∗ (Bv − pn).

(4.145)

Remark 10 Trace condition with the pressure. Here we want to determine the traces of curl v and q = div v on the boundary of M with the conditions v ∈ H 1 (M)3 , Ap v = div σvp = Pv − grad p ∈ L2 (M)3 , p ∈ L2 (M), with given Pv by (4.40). We have curl Ap v ∈ H −1 (M)3 , and thus if μ is constant (or smooth, μ ∈ W 2,∞ (M)), we have curl curl (μ curl v) ∈ H −1 (M)3 , and thus (μ curl v) ∈ H −1 (M)3 with (μ curl v) ∈ L2 (M)3 ,

4.6 Force–Velocity with Constraint on the Divergence

455

and thus (μ curl v)| ∈ H −1/2( )3 , whence ( curl v)| ∈ H −1/2 ( )3 . We also have div Ap v ∈ H −1 (M)3 , and thus (−p + (λ + 2μ)q ) ∈ H −1 (M)3 with p and λq in L2 (M). Then we deduce that (−p + (λ + 2μ)q)| ∈ H −1/2( ). ¶ When the velocity on the boundary of the domain M is given, the relation between force field f ∈ L2 (M)3 and velocity field v ∈ V = H 1 (M)3 is expressed by the primal minimization problem with constraint v| = v ∈ H 1/2( )3 and div v = q ∈ L2 (M) given such that 

 qdx + M

v .n d = 0.

(4.146)



Problem 17 h(q, v ) =

inf

v∈V ,q= div v,γ v=v

(ψ(v) + (f, v)).

Let A be the map v ∈ V → Av = (−div v, γ v) ∈ Y = L2 (M) × H 1/2 ( )3 ). The constraint of (17) is then Av + (q, −v ) = 0. The dual problem is written as follows. Problem 18 e∗ (−f ) =

inf (ψ ∗ (−f − A∗ (p, z)) − (p, q)+ < z, v > )

(p,z)∈Y ∗

with Y ∗ = L2 (M)3 × H −1/2 ( )3 ,

and A∗ (p, z) = A∗ p + γ ∗ z ∈ (H 1 (M)3 )∗ .

The conditions for the application of the Fenchel theorem are −f ∈ Int ((ker )⊥ + (ker A)⊥ ),

(q, z) ∈ Int (AE).

Moreover, we have (ker )⊥ + (ker A)⊥ = (ker  ∩ ker A)⊥ = E ∗ (since (ker )⊥ is of finite codimension), which satisfies the first condition. The second is simply (4.146). This implies the existence of the solutions of Problems (17) and (18) such that (still denoting these solutions respectively by v and (p, z)) − f ∈ ∂ψ(v) + A∗ (p, z).

(4.147)

− f = ∗ An v + A∗ p + γ ∗ z.

(4.148)

We also deduce that

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4 Behavior Laws

By applying (4.148) to w ∈ H01 (M)3 , we obtain the usual equation − Pv + grad p = −f ∈ L2 (M)3 .

(4.149)

Then we can use (4.145) in (4.148), which gives γ ∗ (Bv − pn + z) = 0, and hence Bv − pn + z = 0 on .

(4.150)

Thus z = f is the “exterior surface force” applied to . We observe that Bv −pn ∈ H −1/2( )3 , but a priori, no terms are in this space. We still could use the Calderón operator to remedy this difficulty. However, by Remark 10, we know that if q = div v has a trace in H −1/2( ) (for instance if we have q ∈ H −1 (M)), then it is the same for p, and thus for Bv.

4.6.3 Relation f, v with Constraint, Neumann Condition For a given force f in the dual space of E = H 1 (M)3 , the velocity is obtained by solving the minimization problem h(q) =

inf

v∈E, div v=q

(ψ(v)+ < f, v >),

and the dual problem is written e∗ (−f ) =

inf

(ψ ∗ (−f − A∗ p) − (p, q)).

p∈L2 (M)

Since Dom ψ ∗ = (ker )⊥ , and Im A∗ = (ker A)⊥ , and also ker  ⊂ ker A, the conditions for the application of the Fenchel theorem are −f ∈ Int ((ker )⊥ + (ker A)⊥ ) = (ker )⊥ ,

−q ∈ Int (AE) = L2 (M).

Regular Neumann Framework The force field f ∈ (ker )⊥ is such that f = fM + γ ∗ f with fM ∈ L2 (M)3 (and f ∈ H −1/2( )3 ). For a Newtonian fluid, the relation −f ∈ ∂ψ(v) + A∗ p gives −f = ∗ (An v − pI ). By application to H01 (M)3 , this gives −fM = −Pv + grad p ∈ L2 (M)3 . Therefore, we can apply (4.145), thus −γ ∗ f = γ ∗ (Bv − pn), i.e., Bv − pn + f = 0 on . We have obtained that the velocity field satisfies

4.6 Force–Velocity with Constraint on the Divergence

Pv − grad p − fM = 0 in M,

457

Bv − pn + f = 0 on .

General Neumann Framework (i) Let us specify the element f p = A∗ p ∈ E ∗ . We verify that A∗ p = ∗ (−pI ). Let v p ∈ E = H 1 (M)3 / ker  be a solution of the problem ∀w ∈ H 1 (M)3 .

(v p , w)n = (pI, w), Then we have def

(f 0 , f ) = f p = ∗ (−pI ) = −∗ An (v p ). p

p

Taking w in D(M)3 , we obtain f0 = Pv0 = grad p ∈ H −1 (M)3 . Then taking w in W (see (4.48)), we have, with (4.66), p

p

 < v p  , Bw > = (pI, w). Using the Calderón operator C, we deduce the relation f = R1∗ f p = −Cv , hence f p = (f 0 , f ) = (grad p, −Cv ). (4.151) [Notice that if p ∈ H 1 (M), then application of the Green formula gives p

p

p

p

p



 grad p.v dx +

(pI, v) = (pI, div v) = − M

pn.v d ,

p

p

and then we have −(f p ) = p| n = Bv0 − f .] (ii) Let us verify that A∗ is an isomorphism from L2 (M) onto its image; indeed, p f p = A∗ p = 0 implies f0 = grad p = 0, and thus p is constant, while for p = 1, we have A∗ 1 = f 1 = (0, f 1 ), with  < f 1 , u > = −

u.n d = − < γn u, 1 > ,

whence < f 1 , u > = − < u, γn∗ (1 ) >, and thus f 1 = −γn∗ (1 ) = 0. (iii) For V = H 1 (M)3 (or E = V / ker ), the conjugate function L˜ p∗ of L˜ p is given by (4.129) with f p expressed by (4.151). The formulas (4.130) split according to (4.60). For the “Lagrangian” L˜ p , indeed we have L˜ p (v) = L˜ p (v0 ) + L˜ p (v ), thus L˜ p = L˜ p,0 + L˜ p, ,

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4 Behavior Laws

with 

L˜ p,0 (v0 ) = ψ0 (v0 )+ < p, Av0 >, p L˜ p, (v ) = ψ (v )+ < p, AR1 v > = ψ (v )+ < f , v > .

In a similar way, we have p p L˜ p∗ (−f ) = ψ ∗ (−f − f p ) = ψ0∗ (−f0 − f0 ) + ψ ∗ (−f − f ),

and thus L˜ ∗p = L˜ ∗p,0 + L˜ ∗p, . Then the relations (4.131) split into < −f0 − grad p, v0 > = 2ψ0 (v0 ), p

< −f − f , v > = 2ψ (v ) = < v , Cv >, which yield, by summing these two equalities, < −f, v > + < p, div v > = 2ψ(v). Moreover, the relation −f ∈ ∂ψ(v) + A∗ p of (4.141) splits into −f0 − grad p ∈ ∂ψ0 (v0 ),

p

−f − f ∈ ∂ψ (v ),

or −f0 − grad p = −Pv0 ,

p

−f − f = Cv ,

which is the “Stokes problem” with boundary condition Pv = f0 + grad p,

p

C( v| ) = −f − f .

(4.152)

We can assert (by the previous theorems) the existence and uniqueness of the solution (v, p) of (4.143) in the functional framework E = H 1 (M)3 / ker , and X = L2 (M), for f ∈ E ∗ = (ker )⊥ . Remark 11 On the positivity of the pressure. We obtain a positive pressure by changing the constraint div v = q in Problem 14 for the inequality constraint div v − q ≥ 0. Denoting by K − (respectively K + ) the closed negative (respectively positive) cone of L2 (M), allows us to express the inequality constraint by Av + q ∈ K −. Let us consider the Neumann framework, with the decomposition of the space E according to (4.49) into the product H01 (M)3 × H˜ 1/2( )3 (thanks to the lift R1 ; see

4.6 Force–Velocity with Constraint on the Divergence

459

(4.80)). Thus we have v = v0 + v1 = v0 + R1 v or v = (v0 , v ), ∀v ∈ E, and f = (f0 , f ) ∈ E ∗ . Then with Ad, = AR0 , we have Av = Av0 + Av1 = Av0 + AR0 v = Ad,0v0 + Ad, v ∈ Y = L2 (M). Using the resistance function (4.36) and the decomposition (4.49), (4.58), we consider the minimization problem, with f ∈ E ∗ , h(q) =

inf

Ad,0 v0 +Ad, v +q∈K −

(ψ0 (v0 ) + ψ (v )+ < f, v >).

Let X0 = H01 (M)3 , X1 = H˜ 1/2 ( )3 , Y = L2 (M); the dual problem is e∗ (−f0 , −f ) = inf∗ (ψ0∗ (−f0 −A∗d,0p)+ψ ∗ (−f −A∗d, p)+σK (p)− < p, q >). p∈Y

We easily verify the application conditions of the Fenchel theorem. Then we can assert the existence of the solutions v¯ = v = (v0 , v ) of the problem h(q), and p¯ = p of the problem e∗ (−f0 , −f ). Furthermore, (v, p) is also a solution of the inclusion system 

∗ p), ∗ p), v ∈ ∂ψ (−f − Ad, i) v0 ∈ ∂ψ0 (−f0 − Ad,0 ii) p ∈ NK − (Ad,0v0 + Ad, v + q),

(4.153)

with   NK − (q0 ) = ∂χK − (q0 ) = p ∈ L2 (M), p ∈ (K − )− = K + , < p, q0 > = 0 , for q0 ∈ K − , and the set of solutions v = (v0 , v ) of h(q) (or of (4.153)) is the subdifferential ∂e∗ (−f0 , −f ). The relations (4.153)(i) give also ∗ p ∈ ∂ψ0∗ (v0 ), −f0 − Ad,0

∗ −f − Ad, p ∈ ∂ψ ∗ (v ),

and thus ∗ Pv = Pv0 = f0 + Ad,0 p = f0 + grad p,

p

C( v| ) = Cv = −f − f .

 The relation (4.153)(ii) means that p is a positive function such that M p(−div v + q) dx = 0, with (−div v + q) ∈ K − , whence (−div v + q) ≤ 0. This implies p(−div v + q) = 0, and hence −div v + q = 0 on the support of p. ¶ The conditions of mixed type (Neumann and Dirichlet on complementary parts of the boundary) with pressure do not present new difficulties.

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4 Behavior Laws

4.6.4 Stream with Constraint We return to the situation of a stream with the divergence constraint of previous examples, and notably Problems 5 and 14. Let E = (H 1l )3 = {v ∈ H 1 (M)3 , v| l = 0}. Let v ∈ E, γ− v = v| − , and γn,+ v = n.v| + . Let A be the map v ∈ E → Av ∈ Y , with 1/2

1/2

Av = (−div v, γ− v, γn,+ v) ∈ Y = L2 (M) × H00 ( − )3 × H00 ( + ). 1/2

Let q ∈ L2 (M), v− ∈ H00 ( − )3 ; then let K(q, v− ) = {v ∈ E, div v = q, v| − = v− , n.v| − ≥ 0}. 1/2

We still denote by K1 the positive cone of H00 ( + ). The “viscous relation” between force and velocity is then expressed by the following problem with inequality constraint. Problem 19 inf

(ψ(v)+ < f, v >)

v∈K(q,v− )

or

inf

Av+(q,−v− ,0)∈{0}×{0}×K1

(ψ(v)+ < f, v >).

Then the dual problem is Problem 20 inf

∗ q=(p,z ˜ − ,zn,+ )∈Y

(ψ ∗ (−f − A∗ q) ˜ + σK1 (zn,+ ) − (p, q)+ < z− , v− > − ).

∗ z Naturally we have A∗ q˜ = A∗d p + γ−∗ z− + γn,+ n,+ . Subject to verification of the conditions of the Fenchel theorem, the solutions of the previous problems are linked by −f ∈ ∂ψ(v) + A∗ (q), ˜ and thus ∗ − f = ∗ (An v − pI ) + γ−∗ z− + γn,+ zn,+ .

