*280*
*35*
*900KB*

*English*