(4.154)

In the regular case with f = fM ∈ L2 (M)3 , this gives −f = −Pv + grad p, and ∗ z then γ ∗ (Bv − pn) + γ−∗ z− + γn,+ n,+ = 0, which correspond to the boundary conditions, with = ∂M: 1. γ−∗ (Bv − pn + z− ) = 0, thus Bv − pn + z− = 0 on − , ∗ (Bv) = 0, thus (Bv) = 0 on , 2. γt,+ τ + ∗ (Bv − pn + z 3. γn,+ n,+ ) = 0, thus (Bv)n − p + zn,+ = 0 on + , with zn,+ ∈ NK1 (vn,+ ), i.e., zn,+ ∈ K1− , < zn,+ , vn,+ > = 0.

(4.155)

4.6 Force–Velocity with Constraint on the Divergence

461

There is no condition on z− ; thus the first point does not give any condition on (Bv − pn) − . Therefore, the velocity field v ∈ H 1 (M)3 , v| l = 0 satisfies the conditions Pv − grad p = f in M,

v| − = v− , vn | + ≥ 0,

(Bv)τ = 0 on + , and Bv − pn + zn,+ = 0, with (4.155).

(4.156)

Remark 12 We can also use a lift of the entrance condition of the fluid in order to return to the condition v| − = 0. We can choose the lift vr such that vr ∈ H 1 (M)3 , Pvr = 0 in M,

vr | l = 0, vr | − = v− , vr | + = 0.

Let v = vr + w, and qr = q− div vr ; we transform Problem 19 into the following. Problem 21 inf

(ψ(w + vr )+ < f, w + vr >),

w∈K(q,0)

which is written in the form (up to a constant) inf

(ψ(w)+ < f, w >),

w∈K(q,0)

(the condition Pvr = 0 eliminates the “nondiagonal terms” between vr and w). Then we can change the spaces E and Y for E0 = H 10 (M)3 with 0 = l ∪ − 1/2

and for Y0 = L2 (M) × H00 ( + ), and also we change the map A for the map A0 : v ∈ E0 → A0 v = (−div v, γn,+ v) ∈ Y0 (still with γn,+ v = n.v| + ). ∗ z Then the dual problem is, with A∗0 q˜ = A∗d p + γn,+ n,+ , as follows. Problem 22 inf

∗ q=(p,z ˜ n,+ )∈Y0

(ψ ∗ (−f − A∗0 q) ˜ + σK1 (zn,+ ) − (p, qr )).

Notice that qr ∈ L2 (M) is such that

 M

qr dx =

 M

div w dx =

 1

w.n d 1 ≥ 0.

Below we simply denote by E, Y, A the spaces and map previously denoted by E0 , Y0 , A0 . Conditions of the Fenchel theorem (for the problem with lift). We must verify the following two conditions: 

(i) − f ∈ Int (Dom ψ ∗ + A∗ K˜ 1− ) = E ∗ , (ii) (q, 0) ∈ Int(K˜ 1 − AE).

(4.157)

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4 Behavior Laws

The inclusion (4.157)(i) is trivially satisfied. Let us specify the set Im A in Y . An element (q, y) ∈ AE satisfies 



1/2

qdx + M

1

y d 1 = 0, with q ∈ L2 (M), y ∈ H00 ( 1 ).

(4.158)

Conversely, if (q, y) ∈ Z satisfies (4.158), then there exists v ∈ H 10 (M)3 such that −div v = q, and n.v| 1 = y. It suffices to look for v in the form v = grad ϕ, with ϕ a solution of the problem  ∂ϕ  1/2 = y ∈ H00 ( 1 ). ∂n  1

−ϕ = q ∈ L2 (M),

It remains to specify the set K˜ 1 − AE. We have    qdx + K˜ 1 − AE = (q, y + μ) ∈ Z, μ ∈ K1 , (i.e., μ ≥ 0), M

 yd 1 = 0 .

1

Let us verify that K˜ 1 − AE is the half-space    qdx + P+ = (q, y) ∈ Z, M

 yd 1 ≥ 0 . 1

Let (q, y) ∈ P+ , μ0 ∈ K1 , μ0 = 0, and 





qdx +

a= M

yd 1 , b = 1

μ0 d 1 > 0, Iλ = a − λb. 1

Then we have I0 = a ≥ 0, I+∞ = −∞. Hence there exists λ0 ∈ R such that Iλ0 = 0. Let y0 = y − λ0 μ0 ; then (q, y0 ) ∈ P = AE, (q, y) = (q, y0 ) + (0, λ0 μ0 ), whence the desired property.  Then the condition (4.157)(ii) is equivalent to M qdx > 0.  The case of an incompressible fluid is a limit case. We remark that we must have 1 n.v d 1 = 0, with n.v| 1 ≥ 0, and thus n.v| 1 = 0. Therefore, if there is no fluid incoming into M, there is no outgoing fluid either. Of course, this deals with neither the stability question nor the possibility of the domain’s evolution (free boundary problems, water–air interface, for instance, for a reversed bottle: the condition v.n ≥ 0 is not necessarily everywhere satisfied at the bottle’s exit).

4.6 Force–Velocity with Constraint on the Divergence

463

A Variant of the Stream Problem, with Given Global Incoming Stream Here we do not assume that the incoming velocityor the outgoing velocity of the fluid is known. We assume only that the quantity − n.vd is known, which we  call the global incoming stream (in fact, it is given by − ρn.vd ). Let q ∈ L2 (M), q− ∈ R. We define the affine space   V (q, q− ) = v ∈ H 1l (M)3 , −div v + q = 0, −

−

 n.vd + q− = 0 ;

q− indicates the global incoming stream into the domain M. If the orientation of the normal to the boundary is toward the exterior, with the incoming fluid through − , then q− is negative, and the global outgoing stream is given by  q+ =





n.vd = +



qdx −

n.vd =

M

−

M

qdx − q− .

In the space E = H 1l (M)3 , we consider the following problem: with given f , f ∈ E ∗ = (H 1l (M)3 )∗ , h(q, q− ) =

inf

(ψ(v)+ < f, v >).

v∈V (q,q− )

Let A˜ be the operator defined by ˜ = (− div v, − v ∈ E = H 1l (M)3 → A(v)

 n.vd ) ∈ L2 (M) × R. −

The conjugate operator A˜ ∗ is obtained by ˜ = < p, Av > −λ (A˜ ∗ (p, λ), v) = ((p, λ), Av)

 n.vd . −

Let γ and γn be the trace maps v ∈ E → γ v = v| ,

v ∈ E → γn v = n.v ∈ H 1/2( );

then γn∗ (1 − ) = γ ∗ (n − ), and we have (A˜ ∗ (p, λ), v) = < A∗ p, v > −λ < γn∗ (1 − ), v > . We obtain A˜ ∗ (p, λ) = A∗ p − λγn∗ (1 − ). Then we consider the following dual problem, for f ∈ E ∗ ,

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4 Behavior Laws

e∗ (−f ) =

inf

(p,λ)∈L2(M)×R

(ψ ∗ (−f − A˜ ∗ (p, λ))− < (p, λ), (q, q− ) >).

Let us prove that the operator A˜ is onto (when M has a smooth boundary). Let  1/2 0 = (q, q− ) ∈ L2 (M) × R. Let ζ ∈ H00 ( − ) be such that q− − ζ d = 0. The 1 function ϕ ∈ H (M) satisfying ϕ = q,

 q− ∂ϕ  = 0 ζ, and ϕ| l ∪ + = 0,  ∂n − q−

is such that ϕ ∈ H 2 (M). Then v = grad ϕ ∈ H 1 (M)3 , with v| l = 0; hence v ∈ E, whence the desired property. Thus the conditions of the Fenchel theorem are satisfied:  i) − f ∈ Int (Dom ψ ∗ + Im A˜ ∗ ) = E ∗ ˜ = L2 (M) × R. ii) − (q, q− ) ∈ Int (A˜ Dom ψ) = Int (AE) Therefore, there exist solutions v and (p, λ) of the problems h(q, q− ) and e∗ (−f ), and these solutions satisfy − f ∈ ∂ψ(v) + A˜ ∗ (p, λ),

˜ + (q, q− ) = 0. Av

(4.159)

Let f v be the unique element of E ∗ such that −f v ∈ ∂ψ(v). The relations (4.159) are expressed by − f = −f v + A∗ p − λγn∗ (1 − ),

(4.160)

and the relations (4.160) are also written −f = ∗ (An v − pI ) − λγn∗ (1 − ). By projection onto H −1 (M)3 , (4.160) gives − f0 = Pv + grad p.

(4.161)

“Regular case” f = f0 = fM ∈ L2 (M)3 . Then, besides (4.161), we obtain the boundary conditions γ ∗ (Bv − pn) − λγn∗ (1 − ), and thus (with (Bv)n = n.Bv) Bv − pn = 0 on + ,

(Bv)τ = 0 on − , and (Bv)n − p − λ = 0 on − . 1/2

1/2

General case. We identify H 1l (M)3 with H01 (M)3 × H00 ( − )3 × H00 ( + )3 . 1/2

1/2

Let pn : v − ∈ V − = H00 ( − )3 → n.v − ∈ H00 ( − ) be the orthogonal projection. We remark that

4.6 Force–Velocity with Constraint on the Divergence

465

< pn∗ 1 − , v > = < 1 − , pn (v ) > =

 n.v d , −

whence we deduce R1∗ γn∗ (1 − ) = (γn R1 )∗ (1 − ) = pn∗ (1 − ) = n − . By projection of (4.160) onto H −1/2( ) by R1∗ , we then obtain p

− f = −f v − f − λn − , with f v = Cv .

(4.162)

By projection onto + and − , we obtain, in (H00 ( − )3 )∗ and (H00 ( + )3 )∗ , 1/2

p

1/2

p

− f + = −f v+ − f + , and − f − = −f v− − f − − λn − .

(4.163)

Thus we see that the parameter λ occurs only in the balance of normal components on − .

4.6.5 Application, Viscous Force on an Obstacle Lift, drag. We can also treat the case of a bounded obstacle MS immersed in a fluid, occupying a bounded domain M in R 3 . We are often interested to know the force on the obstacle due to the fluid, and to determine the lift and the drag. We assume that M is connected, with complement set M¯  = M¯ S ∪ M¯ 0 (with M¯ 0 not bounded). Thus we assume that the boundary ∂M has two connected parts, and S = ∂MS , the boundary of the obstacle. There are many applications, such as an object (model) in a pipe. Here the peculiarity is to consider that the obstacle’s velocity as known at initial time (thus the fluid’s velocity on its boundary S), which leads us to determine the surface force. Therefore, we can write the equations of the obstacle’s motion, in order to obtain its velocity and its evolution, for instance in the case in which the obstacle is a rigid medium, or made of several rigid parts (an aircraft with a propeller or with a rotor). We consider the case of an obstacle in a pipe. Let v− be the fluid’s velocity at the entrance − , and vS the fluid velocity on the obstacle. We assume that the fluid’s velocity on the lateral wall l is null, and that at the pipe’s exit, the velocity information is n.v| + > 0. Let H 1l (M)3 = {v ∈ H 1 (M)3 , v| l = 0}. We write γ− v = v| − , γS v = v|S , γn,+ v = n.v| + . These traces belong to the 1/2 1/2 spaces H00 ( − )3 , H 1/2(S)3 , H00 ( + )3 respectively.

466

4 Behavior Laws

For a compressible fluid, the evolution problem with given volume forces f ∈ L2 (M)3 , with boundary conditions indicated below, is identical to Problem 19 by changing for ∪ S. We obtain the relation between force and velocity (4.154), with the additional term γS∗ zS , zS ∈ H −1/2(S)3 : ∗ − f = ∗ (An v − pI ) + γ−∗ z− + γS∗ zS + γn,+ zn,+ .

(4.164)

In the case that f = fM ∈ L2 (M)3 , with the indicated boundary conditions (giving (4.156)), we obtain the condition γS∗ (Bv − pn + zS ) = 0, thus Bv − pn + zS = 0 on S.

(4.165)

Hence zS = f S is the surface force due to the fluid on the obstacle, and this force is expressed as a function of the fluid’s velocity v (solution of (4.156) with v|S = vS ) by f S = (−Bv + pn)|S . This force can also be obtained by solving the dual problem similar to Problem 20 (with its notation), (ψ ∗ (−f − A∗ q˜ − γS∗ zS ) + σK1 (zn,+ )− < q, ˇ y >),

inf

q=( ˇ q,z ˜ S )∈Y ∗ ×H −1/2 (S)

with < q, ˇ y >= (p, q)− < z− , v− > − − < zS , vS >, and y = (q, v− , vS ). The rigid medium case. The action of volume forces f (such as gravity) and of surface forces f S on the obstacle MS is given by the wrench F E = (M E , R E ) (with moment M E and resultant R E ; see Chapter 2): 



ME =

x × f (x) dx + MS

 R =

 f (x) dx +

E

x × f S (x) dS, S

MS

(4.166)

S

f (x) dS. S

Let e3 be the unit vector with direction opposite to that of gravity; the lift is given by 



Po = S

f3S (x) dS = e3 .

(−Bv + pn)|S dS. S

Let evG be the unit vector with direction opposite to the velocity of the center of gravity of the obstacle, the drag is given by 

 T =

f (x).evG dS = evG .

(−Bv + pn)|S dS.

S

S

S

4.6 Force–Velocity with Constraint on the Divergence

467

Then we can write the equation of the rigid medium motion (see Chapter 2) if its velocity is unknown. The lift occurs directly in the equation of motion of the center of gravity by m

d(vG )3 = Po − mg dt

for a gravitational force such that f (x) = −ρge3 , with g the gravitational constant, and m the mass of the rigid medium. The drag occurs notably in the following balance of kinetic energy: d 1 2 ( mv ) = −T .vG − mge3 .vG . dt 2 G We can also treat, in a similar way, the case of joint rigid media, the case in which on the lateral face of the pipe, the condition v| l = 0 is exchanged for n.v| l = 0, (Bv)τ | l = 0, or for n.v| l ≥ 0, which allows us to treat the case of an obstacle in the whole space: we exchange the lateral face of the pipe for a fictitious surface. Remark 13 It would be interesting to study how the lift and the drag depend on the viscosity coefficients (notably with respect to the Reynolds number). We refer to [Piro8, ch. I.6, ch. 2.2] for the calculus of the lift for an aircraft wing in a fluid with a model of the Euler equation in a 2-dimensional geometry with an edge on which a Joukowski condition is imposed.

4.6.6 Superficial Tension Let M be a connected bounded smooth domain in R 3 , filled by two fluids, with boundary that is a rough wall. Let Mi be the domain of the fluid of index i, i = 1, 2, and let S be their common boundary. We assume (in order to simplify) that S is also the boundary of the domain M1 . Thus we assume ∂M1 = S, ∂M2 = ∪S, with ∩ S = ∅, and M = M1 ∪ M2 ∪ S, M1 ∩ M2 = ∅. Let ψi , i = 1, 2, be the resistance function (relative to the velocities) of each part of the fluid with the usual hypotheses of Newtonian fluid, and let ψ be the resistance function of the total set of velocities, whose viscosity coefficients may be discontinuous across S. The functional framework of the velocity fields in M is the Sobolev space H01 (M)3 . For a given force field fM ∈ L2 (M)3 , the relation between fM and the velocity v (with the constraint div v = q, with q given) is Pv − grad p = fM in M, v ∈ H01 (M)3 , with div v = q,

(4.167)

with the usual operator P, which is equal to Pi in the domain Mi for each medium. This is a well-posed problem that has one and only one solution. Yet if the fluids do not mix, we have to add a noncrossing condition over the face S: γn,S (v) = n.v|S = 0.

468

4 Behavior Laws

In the special case in which the specific masses of the media ρ1 and ρ2 are different, the condition of mass conservation (in the stationary case) div (ρv) = 0 gives [ρn.v]S = 0 across S, and since v ∈ H01 (M)3 , we have n.v|S = 0. But the problem (4.167) with this additional condition is a priori ill posed. This problem may be treated by considering this condition a constraint. Then the dual variable (the Lagrange multiplier) intervenes as an additional pressure term, which may be viewed as a superficial tension of the fluid, so that we are led to solve a transmission problem. Thus we consider the (primal) problem Problem 23 h(−q, 0) =

inf

v∈H01 (M)3 ,γn,S v=0,div v=q

(ψ(v) + (fM , v)).

Let A be the map v ∈ H01 (M)3 → (−div v, γn,S v) ∈ Y = L2 (M) × H 1/2(S). The constraint is Av = (−q, 0). Then the dual problem is Problem 24 e∗ (−f ) =

inf (ψ ∗ (−f − A∗ (p, z))+ < p, q >).

(p,z)∈Y ∗

∗ z. Application of We have Y ∗ = L2 (M) × H −1/2(S) and A∗ (p, z) = grad p + γn,S the Fenchel theorem gives ∗ − fM ∈ ∂ψ(v) + A∗ (p, z), i.e., − fM = ∗ (An v) + grad p + γn,S z.

(4.168)

By this formula, only the restrictions of ∗ (An v) + grad p I to M1 and M2 are in L2 (M1 ) and L2 (M1 ), which allows us to apply the formula (4.145) for each of these spaces. Thus we obtain the relations −Pv + grad p = −fMi , on each Mi , and the joint conditions on S ∗ ∗ ∗ γ∂M (Bv − pn) + γ∂M (Bv − pn) + γn,S z = 0. 1 2 ∩S

(4.169)

We then deduce the joint relations of tangential forces and jumps of the normal components (with the orientation of the normal toward the outside of M1 ) (i) (Bv)t,1,S + (Bv)t,2,S = 0, (ii) ((Bv)n,1,S + (Bv)n,2,S ) + (p2,S − p1,S ) + z = 0.

(4.170)

Thus the velocity field v satisfies the following relations: v ∈ H01 (M)3 ,

γn,S v = 0, div v = q,

− Pv + grad p = −fMi in Mi , i = 1, 2,

(Bv)t,1,S + (Bv)t,2,S = 0. (4.171)

4.6 Force–Velocity with Constraint on the Divergence

469

The superficial tension z is then obtained by (4.170)(ii), or by directly solving Problem 24. In the case v = 0, we thus obtain [p]S + z = 0 according to the interpretation of z as a superficial tension. Under stability conditions with respect to domain variations, we can verify that this term of superficial tension is proportional to the inverse of the radius of curvature of the surface (see, for instance, [Lan-Lif2]). Remark 14 Writing equation (4.171) in a weak sense, i.e., < An v − pI, w >= − < fM , w >,

∀w ∈ H01 (M)3 , div w = 0, γn,S w = 0,

allows us to avoid the calculus of the superficial tension when we determine the solution v of the problem (4.171). ¶

Appendix A

The goal of this appendix is essentially to recall some fundamental notions in differential geometry, from references [Mall, Arn1, Arn2, Bour.var]. You will find mainly definitions along with theorems without proofs. We assume that the usual notions of manifold and submanifold are known (see [Mall]).

A.1 Some Notation and Definitions Let V be a manifold of class C r , r ≥ 1, with generic element denoted by x. We employ the following notation: • • • • • •

Tx V , the tangent space at point x of V ; T V , the tangent bundle to the manifold V ; Tx∗ V , the set of covectors at point x of V , i.e., of linear forms on Tx V ; T ∗ V the cotangent bundle, or phase space; ⊗1 V or 1 V , the set of vector fields on V , or sections of T V ; ⊗1 V or 1 V ∗ , the set of differential forms on V , or sections of T ∗ V ).

A differential form, or 1-form, on V is a field of covectors on V ; we denote by dω the differential exterior of the differential form ω. Let V and V1 be two manifolds of class C r , and let f be a map f ∈ C 1 (V , V1 ); we use the following notation: • Tx f = f  (x) (or f∗ ), the derivative map of f at x ∈ V , from Tx V into Tf (x) V1 ; • Tf = f  , the derivative map of f , which is a map from T V into T V1 ; • f∗ (X) = Tx f (X) = f  (x).X or (f∗ (X))f (x) = f  (x).Xx , the image (called the direct image) of the vector X ∈ Tx V by Tx f , also defined by (f∗ (X))g = X(g ◦ f ) for every real function g differentiable in a neighborhood of f (x) ∈ V1 . In a coordinate system (x i ) (respectively y j ) of V (respectively of V1 ), for Xi = ∂x∂ i , with y = f (x), x ∈ V , we have © Springer International Publishing AG, part of Springer Nature 2018 M. Cessenat, Mathematical Modelling of Physical Systems, https://doi.org/10.1007/978-3-319-94758-7

471

472

Appendix A

(f∗ Xi )y g =

 ∂f j ∂g ∂ (g ◦ f (x)) = (y). ∂x i ∂x i ∂y j

Let f be a diffeomorphism, and X a vector field on V . Then f∗ X is the vector field on V1 defined by (f∗ X)y = f  (f −1 (y)) ◦ Xf −1 y , ∀y ∈ V1 . • f ∗ ω the pullback of the differential form ω on V1 by the map f , that is, the differential form on V defined by (f ∗ ω)x = t f  (x).ωf (x) , i.e., < f ∗ ω, X > = < ω, f∗ X >,

∀X ∈ Tx V .

More generally, if ω is a p-form on V1 , we define its pullback by f as the p-form on V < f ∗ ω, (X1 , · · · , Xp ) > = < ω, (f∗ X1 , · · · , f∗ Xp ) >, ∀X1 , · · · , Xp ∈ Tx V . Note that f ∗ dω = d f ∗ ω. Let π, π1 be the canonical projections from T V and T V1 onto V and V1 respectively. The following diagrams are commutative: f

tf 

T V −→ T V1 T ∗ V ←− T ∗ V1 s ↑↓ π f∗ s ↑↓ π1 f ∗ ω ↑ ω↑ V

f

−→

V1

V

f

−→ V1

Let X be a vector field on V . We denote by (Ut ) the 1-parameter pseudogroup of transformations generated by X. • iX is the contraction with X (or the interior product with X) of a differential form (of degree k): iX ω(X1 , . . . , Xk−1 ) = ω(X, X1 , . . . , Xk−1 ), ∀Xi ∈ ⊗1 V , i = 1, . . . , k − 1. • LX is the Lie derivative with respect to the vector field X; these notions are linked by the Cartan relation given for every differential form ω of degree k by LX ω = iX dω + diX ω. • ∇X Y is the covariant derivative of the vector field Y on V in the direction of the vector X, with V equipped with a connection. • [X, Y ] is the bracket operation between two vector fields X, Y on V . We have [X, Y ] = LX Y. In a coordinate system (x i ) of V , where X, Y are given by X = Xi ∂/∂x i , Y = Y i ∂/∂x i , [X, Y ] is given by

Appendix A

473

[X, Y ] = XY − Y X =

 ∂Y j ∂Xj ∂ (Xi i − Y i ) . ∂x ∂x i ∂x j

(A.1)

Some other definitions.1 Let f ∈ C 1 (V , V1 ); then f is said to be: • regular at x0 if the rank of f at x0 (that is, dim Tx0 f (Tx0 V )) is equal to dim Tx0 V ; that is, if Tx0 f = f  (x0 ) is injective; f is said to be regular if it is regular at every point. • an immersion if f is regular and injective. • an embedding if f is an immersion and moreover is a homeomorphism from V onto f (V ) equipped with the induced topology of V1 . The canonical injection of a submanifold V0 of V into V is an embedding. • a submersion if f is such that Tx0 f (Tx0 V ) = Tf (x0 ) V1 , that is, its derivative function is surjective and its kernel ker Tx0 f admits a topological supplement at every point x0 ∈ V . Lie derivative with respect to a vector field X. Let X be a vector field on M, and let (φt ) be the local group generated by X. Let K = ω be a differential form on M, or let K = f be a real derivable function on M. Let φ˜ t ω = φt∗ ω, φ˜t f = f ◦ φt , which defines a map K → φ˜ t K for every covariant tensor field. Let K = Y be a vector field on M. Then φ˜ t Y is defined as the direct image of Y under φ−t = (φt )−1 ;  (φ x)Y thus for every x ∈ M, (φ˜ t Y )x = φ−t t φt x , which defines a map K → φ˜ t K for every contravariant tensor field, and thus for every tensor field K by composition of tensor products. The Lie derivative LX K of every tensor field K on M with respect to the vector field X is then defined by 1 (LX K)x = lim ((φ˜ t K)x − Kx ), t →0 t

(A.2)

where the limit must be taken at every point x ∈ M. Thus we have for K = f or K = ω, then K = Y , 1 LX K = lim (φt∗ K − K), with K = f or ω, t →0 t 1 LX Y = lim ((φ−t )∗ Y − Y ) = [X, Y ]. t →0 t

(A.3)

The differential df of a real function f and the Lie derivative with respect to a vector field X are linked by LX f = < X, df > . Other definitions may be given; see notably [Kob-Nom]. 1 We

follow essentially [Mall], which is different from [Kob-Nom].

(A.4)

474

Appendix A

A.2 Integral Manifold A.2.1 Integrable k-Field Definition 1 k-field2 on an n-dimensional manifold V . Let k be an integer, k < n. A k-field is the assignment P to each point x ∈ V of a k-dimensional subspace Px (or P (x)) of the tangent space Tx V of V at x. The main notions relative to vector fields are transposed to k-fields: restriction of a k-field to an open set  of V , transport of a k-field by a diffeomorphism. Definition 2 Differentiable k-field (or k-field of class C 1 ). A k-field P on a manifold V is differentiable if for every x ∈ V , there exist a neighborhood Ux of x and k differentiable vector fields X1 , . . . , Xk on Ux giving a basis of Px ; then X1 , . . . , Xk is called a local basis of P in Ux . Notice that P is a differentiable map from V into the manifold called the Grassmannian, denoted by Gk (T V ) (which is the union of k-dimensional vector subspaces of Tx V ), such that x ∈ V → Px ∈ Gk (Tx V ); see [Dieud3, ch. 18.8]. Notice that there does not necessarily exist a frame field on the whole space V giving a basis of Px , ∀x ∈ V . But the local existence of frames implies that (V , P ) may be considered a vector subbundle of T V by taking here Px as a subspace of Tx V ; see [Dieud3, ch. 18.8]. Let O be an open set of a vector space E. A k-field P defined on O is constant if for every x ∈ O, P is a fixed space of E. Definition 3 Adapted chart to a k-field P . Let P be a k-field on a manifold V ; a chart (U, φ, E) with an open set U of V is said to be adapted to P if P is read in this chart as a constant k-field, i.e., if in this chart the image of the restriction P |U of P to U by (the derivative of) φ, φ∗ (P |U ) is a fixed space of E. Let P be a k-field of class C 1 in V . The space P˜ of vector fields associated to P is the set of vector fields A of class C 1 such that Ax is in Px , x ∈ V :   P˜ = A ∈ ⊗1 V , of class C 1 , Ax ∈ Px , ∀x ∈ V .

(A.5)

Definition 4 Involutive k-field. A k-field P on V is said to be involutive if for all vector fields X and Y on V in P , we have [X, Y ] ∈ P , that is, P˜ is an algebra. Let (Ut ) be a transformation group defined on V . We say that a k-field P is invariant under (Ut ) if (Ut )∗ P = P , i.e., Ut (x).Px = PUt (x) for every x ∈ V . The following notion generalizes that of integral of a vector field.

2 The term “distribution” is also used, see [Kob-Nom], or the field of k-directions see [Dieud3]; see also [Mall, Arn2, Kob-Nom, Bour.var] about this section.

Appendix A

475

Definition 5 Integral manifold of a k-field. We call a submanifold3 W of V such that Tx (W ) ⊂ Px , ∀x ∈ W

(A.6)

an integral manifold of a k-field P on V . An integral manifold W such that dim W = k is said to be of maximal dimension. An integral submanifold W of V with Tx (W ) = Px , ∀x ∈ W „ and such that there does not exist any other integral submanifold containing W is said to be a maximal integral submanifold. Definition 6 Integrable k-field.4 A k-field P on V is said to be integrable if at every x ∈ V , there is a unique maximal integral submanifold of P . The existence of maximal integral submanifolds of a k-field is the goal of the following the Frobenius theorem.

A.2.2 Frobenius Theorem The following theorem is a fundamental theorem of differential geometry (see [Mall, ch. II.5.2.5], [Arn1, App. 4], [Bour.var, 9.3.5]). Theorem 1 The Frobenius theorem. Let V be an n-dimensional manifold (with finite n), of class C r , and let P be a k-field of class C s , s ≤ r − 1, on V . The following statements are equivalent: 1. 2. 3. 4.

P is integrable. On a neighborhood of every point x of V , there exists an adapted chart for P . For every A ∈ P˜ , the associated pseudogroup Ut to A keeps P invariant. The k-field P is involutive.

Theorem 2 The Frobenius theorem (with differential forms). Let ω = (ωj ), j = 1, . . . , n − k, be an R n−k -valued differential form such that Px = ker ωx . The previous statements are also equivalent to the following: 1. (ker ωx ) ∧ (ker ωx ) ⊆ ker dωx ;

(A.7)

dωi ∧ ω1 ∧ . . . ∧ ωn−k = 0, i = 1, . . . , n − k,

(A.8)

2.

3 Sometimes 4 We

we also impose that W be connected; see [Kob-Nom]. also say completely integrable k-field (see, for instance, [Mall]).

476

Appendix A

and thus in the case k = n − 1 (hyperplanes), dω ∧ ω = 0;

(A.9)

j

3. there exist αi , i, j = 1, . . . , n − k, differential forms such that dωi =



αji ∧ ωj ,

(A.10)

j

which may also be written in vector form dω = α ∧ ω.

(A.11)

Pfaff system [Mall, ch. II.5.4]. A k-field P on V may be locally defined by Px = ∩ ker(ωxi ), with n − k independent differential forms ωi . The problem of finding an integral manifold W of P is then identical to solving the Pfaff system (or Pfaff equations): to find W such that ωi |W = 0, i = 1 à n − k (we also write ωi = 0). Here we verify only the following points of the Frobenius theorem with k = n − 1: 1. the equivalence of point 4 of Theorem 1 with point 1 of Theorem 2, 2. the equivalence of point 1 with point 2 of Theorem 2. Proof 1. It is a straightforward consequence of the so-called cobordism formula (see [Mall, ch. II.4.3]), with usual notation Xi , i = 1, 2, vector fields on V , ω a 1-form on V such that ker ωx = Px , < dω, X1 ∧ X2 > + < ω, [X1 , X2 ] > = LX1 < ω, X2 > +LX2 < ω, X1 >, (A.12) where LX is the Lie derivative with respect to the vector field X. If (Xi )x ∈ Px , with i = 1, 2, the right member is null, which proves the equivalence. 2. This results from the formula  (−1)ν dω(Xi1 , Xi2 )ω(Xi3 ), (A.13) dω ∧ ω(X1 , X2 , X3 ) = where i1 , i2 , i3 is a permutation of the indices 1, 2, 3, with i1 < i2 and where ν is the signature of this permutation. ¶ The notion of integrable k-field P on V , or of integrable vector subbundle of T V , may be defined from the notion of foliation of the manifold V (see [Bour.var, 9.2.2, 9.3.2]):

Appendix A

477

Definition 7 A foliation of a manifold V is a manifold W whose elements are those of V and that satisfies the property that for every x ∈ V , there exist an open submanifold U of V with x ∈ U , a manifold S, and a submersion π from U on S such that the manifold Uπ , the sum of the spaces Vs = π −1 (s), s ∈ S, is an open submanifold of W . The pair (V , W ) is then said to be a foliated manifold. The spaces Tx W, for x ∈ V , are the fibres of a vector subbundle of T V , called the subbundle of T V , which is tangent to the foliation W , and denoted by T (V , W ). In order that a submanifold Z of V be a leaf of T (V , W ), it is necessary and sufficient that Tx Z = Tx (V , W ), ∀x ∈ Z. Recall the following definition (see [Bour.var, 9.3.2]), corresponding to Definition 6. Definition 8 A vector subbundle P of T V is said to be integrable if there exists a foliation W of V such that P = T (V , W ). We give the simpler case of foliated manifold (V , W ) in which there exist a manifold S and a submersion π from V onto S such that W = Vπ . This is equivalent to (see [Bour.var, 9.2.9]) the following. For every x ∈ V , there exists a submanifold Sx of V with the two following properties: 1. Tx (Sx ) is a topological supplement of Tx (V , W ) in Tx V . 2. Every connected leaf of (V , W ) has at most one point in common with Sx . We refer to [Bour.var, 9.3.3, 9.3.5] for equivalences of the Frobenius theorem in this frame, and for specifying regularities. Let A be a given vector field on a Riemannian manifold V , with metric g, with Ax = 0, ∀x ∈ V . The field A generates a family of trajectories or integral curves, called rays. When does this family admit a wave function? ⊥ Let A⊥ x be the orthogonal space to Ax in Tx (V ), and A the corresponding (n − ⊥ 1)-field. This (n − 1)-field is defined by Ax = ker(ωA )x , with the 1-form ωA on V , also denoted by ωA = GA, such that (ωA )x (X) = gx (Ax , X),

∀X ∈ Tx (V ).

With |A| = 1, (ωA )x (B) is the orthogonal projection of the vector B on Ax . The previous question is whether the (n − 1)-field A⊥ is integrable. Thanks to the Frobenius theorem, the answer is yes if and only if ωA satisfies dωA ∧ ωA = 0, or equivalently, if and only if there exists a 1-form α on V such that dωA = α ∧ ωA . If V is the space R 3 , these conditions amount to A.curlA = 0. Indeed, dωA is the 2-form with components (curl A)i , and then ωA ∧ dωA = (A.curl A)v g , with v g = dx1 ∧ dx2 ∧ dx3 . ¶ Remark 1 Note that if A⊥ is not integrable, then we have the following situation. Let x0 ∈ R 3 be taken as the coordinate origin, and let (e3 ) be the axis of Ax0 . A curve γ with origin x0 such that its tangent vector satisfies v(x) ∈ A⊥ x , ∀x ∈ γ , and

478

Appendix A

whose projection on the plane (e1 , e2 ) (i.e., on A⊥ x0 ) is a cycle, a priori is not a cycle 3 in R , i.e., there exists t > 0 such that (γ1 (t), γ2 (t)) = (0, 0), but γ3 (t) = 0. In contrast, if A⊥ is integrable, we necessarily have γ3 (t) = 0, and the curve γ (on an integral surface of A⊥ ) is a cycle. Let S be a surface with boundary γ . By application of the Stokes theorem (with n = A, the normal to the surface S, with |A| = 1), we have 

 curl A. ndS = S

 ωA =

γ

t

A.Xdt = 0,

0

X a tangent vector to γ , thus orthogonal to A. We again obtain the integrability condition A.curl A = 0. ¶

A.2.3 Characteristics Generally, a k-field P does not admit a k-dimensional integral manifold. Definition 9 Characteristic field of a k-field P . A vector field X defined on an open set U of V is said to be characteristic with respect to P if Xx ∈ Px , and [X, Y ]x ∈ Px , ∀x ∈ U,

(A.14)

for every vector field Y defined on U such that Yx ∈ Px , ∀x ∈ U , or if P is invariant under the flow of the field X. Thus the characteristic fields of P are the fields that leave P invariant and belong to P . Let PU0 be the space of 1-forms ω defined on an open set U of V , null on Px for every x ∈ U , and let P 0 be the union of these spaces:   PU0 = ω 1-form on U such that ωx |Px = 0, ∀x ∈ U ,

P 0 = ∪PU0 ;

(A.15)

Px0 is an (n − k)-dimensional vector space. Using the Lie derivative LX , or the contraction iX with X, we can give the following theorem [Mall, ch. II.5.5] Theorem 3 Let X be a vector field defined on an open set U of V ; X is characteristic with respect to P if and only if it satisfies Xx ∈ Px , ∀x ∈ U,

and LX ω ∈ P 0 , ∀ω ∈ P 0 ,

(A.16)

or equivalently, if and only if for every 1-form ω ∈ PU0 , we have iX dω ∈ PU0 .

(A.17)

Appendix A

479

Proof (in U ) Let ω ∈ P 0 ; we have < ω, ξ > = 0, ∀ξ ∈ P , and thus < ω, X > = 0. Then < LX ω, ξ > = LX < ω, ξ > − < ω, LX ξ >, and LX ξ = [X, ξ ] = 0, whence < LX ω, ξ > = 0, ∀ξ ∈ P . Then the Cartan relation LX ω = iX dω + diX ω implies < iX dω, ξ > = 0, ∀ξ ∈ P . ¶ Let Pxc be the set of characteristic fields with respect to P , and let r be its dimension. We assume that r is independent of x in V . Then P c : x → Pxc is differentiable, and is called the r-characteristic field with respect to P . Definition 10 Cauchy characteristic. The integral manifolds of P c are called the Cauchy characteristics of P . Main property: P c is an integrable field, and P admits an r-dimensional Cauchy characteristic (with r ≤ k). Consequences [Mall, ch. II.5.5]. For every x ∈ V , there exist a local coordinate system (U ; x 1, . . . , x n ) and a “local frame”5 (U ; ω1 , . . . , ωq ) of P 0 (q = n − k) such that: 1. the submanifolds V(cr+1 ,...,cn ) (with cr+1 , . . . , cn constants) of U defined by the equations x r+1 = cr+1 , . . . , x n = cn , are integral manifolds of P c , and then form a foliation (Vc )c∈R n−r of U ⊂ V ; 2. the forms ωλ , λ = 1 to q, are written only thanks to the coordinates x r+1 to x n and their differentials by ωλ =



fjλ (x r+1 , . . . , x n ) dx j .

j =r+1,...,n

Remark 2 Existence of an integrating factor. Let ω be a differential form on V . Does there exist a positive function f , i.e., f (x) > 0, ∀x ∈ V , of class C 1 , such that d (f ω) = 0 ?

(A.18)

In that case, the differential form f ω is said to be closed. Using the differential relation d(f ω) = df ∧ ω + f dω, we see that (A.18) is satisfied if df ∧ ω + f dω = 0.

(A.19)

Taking the exterior product with ω, we obtain dω ∧ ω = 0. Thus in order that ω admits an integrating factor, it is necessary that ker ω be integrable (see the Frobenius theorem). But this is not an equivalence. Since f (x) > 0, the function φ = − log f is such that dφ = −df/f , and thus dividing the relation (A.19) by f , we obtain 5 This

means the linear independence of ωj , j = 1, · · · , q, at every point of U .

480

Appendix A

dω = dφ ∧ ω,

(A.20)

which is of type (A.11), but with the so-called exact differential form α = dφ. Conversely, if ω is such that (A.20) is satisfied, by taking f (x) = exp −φ(x), we obtain (A.19), thus (A.18), whence f is a positive integrating factor of ω. Therefore, there exists (at least locally) a function ϕ on V such that ω=

1 dϕ. f

(A.21)

Notice that if V is a Riemannian manifold, and if A is a vector field on V such that ω = ωA = GA admits a positive integrating factor, we also say that the vector field A admits a positive integrating factor. From (A.21), A is such that A = f1 grad ϕ. In the space R 3 , the condition that A admits a positive integrating factor is curl (f A) = 0, which is also (grad f ) × A + f curl A = 0, thus curl A = (grad φ) × A, with φ = − log f . ¶

A.2.4 Applications Relation between an orthonormal frame of a Riemannian manifold V and an orthogonal coordinate system. Let x ∈ X → ux = (A1 , . . . , An )x be a field of orthonormal frames on V , thus such that gx (Ai , Aj ) = δij , x ∈ V . Does there exist a coordinate system (ϕ j ) of V such that the Aj are given (up to a factor) by partial derivatives with respect to coordinates? It is easier to work with the differential forms associated through the metric g. Let ωj = ωAj = GAj , and ω = (ω1 , . . . , ωn ). With ξ = (ξ j ) ∈ R n , we have ωx (ux (ξ )) = ωx (



ξ j Aj ) = (



ξ j δj i ) = (ξ i ) = ξ,

and thus ωx ◦ ux = I , the identity map in R n . Then we have ωx = u−1 x . The desired orthogonal coordinate system (ϕ j ) must be such that ωj = (1/fj ) dϕ j ,

(A.22)

with coefficients 1/fj that are positive functions on V . Hence the differential forms ωj must admit integrating factors. Let ϕ = (ϕ j ) be a map from V into R n . There exists an open set U in V such that (U, ϕ|U ) is a chart of V ; this is equivalent to proving (see [Dieud3, ch. 16.5.9]) that the sequence of differentials dx ϕ j is a basis of (Tx V )∗ . Moreover, the independence of the family (Aj ) implies that of the (ωj ), and thus of the (dϕ j ) (for every x ∈ U ). Hence the inverse map ϕ −1 of ϕ gives a parametrization of V (in fact of U ). Notice

Appendix A

481

that if x i is a coordinate function of R n (that is, the projection x = (x 1 , . . . , x n ) → x i ∈ R), then ϕ ∗ (dx i ) = dϕ ∗ (x i ) = d(x i ◦ ϕ) = dϕ i , thus ϕ ∗ (dx i ) = dϕ i . The dual basis of (dx i ) (and of (dϕ i )), respectively at x ∈ R n and x˜ ∈ V (such that ϕ(x) ˜ = x) is usually denoted by (∂/∂x i )x ) (respectively (∂/∂ϕ i )x˜ ), < ϕ ∗ (dx i ), ∂/∂ϕ i )x˜ > = < dx i , ϕ∗ (∂/∂ϕ i )x˜ ) > = δij , which proves that ϕ∗ (∂/∂ϕ i )x˜ ) = (∂/∂x i )x . Notice that the commutation relations [∂/∂x i , ∂/∂x j ] = 0, i = j imply ϕ∗ ([∂/∂x i , ∂/∂x j ]) = [ϕ∗ (∂/∂x i ), ϕ∗ (∂/∂x j )]) = [∂/∂ϕ i , ∂/∂ϕ j ] = 0, i = j. Since (∂/∂ϕ i ) is the dual basis of (dϕ i ), we deduce that (fi ∂/∂ϕ i ) is the dual basis of ((1/fi )dϕ i ), i.e., of the basis (ωi ). Thus we have proved that Ai = fi ∂/∂ϕ i , ∀i = 1, . . . , n.

(A.23)

Hence we have g(∂/∂ϕ i , ∂/∂ϕ j ) = (fi fj )−1 g(Ai , Aj ) = (fi )−2 .δij . The Riemannian metric g is then written g=



ωj ⊗ ωj =



fj−2 dϕ j ⊗ dϕ j ,

or ds 2 = fj−2 (dϕ j )2 . Thus (ϕ i ) is an orthogonal coordinate system corresponding to the orthonormal field (Ai ). Remark 3 The parametrizations of integral curves of Ai by ϕ i and by the distance s correspond through ∂ϕ i /∂s = fi > 0. This implies that if we have (for instance, with i = n) fn = 1, the integral curves of An are geodesics. Furthermore, we have ωn = dϕ n , and ϕ n is then the “wave function of the field of normals” (or of geodesics) An . We thus obtain a known result (see [Mall, ch. IV.4.2, Thm. 4.2.3]): a geodesic field A = An such that A⊥ is integrable admits a wave function ϕ that gives ωA = dϕ. The conditions ∇A A = 0, |A| = 1 (unit geodesic field), dωA ∧ ωA = 0 (integrable orthogonal field) then imply (at least locally) that ωA is an exact differential form. (In the case that V is the space R 3 , these conditions are (A.grad )A = 0, |A| = 1, A.rot A = 0.) ¶ Remark 4 The commutation relations between vector fields Ai , Aj give (with the notation ∂/∂ϕ i = ∂i ) [Ai , Aj ] = [fi ∂i , fj ∂j ] = fi (∂i fj )∂j − fj (∂j fi )∂i = fi

∂i fj ∂j fi Aj − fj Ai . fj fi

Thus the bracket of Ai , Aj is a linear combination (with coefficients that are functions on V ) of Ai and Aj . This is verified from the formula (A.12):

482

Appendix A

dωi (Aj , Ak ) = LAj (< ωi , Ak >) − LAk (< ωi , Aj >) − ωi ([Aj , Ak ]). Taking account of the relations ωi (Aj ) = δij and dωi = dϕ i ∧ ωi , which is the integrability condition, we obtain ωi ([Aj , Ak ]) = 0 if i = j, k, which corresponds to the advertised property. ¶

A.3 Symplectic Structure Definition 11 Symplectic structure. Let M be an even-dimensional differentiable manifold; M is equipped with a symplectic structure if there exists a 2-form ω on M (called symplectic) such that dω = 0 and nondegenerate, that is, ∀ξ ∈ Tx M, ∃η ∈ Tx M, such that ω(ξ, η) = 0.

(A.24)

Then (M, ω) is called a symplectic manifold. We denote by Iˆ the isomorphism from T M onto T ∗ M induced by ω: ωx (ξ, η) = < Iˆx (ξ ), η >,

∀ξ, η ∈ Tx M.

(A.25)

Recall that if V is a manifold, then there exists on the phase space M = T ∗ V a canonical 1-form θ such that < θx,p , ξ > = p(π∗ (ξ )),

∀(x, p) ∈ T ∗ V ,

∀ξ ∈ Tx M,

(A.26)

and such that dθ = ω is a symplectic form. In a local chart system of V , θ and ω are given by θ = p.dx =



pj dx j ,

ω = dp ∧ dx =



dpj ∧ dx j .

Let F be a vector subspace of Tm M. Then we define the orthogonal space F ⊥ of F with respect to ω, the vector subspace of Tm M such that F ⊥ = {ξ ∈ Tm M, ω(ξ, η) = 0,

∀η ∈ F }.

This set is identified with the polar set of F , denoted by F 0 , defined by F 0 = Iˆ(F ), or F 0 = {η ∈ Tm∗ M,

∃ξ ∈ F, η = Iˆ(ξ ) = iξ ω} ⊂ Tm∗ M.

Thus < η, ξ˜ > = ω(ξ, ξ˜ ), ∀ξ˜ ∈ Tm M, and then the orthogonal space of ξ with respect to ω is

Appendix A

483

(Rξ )⊥ = ker η = ker Iˆ(ξ ). Let (ξj )j =1,...,k be a basis of F ; the previous orthogonal space of F is given by F ⊥ = ∩j =1,...,k ker ηj ,

with ηj = Iˆξj .

The dimension of F ⊥ is 2n − k. If ωm (ξ, ξ˜ ) = 0, ∀ξ, ξ˜ ∈ F (i.e., if ω is null on F ), then F ⊂ F ⊥ , and F is said to be isotropic. We have k ≤ 2n − k, which implies k ≤ n. Then the maximal dimension of an isotropic space is n. Definition 12 Lagrangian submanifold. A submanifold V of (M, ω) is called Lagrangian if Tm V is an n-dimensional isotropic space for every m ∈ V , and is maximal in the sense that if V˜ is another submanifold containing V , then V = V˜ . Definition 13 Lagrangian foliation. A Lagrangian foliation of a symplectic space (M, ω) is a foliation W whose leaves are Lagrangian submanifolds of M. Theorem 4 The Darboux theorem. Let (M, ω) be a 2n-dimensional symplectic manifold. For every m ∈ M, there exist a neighborhood V of x and a coordinate system ((x 1 , . . . , x n ); (p1 , . . . , pn )) such that in V, the symplectic form ω is ω =  dpj ∧ dx j . Definition 14 The Poisson bracket of two functions, of two vector fields. The Poisson bracket of two vector fields ξ, η (respectively of two real differentiable functions f and g) on a symplectic manifold (M, ω) is defined by respectively {f, g} = ω(Iˆdf, Iˆdg).

{ξ, η} = ω(ξ, η),

Other expressions for the Poisson bracket of two functions may be given, such as {f, g} = df (Iˆdg). If M = T ∗ V , with a coordinate system (x j , pj ), we have {f, g} =

 ∂f ∂g ∂f ∂g − j . ∂pj ∂x j ∂x ∂pj

Foliation due to a family of first integrals. Let (Hj ), j = 1, . . . , k, be a family of real independent differentiable functions on (M, ω), with M = T ∗ V . We assume that the family of the (Hj ) is involutive and commutative with respect to the Poisson bracket: {Hj , Hi } =

 ∂Hj ∂Hi ∂Hj ∂Hi − = 0. ∂pl ∂x l ∂x l ∂pl l

Let h be the product map: h(m) = (H1 (m), . . . , Hk (m)) ∈ R k . To every function Hj corresponds the vector field XHj on M defined by

484

Appendix A

XHj = Iˆ−1 (dHj ) =

 ∂Hj ∂ ∂Hj ∂ − . ∂pl ∂x l ∂x l ∂pl l

The map h : M → R k defines a foliation Mh of M whose (2n − k)-dimensional leaves are given by Me = h−1 (e), with e = (e1 , . . . , ek ) ∈ R k . The vector fields XHj are tangent to the leaves, and then the k-field P generated by these vector fields is integrable and admits these leaves as integral manifolds. When k = n, the foliation is said to be Lagrangian. Foliation due to a symmetry group. Momentum due to the Poisson action of a connected Lie group P : M → g∗ , the Noether theorem, see Chapter 2, and [Arn1, App. 5]. Lagrangian foliation for an integrable system (with an involutive family of n first integrals). The Liouville theorem. Coordinates with time (e1 , . . . , en ; t1 , . . . , tn ); action-angle variables (see [Arn1, chap. 10.49, 50]). Some examples. 1. In T ∗ (R 3 ), Hamiltonian invariant by rotation. 2. Involutive family of n quadratic Hamiltonians Hj = 12 (pj2 + ωj2 qj2 ). Means for  H = Hj , case n = 2, with ω1 /ω2 irrational. Ergodicity. 3. Solar system by perturbation of an integrable system.

A.4 Contact Structure The notion of contact structure6 for odd-dimensional manifolds corresponds to the notion of symplectic structure for even-dimensional manifolds.

A.4.1 Definitions. Contact Structure Definition 15 Contact structure, contact manifold (M, P ). A manifold M = M (2n+1) is equipped with a contact structure if there exists a differentiable 2n-field P (or tangent hyperplane field) satisfying the following property: There exist an open covering (Ui , i ∈ I ) of M and on each Ui , a differential 1-form θi such that ker (θi )m = Pm , ∀m ∈ Ui , and rank (dθi |Pm ) = 2n, i.e., ∀ξ ∈ Pm , ξ = 0, ∃η ∈ Pm , so that dθi (ξ, η) = 0. (A.27) The pair (M, P ) is called a contact manifold.

6 In

this section, we refer essentially to [Arn1] and [Arn2]; see also [Kob-Nom, Vol. 2, Note 28].

Appendix A

485

The fact that the dimension of a contact manifold is odd is due to the condition (A.27) of being nondegenerate. This condition implies that every integral manifold of P is n-dimensional at most. Definition 16 In a contact manifold (M, P ), we have the following terminology: • A contact element (or contact hyperplane) is a hyperplane Pm (of dimension 2n) that is tangent to the manifold M at a point m ∈ M called the contact point; • A contact covector is a covector p ∈ Tm∗ M such that ker p = Pm . • A contact form is a differential form θ (globally defined) on M such that ker θm = Pm , ∀m ∈ M. It satisfies, at every point m ∈ M, θ (∧dθ )n = θ ∧ dθ . . . ∧ dθ = 0. Such a form does not necessarily exist. Definition 17 Contact diffeomorphism. A contact diffeomorphism on a contact manifold (M, P ) is a diffeomorphism φ on M that satisfies φ∗ (Pm ) = Pφ(m) , ∀m ∈ M.

(A.28)

A.4.2 Projective Contact Manifold The notion of contact manifold is directly linked to that of projective manifold. Let E be a vector bundle of base V ; we define the projective bundle from E, denoted7 by PE, whose fibre at x ∈ V is the quotient of Ex \{0} by the equivalence relation p1 ≈ p2 ⇐⇒ there exists λ ∈ R, λ = 0, such that p1 = λp2 .

(A.29)

For an n-dimensional manifold V , we thus define a bundle M = PT ∗ V , of base V , of dimension 2n − 1, whose fibre at a point x of V is the projective space Mx , quotient space of Tx∗ V without 0, by this equivalence relation; we write Mx = PTx∗ V = (Tx∗ V \{0})/R∗ (identifiable with R∗n /R∗ ). This manifold M = M (V ) is called the manifold of all contact elements of the manifold V . Let π be the canonical projection of M on V : π(x, p) = x. We then define a k-field on M, with k = 2n − 2, by   PmM = ξm ∈ Tm M, πm (ξm ) ∈ ker p = (πm )−1 (ker p), ∀m = (x, p), p ∈ Mx . We verify that this is a contact structure P M on M. Consider a coordinate system of T ∗ V : (x 1 , . . . , x n−1 , x n ; p1 , . . . , pn−1 , pn ). We choose a chart of PT ∗ V such that pn = 1, and then let u = x n ; thus the coordinate system of PT ∗ V is denoted by 7 See

[Dieud3, ch. 16.20, ex. 9].

486

Appendix A

(x  , u, p ), x  = (x 1 , . . . , x n−1 ), p = (p1 , . . . , pn−1 ). The vector ξm ∈ PmM is a priori of the form ξm = ξx  ∂x  +ξu ∂u +ξp ∂p . We then have πm (ξm ) = ξx  ∂x  +ξu ∂u , and πm (ξm ) must satisfy < p, πm (ξm ) > = p .ξx  + ξu = 0. Moreover, with the natural differential form (see below) θ˜m = p .dx  + du on PT ∗ X, we also have < θ˜m , ξm > = p .ξx  + ξu . Thus we have defined on M a natural contact structure. The canonical differential form θ = p.dx on T ∗ V induces on M a (global) differential form θ˜ such that ker θ˜m = PmM . Proof Let π V be the canonical projection T ∗ V on V , and let π˜ be the projection from T ∗ V onto M = PT ∗ V , linked with the projection π from M onto V by π V = π ◦ π˜ , and then for every vector field ξ on T ∗ V , we have π∗V (ξ ) = π∗ (π˜ ∗ (ξ )) = π∗ (ξ P )) with ξ P = π˜ ∗ (ξ ) ∈ Tm M. For every (x, λp) ∈ T ∗ V , λ ∈ R∗ , we have θ(x,λp)(ξ ) = λp(π∗V (ξ )) = λp(π∗ (ξ P )). Thus by passing to the quotient, we have defined a (global) differential form θ˜ on M, and by taking λ = 1/pn , we can write θ˜(x  ,u,p ) = du + p dx  . ¶ We also say (see [Arn1]) that M is the bundle of all contact elements of V (here we call every tangent hyperplane to V a contact element of V ). In the manifold M = M (V ) , to each contact element p of V we can assign a positive side by choosing one of the sides of ker p in R n . Such an element is called M+ an oriented contact element and denoted by P(x,p) = (π  (x, p))−1 p+ . The set of oriented contact elements constitutes a (2n−1)-dimensional differentiable manifold, ˜ with a natural contact structure. This is a double covering of M. ¶ denoted by M,

A.4.3 Darboux Theorem for Contact Structures We can also define a contact structure on a (2n + 1)-dimensional manifold M in the following way. Definition 18 Let (Ui , φi )i∈I be a family of charts of a manifold M such that (Ui )i∈I is an open covering of M and such that (φi ) is a diffeomorphism from Ui into R 2n+1 with the following properties. For every function fj i = φj ◦ φi−1 on R 2n+1 (from φi (Ui ) onto φj (Uj )), there exists a (differentiable) function αj i from φi (Ui ) with values in R∗ such that

Appendix A

487

fj∗i (θ ) = αj i θ,

with

αij (m) = 0, m ∈ R 2n+1 ,

where θ is the differential form on R 2n+1 given at m = (x, y, z) ∈ R n × R n × R by  θ(x,y,z) = y.dx + dz = yk dx k + dz. Then the family of differential forms (θi = φi∗ θ ) is such that θj = aj i θi with aj i (m) = 0 for every m ∈ Ui ∩ Uj , and thus defines a contact structure by Pm = ker (θi )m ,

m ∈ Ui .

The equivalence between the two definitions may be viewed as a form of the Darboux theorem (or also of that of É. Cartan), which is recalled below. We verify the relation between the forms θj and θi . We have φi∗ fj∗i (θ ) = φj∗ (θ ) = θj = φi∗ (αj i θ ) = (αj i ◦ φi )θi , and thus aj i = αj i ◦ φi = 0 with aj i (m) = 0, ∀m ∈ M.   Let Uˆ k = (p1 , . . . , pn+1 ) ∈ R n+1 , pk = 0 . Then we define a family of open sets Uˆ k , k = 1, . . . , n + 1, in R n+1 , and an atlas on the projective space PR n+1 by the maps φˆ k defined from Uˆ k onto R n by φˆk (p1 , . . . , pn+1 ) = ((p1 /pk , . . . , pn+1 /pk ). By choosing the chart (Uˆ n+1 , φˆn+1 ), we obtain a coordinate system denoted by (x, y, z) with x = (x 1 , . . . , x n ), y = p = (p1 , . . . , pn ), z = x n+1 . In this chart, the differential form θ = pk dx k is transformed into   (pk /pn+1 )dx k + dx n+1 = yk dx k + dz, on φi (Ui ) ∩ φˆ n+1 (Uˆ n+1 ) × R n+1 . k

k

This corresponds to the Darboux theorem for the contact structures: Theorem 5 The Darboux theorem (see [Arn1, App. 4.G]). 1. The contact manifolds of the same dimension locally correspond through contact diffeomorphisms. 2. For every contact manifold (M, P ) there is a contact diffeomorphism that applies (M, P ) to a manifold (M (V ) , P M ). 3. Every differential form θ that defines a contact structure on a (2n + 1)dimensional manifold can be written, in a coordinate system x = (x 1 , . . . , x n ), with y = (y1 , . . . , yn ), and z, in the “normal form”  θ = ydx + dz = yj dx j + dz. (A.30)

488

Appendix A

A.4.4 Symplectification of a Contact Manifold (M, P ) It is interesting to exchange a contact structure for a symplectic structure, and conversely. Let (M, P ) be a contact structure. To every Pm contact hyperplane corresponds, in a unique way, the element, denoted by Pm0 , that is the set of contact covectors at m (see (A.15)). This is an equivalence class of covectors at m for the relation (A.29), thus an element of PT ∗ M. Then the map σP : m ∈ M → Pm0 is a (global) section of the bundle PT ∗ M. Let   P 0 = (m, p), p ∈ Tm∗ M, ker p ⊇ Pm , m ∈ M ;

(A.31)

P 0 has the structure of a line fibre on M, i.e., of a vector bundle of base M (or on σP (M)), with fibre Pm0 identifiable with R. We also define   P∗0 = (m, p), m ∈ M, p ∈ Tm∗ M, ker p = Pm , that is, P 0 without the null section of T ∗ M; P∗0 has the structure of a principal bundle, of base M (or σP (M)), with fibre identifiable with R∗ . These manifolds are of even dimension, dim M + 1 = 2n + 2, and are naturally submanifolds of the phase space T ∗ M. Hence there exists on P 0 (and thus on P∗0 ) a canonical differential form θ P induced by the canonical differential form θ on T ∗ M, and thus defined by P (ξ ) = p(πP (ξ )) = p(π  (jP (ξ ))), θ(m,p)

∀ξ ∈ T(m,p) P 0 , thus θ P = jP∗ θ, (A.32) where πP = π ◦ jP is the canonical projection from P 0 onto M, and jP is the canonical embedding of P 0 in T ∗ M. Then we have the following theorem. Theorem 6 ( [Arn1, App. 4.E]) The exterior derivative of θ P is a nondegenerate 2-form dθ P that defines on P 0 a canonical symplectic structure. Definition 19 The symplectic space defined from a contact manifold (M, P ) is the pair (P 0 , dθ P ), with P 0 and θ P given below. Remark 5 Notice that if there exists a contact form θ onto M, then the principal bundle P∗0 is trivial, i.e., P∗0 = σP (M) × R∗ . This is a consequence of [Kob-Nom, ch. I.5] and [Kob-Nom, Vol. 2, Note 28].Then the map m → θm is an embedding of M into its symplectic space P 0 , which is called a θ -embedding. ¶ Remark 6 The family of functions (aj i ) that occurs in Definition 18 naturally satisfies the relations aj i .aik = aj k

on Uj ∩ Ui ∩ Uk ,

Appendix A

489

and thus is a cocycle on M with values in R∗ , and then it defines a principal bundle λ = (P˜ , R∗ , M, π). We can identify P˜ with P∗0 , and for all i, the differential form θi is a section of the principal bundle λ on Ui . ¶ A diffeomorphism φ on a manifold V induces (by pullback) a diffeomorphism φ ∗ on T ∗ V that preserves the canonical form α of T ∗ V . Let (M, P ) be a contact manifold, and φ a contact diffeomorphism of (M, P ). Then φ ∗ maps the submanifold P 0 into itself. We can also use the pullback of a contact covector p (at v) on (M, P ) by a contact diffeomorphism φ of (M, P ); this is the contact covector “obtained by transport” by φ, defined by φ∗ p = (φ −1 )∗ p (this is a contact covector, since φ is a contact diffeomorphism). The main properties are given in the following theorem and diagram. Theorem 7 A contact diffeomorphism φ on (M, P ) induces a diffeomorphism on P∗0 , φ∗ : p → φ∗ p that commutes with the real multiplicative group R∗ and preserves the canonical form θ P of P∗0 . The map φ∗ is called a symplectification diffeomorphism of φ: φ∗

T ∗ M −→ T ∗ M P∗0 ↓ ↓ ↓

(φ −1 )∗ =φ∗

PT ∗ M −→ PT ∗ M PP∗0

−→

Pφ ∗

−→

P∗0 ↓

Pφ∗

PP∗0

Remark 7 “Contactification” of a symplectic manifold. Let (M, ω) be a 2ndimensional symplectic manifold; we can define a (2n + 1)-dimensional contact manifold, which is a bundle on M, with fibre R, called the contactification of (M, ω). This problem is naturally posed for the quantization of a mechanical system (see [Kir, §15]). If ω is exact, i.e., there exists a differential form θ such that ω = dθ , then on the π trivial bundle M × R → M, the differential form θ˜ = π ∗ θ + dφ on M × R (where φ is a coordinate of R) is a contact form such that d θ˜ = π ∗ dω. If ω is not exact, the problem comes under notions of algebraic geometry. For application to quantum physics, we refer, for instance, to [Kir, §15]. ¶

A.4.5 Integral Submanifolds and Hamiltonian Definition 20 Contact vector fields. A vector field X on a contact manifold (M, P ) is said to be a contact field if it generates a 1-parameter pseudogroup (Ut ) of contact diffeomorphisms. Note that such a vector field X cannot be a vector field of P , since X would be a characteristic vector field, which is impossible due to the nondegenerate condition of P . Let X be a contact vector field on a contact manifold (M, P ); then let (Ut ) be the 1-parameter pseudogroup generated by X, and (Ut )∗ the family of

490

Appendix A

symplectifications of (Ut ), which is also a 1-parameter pseudogroup. The vector field on P∗0 that generates (Ut )∗ is called the the symplectified of X, and denoted by X∗ . Using the isomorphism Iˆ of (A.25), we have the following result. Theorem 8 The field X∗ is a Hamiltonian vector field: there exists a real function H on P∗0 , with H (m, λp) = λH (m, p), ∀λ > 0, (m, p) ∈ P 0 (H (m, p) is a first-order homogeneous Hamiltonian), such that X∗ = XH = Iˆ−1 (dH ). Conversely, every Hamiltonian vector field defined on P 0 corresponding to a firstorder homogeneous Hamiltonian H gives by projection a contact vector field on (M,P). These properties are consequences of the Darboux theorem. Since (Ut )∗ preserves θ P , it follows that LX∗ θ P = 0, and thus iX∗ dθ P = −d(iX∗ θ P ). Hence X∗ is a Hamiltonian vector field (and not only locally Hamiltonian), and we can take P H = iX∗ θ P , i.e., H (m, p) = θ(m,p) (X∗ ) = p(πP (X∗ )) = p(Xm ).

(A.33)

Main properties: The set of contact vector fields on (M, P ) has a Lie algebra structure. The Poisson bracket of two contact vector fields is a contact vector field. The map X → X∗ is a Lie algebra isomorphism. With the hypothesis of existence of a (global) contact form θ on V , the canonical P form θ P may be written θ(m,p) = λπ ∗ θm , with (m, p) = (m, λθm ) (see Remark 5 and [Arn1, App. 4.E]). Furthermore, we see by (A.33) that H (λθm ) = λθm (Xλθm ) = λθm (Xθm ) = λH (θm ), which is the desired homogeneity of H . Then the function K on (M, P ) defined thanks to H by K(m) = H (θm ) is said to be a contact Hamiltonian of a contact vector field X on (M, P ) (see [Arn1, App. 4.I]). Definition 21 Legendre submanifold. An integral manifold with maximal dimension of a contact manifold (M, P ) is called a Legendre submanifold of (M, P ). This maximal dimension is n if the dimension of V is 2n + 1. Example 14 ( [Arn1, App. 4]) Let V be an (n + 1)-dimensional manifold, and Z a k-dimensional submanifold of V , with k ≤ n. The set Z˜ of all the contact elements of V that are “tangent” to Z, that is, Z˜ = {(z, Pz ), z ∈ Z, Tz Z ⊂ Pz ⊂ Tz X, dim Pz = n} is a Legendre submanifold (of dimension n) of M = M (V ) . Here Z˜ must be viewed as a submanifold of the Grassmann manifold Gn (T V ) (it is not a bundle of base Z). ¶

Appendix A

491

A.4.6 Contact Manifold of Jets On the set of real functions defined on a neighborhood of a point x ∈ V , of class C 1 , we consider the equivalence relation f * g if f (x) = g(x), df (x) = dg(x) : f and g have a contact at x of order greater than or equal to 1. Definition 22 1-jet and 1-graph of a function.8 We call the equivalence class, jx1 (f ) of f , which is given by (x, u, p) with u = f (x), p = df (x), the 1-jet of f. We call every element (x, u, p) for which there exists a differentiable function f such that u = f (x), p = df (x) a 1-jet of the function f . We define the 1-graph of a real differentiable function f on an n-dimensional manifold V to be the set J 1 (f ) = {(x; u; p), x ∈ X, u = f (x), p = df (x)} .

(A.34)

In a coordinate system (x i ) in V , we change p = df (x) into p = (pi ) ∈ R n , with ∂f pi = < df (x), ∂x i > = ∂x i. Definition 23 Manifold of 1-jets of a function. The set of 1-jets of real differentiable functions on V has a manifold structure, called the manifold of 1-jets of functions on V , and denoted by J 1 (V , R). The manifold J 1 (V , R) has naturally the structure of a bundle on V , on V × R, and on T ∗ (V ), with respective fibres R × Tx∗ V , Tx∗ V and R. Then we denote by π, , and 0 the natural projections of the space J 1 (V , R) on respectively V , T ∗ V , and V × R. The 1-graph of a real differentiable function f on V is an n-dimensional submanifold, called the Legendre manifold, of M = J 1 (V , R), which is of dimension 2n + 1. A differentiable function f on V defines a map fˆ = j 1 (f ): x ∈ V → (x, f (x), Df (x)) ∈ J 1 (f ) ⊂ J 1 (V , R), which is a section of the bundle J 1 (V , R) on V . The manifold J 1 (V , R) is equipped with the following contact structure. Definition 24 Standard contact 1-form on J 1 (V , R). We define the standard contact 1-form on J 1 (V , R) to be the differential form α, usually defined9 by αm = αx,u,p = p.dx − du. 8 For 9 Or

a more general definition of the jet notion, see [Bour.var, 12.1]. its opposite du − p.dx.

(A.35)

492

Appendix A

A standard contact structure is defined on M = J 1 (V , R) (the manifold of jets) by the field of hyperplanes m ∈ M → P (m) = ker αm = {ξ ∈ Tm M, αm (ξ ) = 0}.

(A.36)

The 1-form α, in a coordinate system (x 1 , . . . , x n ) of V , is then given by αx,u,p = p1 dx 1 + · · · + pn dx n − du, but it does not depend on the choice of  coordinates and may be globally defined by αx,u,p = p ◦ πx,u,p − du. The 1-form α is null on every tangent space to a 1-graph of a function: Tm J 1 (f ) ⊂ P (m),

∀m ∈ M, f ∈ C r (X), r ≥ 1.

Remark 8 The set J 1 (V , R) may be identified with an open set of the projective manifold PT ∗ (V × R) using a chart Un+1 , where x n+1 = −u and pn+1 = 1. The differential form θ on PT ∗ (V × R) is then changed into the differential form α of J 1 (V , R). ¶

A.4.7 Legendre Involution, Legendre Transformation Let M = J 1 (R n , R) be identified with R n × R × R n (or PT ∗ R n+1 ). Definition 25 Legendre involution. The map L defined by m = (x, u, p) ∈ M → m = (x  , u , p ) ∈ M with

p = x, x  = p, u = p.x − u,

(A.37)

is called a Legendre involution. Theorem 9 The Legendre involution is a diffeomorphism of the contact structures (such that L(α) = −α) that transforms Legendre submanifolds into Legendre submanifolds: L

(x, u, p) ∈ J 1 (f ) ⊂ M = J 1 (R n , R) −→ (x  , u , p ) ∈ J 1 (R n , R) ↓π ↓π (x, u) ∈ G(f ) ⊂ R n × R

L˜

−→

(x  , u ) ∈ R n × R

The image L(J 1 (f )) of the 1-graph of the function f is called the Legendre transform of the manifold J 1 (f ). The dimension of this image is n. The projection of L(J 1 (f )) onto the space Rxn × Ru (parallel to the direction n Rp ) has singularities and is not necessarily a differentiable manifold. If f is a convex (respectively strictly convex) function, then the projection of L(J 1 (f )) is also the graph of a convex (respectively strictly convex) function g, which is called

Appendix A

493

˜ . This function the Legendre transform of the function f , also denoted by f ∗ = Lf is directly defined from f by g(p) = f ∗ (p) = sup (p.x − f (x)), x∈V

and we have f ∗ (p) + f (x) = p.x. Definition 26 Legendre foliation, Legendre singularities. Let (M, P ) be a contact manifold of dimension (2n + 1). • A Legendre foliation of (M, P ) is a foliation MP whose leaves are Legendre manifolds. The pair (M, MP ) is said to be a Legendre foliated manifold. • Let M = PT V (or M = J 1 (V , R)) with the (n + 1)-dimensional manifold V ; we define the Legendre singularities of a Legendre submanifold Z of M to be the singularities of the projective map π|Z on V . Let (M, P ) and (M  , P  ) be two (2n + 1)-dimensional contact manifolds with Legendre foliations respectively MP and MP  . The Legendre foliated manifolds (M, MP ) and (M  , MP  ) are said to be equivalent if there exists a diffeomorphism f between the two Legendre foliated manifolds with conservation of the contact structures and their leaves.10 Example 15 Let M = R 2n+1 equipped with the contact structure defined by the form α = p.dx − du, with x = (x1 , . . . , xn ), p = (p1 , . . . , pn ). Let Hx0 ,u0 ,p = {(x, u, p) ∈ R n × R × R n , p.(x − x0 ) − (u − u0 ) = 0}. This is an n-dimensional plane such that p.x − u = p.x0 − u0 is constant, which contains the point (x0 , u0 , p). Thus p.dx − du = 0 on Hx0 ,u0 ,p , which is then a Legendre submanifold of M. Let us take x0 = 0, and set Hu0 ,p = H0,u0 ,p . The planes (Hu0 ,p ), (u0 , p) ∈ R n × R are the leaves of a Legendre foliation of M. Let π be the projection (x, u, p) → (u−p.x, p) = (u0 , p); we have Hu0 ,p = π −1 (u0 , p). Then the family Hu0 ,p , with (u0 , p) ∈ R n × R is a partition of M. Every Legendre foliation is locally equivalent to the previous Legendre foliation M = R 2n+1 . ¶ Remark 9 Partial Legendre involution and construction of Legendre submanifolds M = R 2n+1 thanks to generating functions. Let (I, J ) be a partition of {1, . . . , n}. Let xI = (xi )i∈I , x J = (x j )j ∈J and similarly for p. We define the partial Legendre involution LJ,I by m = ((x J , x I ), u, (pJ , pI )) ∈ M → m = (x J , pI , S, −pJ , x I ) ∈ M, with S = pI .x I − u.

10 Thus

f  = Tf “interchanges” the bundles tangent to the foliation and the contact fields.

(A.38)

494

Appendix A

This map transforms every Legendre submanifold of M into a Legendre submanifold. Indeed, we have d(pI .x I ) − du − dS = 0 and pJ .dx J + pI .dx I − du = 0, and thus by difference, we obtain −pJ .dx J + x I .dpI − dS = 0. Now we assume that S = S(x J , pI ) is a regular function of n variables x j , pi , with j ∈ J, i ∈ I . Then it defines the 1-graph J 1 (S), which implies the formulas xi =

∂S , i ∈ I, ∂pi

pj = −

∂S , j ∈ J. ∂x j

The inverse of the map LJ,I defines u = pI .x I − S =

(A.39) 

∂S pi ∂p − S as a i

1 “pseudofunction” of x = (x J , x I ), but recall that a priori, L−1 J,I (J (S)) is not I a 1-graph of a function. However, if S = S(x , pJ ) is a convex function, then 1 1 L−1 J,I (J (S)) = J (u) is the 1-graph of a convex function u that is defined by

u(x J , x I ) = sup (pI .x I − S(x J , pI )). pI ∈R I

(Observe that the space R J × R I , which is the base of J 1 (S), which is the set of the (x J , pI ), has a fibre space interpretation as a transverse fibre space (or normal space) to the manifold γ = R J of the x J .) Conversely, every Legendre submanifold of R 2n+1 is defined in a neighborhood of one of its points by the formula (A.39). The function S is called a generating function. Classification of the Legendre singularities may be realized thanks to generating functions (see [Arn1, App. 4.J]). ¶

A.4.8 Characteristics of a Contact Structure Let W = W (2n) be a differentiable hypersurface in M = M 2n+1 = J 1 (V , R). Definition 27 Noncharacteristic hypersurface. Characteristic plane. Characteristic direction at m in W . Noncharacteristic point. • A hypersurface W in M is said to be noncharacteristic for a contact structure (M, P ) if Tm W and Pm are transverse for every m in M, i.e., Tm W ⊕ Pm = Tm M, or if the dimension of PmW = Tm W ∩ Pm is 2n − 1, ∀m ∈ M. • Then PmW is called a characteristic plane.

Appendix A

495

• If α is a differential form on M locally defining P , then a direction ξ at a point m of W is said to be characteristic if ξ ∈ PmW , dα(ξ, η) = 0, ∀η ∈ PmW .

(A.40)

• The integral curves of a characteristic direction field (defined on a noncharacteristic hypersurface W of a contact manifold (M, P )) are said to be characteristics of W . • Let N be an integral manifold of (M, P ) contained in W . A point x of N is said to be noncharacteristic if Tx N does not contain the characteristic direction. The characteristics that pass through points of a noncharacteristic integral manifold N = N k locally constitute a (k + 1)-dimensional integral submanifold of (M, P ) containing W . Explicit formulas for the equation of characteristics for (M, P ) in W defined by a function of class C 2 , W = {m = (x, u, p) ∈ M, (m) = 0}, with d (m) = 0, ∀m ∈ W.

(A.41)

We verify that a characteristic vector ξ = ξx ∂x + ξu ∂u + ξp ∂p ∈ Tm M is such that ξx = p , ξp = −( x + p u ), ξu = p. p .

(A.42)

Proof A vector ξ = (ξx , ξu , ξp ) is characteristic if it satisfies the following three conditions: • i(ξ )α = 0 (i.e., ξ ∈ P (m)), with α = pdx − du, and thus ξu = p.ξx . • < d , ξ > = 0 (i.e., ξ ∈ Tm W ), and thus x ξx + u ξu + p ξp = 0. Thus ξ ∈ P (m) ∩ Tm W is equivalent to ξ = (ξx , p.ξx , ξp )

and ( x + p u ).ξx + p ξp = 0.

(A.43)

• dα(ξ, ξ˜ ) = 0, ∀ξ˜ ∈ P (m) ∩ Tm W (thus ξ˜ = (ξ˜x , p.ξ˜x , ξ˜p )), ξx .ξ˜p − ξp .ξ˜x = 0,

∀ξ˜ ∈ P (m) ∩ Tm W.

(A.44)

Moreover, ξ˜ also satisfies (A.43). We deduce the proportionality of the coefficients between (A.43) and (A.44). Thus we can take ξx = p ,

ξp = −( x + p u ),

ξu = p. p .

(A.45)

Remark 10 Then we can verify that ξ satisfies the relation i(ξ )dα = d − α u .

(A.46)

496

Appendix A

Indeed, we have i(ξ )dα = i(ξ )(dx ∧ dp) = (i(ξ )dx)dp) − dx(i(ξ )dp) = ξx dp − ξp dx = p dp + ( x + p u )dx, whence (A.46).

(A.47)

¶

Application with M = J 1 (V , R): let (x, u, p) = ai (x)pi . Then we have ∂ai ∂ ξx = A(x) = (aj (x)), ξu = , and ξpj = − ∂x j = −( ∂x j )pi . Thus the projection of the characteristic vector ξ on the space Tx V is the vector A(x) = (aj (x)).

A.4.9 Cauchy Problems, First-Order Partial Differential Equations Let W be a hypersurface defined in M = J 1 (V , R) by a function of class C 2 : W = {m = (x, u, p) ∈ M, (m) = 0}, with d (m) = 0, ∀m ∈ W.

(A.48)

In a coordinate system (xi ) of V , we consider (x, u, p) = 0,

(A.49)

a nonlinear equation with partial derivatives of first order in the unknown function U such that U (x) = u, ∂U/∂x i (x) = pi , i = 0, . . . , n − 1. Finding the solutions of this equation is equivalent to determining the integral surfaces of (M, P ) in W . This problem goes back to the determination of characteristics in W . Let us assume that W is a noncharacteristic hypersurface of (M, P ). We define initial conditions as follows: • γ = γ (n−1) is an (n − 1)-dimensional submanifold of V = V (n) , • φ : γ → R is a differentiable function. Let J 1 (φ) be the 1-graph of the function φ, which is a submanifold of J 1 (γ , R), see (A.34), and φˆ = j 1 (φ), the map x ∈ γ → (x, φ(x), dφ(x)) ∈ J 1 (φ). The canonical injection j of the manifold γ in V induces the map j ∗ from T ∗ V into T ∗ γ and then from PT ∗ (V × R) into PT ∗ (γ × R), thus from J 1 (V , R) into J 1 (γ , R). (Thus J 1 (γ , R) is not a submanifold of J 1 (V , R).) We construct a submanifold, denoted by JU0 , of J 1 (V , R), called the initial manifold associated with (γ n−1 , φ), as the set ( -dependent) such that if γ is locally defined (in an open set O of V ) in a coordinate system (x0 , x  ) by x0 = 0, then JU0 ∩ J 1 (O, R) = {(x, u, p) ∈ J 1 (O, R), x = (0, x  ) ∈ γ ∩ O, u = φ(x), p = (p0 , p ), p = dφ(x),

(x, u, p) = 0}. (A.50)

Appendix A

497

In the open set O, with the previous coordinate system, we identify p with p = ∂φ  ∂x  (x ). The equation (x, u, p) = 0 determines p0 , thanks to the implicit function theorem (see [Dieud2, ch. 10.2]), under the condition ∂ ˆ (φ(x)) = 0, ∂p0

x = (0, x  ) ∈ γ ∩ O,

(A.51)

and its derivative with respect to x  is given by p0 (x  ) = −[

∂ ˆ  ))]−1 ∂x  (φ(x ˆ  )) : (φ(x ∂p0 j∗

J 1 (φ) ⊂ J 1 (γ , R) ←− J 1 (U ) ⊂ J 1 (V , R) Uˆ ↑↓ πV γˆ ↑↓ πγ j

−→

γ

V

Let J0 = J0(n−1) be an integral submanifold (M, P ) contained in W . Definition 28 Cauchy problems. • The Cauchy problem for W in (M, P ) with the initial manifold J0 consists in finding an integral manifold Y = Y (n) of (M, P ) such that J0 ⊂ Y ⊂ W. • The Cauchy problem for (A.49) consists in finding the 1-graph of a function U : V → R satisfying (A.49) that is identical to φ on γ , thus such that γˆ = j ∗ ◦ Uˆ ◦ j,

JU0 ⊂ J 1 (U, R) ⊂ W.

Definition 29 Noncharacteristic point (for relative to γ ). A point w, w = (x, u, p), of an initial manifold JU0 associated with (γ , φ) is said to be noncharacteristic for relative to γ if the projection of the characteristic direction at this point w on V is transverse to γ . The condition that (x, u, p) is noncharacteristic for relative to γ is equivalent to p (x, u, p) ∈ Tx γ , which is the condition (A.51) in the open set O with the chosen coordinate system. Then we can give some results of local unicity of the solution of the Cauchy problem; see [Arn2, ch. II.8, G, K]. Theorem 10 Existence, local uniqueness of the solution of Cauchy problems.

498

Appendix A

• Let w ∈ W be a noncharacteristic point of an initial manifold J0 . Then there exists a neighborhood O ⊂ W of w such that there exists a solution of the Cauchy problem in W with the initial condition J0 ∩O, and this solution is locally unique (in O). • Let w = (x, u, p) be a noncharacteristic point (for γ ) of the initial manifold JU0 . Then equation (A.49) with the initial condition JU0 admits, in a neighborhood V of x, a (locally) unique solution. The local uniqueness of the solution means that if two solutions U1 and U2 satisfy the initial condition U |O∩γ = φ|O∩γ , U (x) = u, dU (x) = p, then they are identical in a neighborhood of x. The main point is that the characteristics that come across the initial manifold at a neighborhood of the point (x0 , u0 , p0 ) constitute in this neighborhood a differentiable manifold that is the 1-graph of a function [Arn2, ch. II.8, L].

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502

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Further Readings of Works More or Less Interested in Mathematical Modelling [Ab-Mar] Abraham, R., Marsden, J.E.: Foundations of Mechanics. Library of Congress Cataloging in Publication Data (1977) [Bahouri] Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011) [Berndt] Berndt, R.: An Introduction to Symplectic Geometry, vol. 26. AMS, Providence (2000) [Chemin] Chemin, J.-Y.: Fluides parfaits incompressibles, Astéristique, 230 (1995). English translation Incompressible perfect fluids, Oxford University Press (1998) [Chemin2] Chemin, J.-Y., Desjardins, B., Gallagher, I., Grenier, E.: Mathematical Geophysics; An Introduction to Rotating Fluids and Navier–Stokes Equations. Oxford Lecture Series in Mathematics and Its Applications, vol. 32. Oxford University Press, Oxford (2006) [Chetayev] Chetayev, N.G.: The Stability of Motion. Pergamon Press, New York (1961) [Gay-Bal] Gay-Balmaz, F., Yoshimura, H.: Dirac Structures in Non Equilibrium Thermodynamics. J Math. Phys. 59, 012701 (2018) [Glik] Gliklikh, Y.: Global Analysis in Mathematical Physics. Geometric and Stochastic Methods. Applied Mathematical Sciences, vol. 122. Springer, Berlin (1997) [Godbillon] Godbillon, C.: Géométrie différentielle et Mécanique analytique. Hermann, Paris (1969) [Hairer] Hairer, E., Lubich, C., Wanner, G.: Geometrical Numerical Integration Structure preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer, Berlin (2006) [Holm] Holm, D.D.: Geometric Mechanics. Part I: Dynamics and Symmetry. Part II: Rotating, Translating and Rolling. Imperial College Press, London (2011) [Ho-Sch-ST] Holm, D.D., Schmah, T., Stoica, C.: Geometric Mechanics and Symmetry. From Finite to Infinite Dimensions. Oxford University Press, Oxford (2009) [Lema] Lemarié-Rieusset, P.-G.: The Navier–Stokes Problem in the 21st Century. CRC Press, Boca Raton (2016) [Mack1] Mackey, G.W.: Induced Representations of Groups and Quantum Mechanics. Benjamin, New York (1968) [Mack2] Mackey, G.W.: Unitary Group Representations in Physics, Probability and Number Theory. Benjamin, New York (1978) [Mar-Rat] Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems Text in Applied Mathematics, vol. 17. Springer, Berlin (1999) [PLL] Lions, P.L.: Cours 2017–2018 du Collège de France [Rue] Ruelle, D.: Statistical Mechanics. Rigorous Results. Benjamin, New York (1969) [Sou] Souriau, J.M.: Structure of Dynamical Systems. A Symplectic View of Physics. Birkhäuser, Basel (1997) [Sou1] Souriau, J.M.: Thermodynamique et Géométrie. In: Differential Geometrical Methods in Mathematical Physics II, Proceedings, Bonn. Lecture Notes in Mathematics, vol. 676, pp. 369–398. Springer, Berlin (1977) [Stu-Sch] Stueckelberg, E.C.G., Scheurer, P.B.: Thermocinétique Phénoménologique Galiléenne. Birkhäuser, Basel (1974) [Sud-Mu] Sudarshan Mukunda: Classical dynamics: a modern perspective. TRIPS 17. Hindoustan Book Agency (2015) [Villani] Villani, C.: Optimal Transport. Old and New, vol. 338. Springer, Berlin (2009) [Whitt] Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1917)

Index

A Activity of a constituent, 103 Adapted frame, 342 Additivity properties, 78 Adiabatic evolution, 15 Adjoint representation, 250 Affinity of a reaction, 110 θ-Admissible evolution, 16

B Backward Kolmogorov equation, 125 Barotropic fluid, 29 Basic differential forms, 10 Behavioral laws, 407 Biconjugate, 55 Bingham fluid, 418 Bitangent space, 236

C Canonical form, 321 Carnot cycle, 43 Cartan formula, 183 Cauchy characteristic, 479 Center of mass, 258 Characteristic, 31 Characteristic field, 478 Chemical kinetics, 111 Chemical potential, 11 Chemical reactions, 106 Christoffel symbols, 239, 323, 348 Clausius–Clapeyron equation, 64 Closed differential form, 479 Coadjoint representation, 253, 317

Coercive operator, 411 Complete vector field, 338, 362 Conjugate function, 19, 409 Connection, 332 form, 321 map, 321 Contact diffeomorphism, 485 Contact form, 485 Contact Hamiltonian, 101 Contact manifold, 484 Contact structure, 8, 484 Contact transformation, 8 Contact vector fields, 489 Convex analysis, 19 Convexification, 55 Convexity, 39 Convexity property, 19 Coriolis force, 295 Covariant derivative, 320, 324, 349 Critical point, 62 Cyclic evolution, 16

D d’Alembert–Lagrange principle, 159 d’Alembert principle, 242 Darboux’s theorem, 486 Degree of reaction, 109 Descent of a vertical vector, 322 Deviator, 416 Diffusion process, 125 Domain, 19, 412 Drag, 465 Dual problem, 413 Duhem differential form, 82

© Springer International Publishing AG, part of Springer Nature 2018 M. Cessenat, Mathematical Modelling of Physical Systems, https://doi.org/10.1007/978-3-319-94758-7

503

504 E Effective action of a group, 306 Energy, 10 Enthalpy, 10 Entropic foliation, 17 Entropy, 10 Epigraph, 19 Equation of polytropic gas, 34 Euler angles, 276 Euler equation, 271 Euler–Lagrange equation, 158 Euler metric, 340 Euler operator, 82 Exact differential form, 17, 480 Extensive variable, 80

F Fenchel’s theorem, 413 Fenchel transformation, 19 Fibre bundle, 320 Fibre bundle associated, 331 Fick’s law, 124 First integral, 101, 309 Flux, 124 Fokker–Planck equation, 125 Foliated manifold, 14 Foliation, 3, 34, 337 Form volume, 366 Fourier’s law, 124, 373 Free action of a group, 306 Free energy, 11 Free Gibbs energy, 11 Frobenius theorem, 475 Function, l.s.c., 409 Functional modelling, 135

G Gâteaux differential, 408 Generalized flow, 363 Geodesic, 158 Gibbs-Duhem equation, 83 Global modelling, 2, 78 Green–Lagrange strain tensor, 346 Green–Lagrange tensor, 346

H Hamiltonian field, 165 Hamilton mechanics, 159 Heat, 12 Hessian matrix, 23 Horizontal form, 179, 322

Index Horizontal lift, 321 Horizontal tangent vectors, 321 Hugoniot relation, 401

I Ideal gas, 14, 30 Ideal gas law, 31 Ideal solution, 103 Indicator function, 410 Integrable map, 5 Integrable subbundle, 4 Integral curve, 336 Integral flow, 336 Integral manifold, 3 Integrating factor, 479 Intensive variable, 80 Internal energy, 157 θ-Irreversible evolution, 16 Isentropic evolution, 15 Isochronic fields, 163, 166

J Jacobian matrix, 7 Jets of functions, 5, 491 Jump relations, 399

K K-field on a manifold, 474

L Lagrange metric, 340 Lagrangian, 158 Lamé elasticity coefficient, 415 Least action principle, 159 Leaves of a foliation, 5 Legendre involution, 492 Legendre transformation, 19 Lewis formula, 102 Lie algebra, 315 Lie derivative, 473 Lie group, 315 Lifetime interval, 336 Lift, 465 Linear connection, 321 Locally Hamiltonian field, 165

M Maurer–Cartan form, 316 Maximal integral manifold, 14

Index Maximum parabolic principle, 127 Mixture, 89 Momentum map, 310

N Newtonian fluid, 416 Newtonian mechanics, 158 Noether’s theorem, 309

O Odd form, 366

P Parallelizable manifold, 318 Parallel transport, 322 Pendulum, 235 Pfaff system, 476 Phase transition, 69 Piece, 366 Planck’s law, 143 Poisson action, 313 Poisson bracket, 307 Positively homogeneous function, 36 Primal problem, 413 Principal fibre bundle, 330 Pseudogroup, 336 Pull back, 472

R Rate of strain tensor, 347 Reaction rate, 111 Real solutions, 102 Real van der Waals solutions, 104 Relative motion, 290 Resistance function, 414 θ-Reversible evolution, 15 Ricci tensor, 356 Riemann–Christoffel tensor, 331

505 S Saturation curve, 57 Schwarz conditions, 5 Second fundamental form, 359 Second law, 16 Semiglobal modelling, 3 Speed of sound, 25 Spray, 181 Stability, 26 State equations, 13 Stefan–Boltzmann law, 147 Subadditive function, 84 Subdifferentiable function, 19 Subdifferential, 19, 408 Subgradient, 408 Superficial tension, 91, 467 Support function, 410 Symplectic form, 165 Symplectic structure, 482 System of ideal gases, 93

T Thermal radiation, 143 Thermodynamic field, 122 Thermodynamic forces, 129 Thermoelectromagnetism, 139 Thermosystem, 3 Transpose, 412

V van der Waals equation, 45 Variational principle, 158 Vertical descent, 323 Vertical form, 179, 322 Virtual work principle, 159 Viscous coefficient, 416

W Work, 12 Wrench, 250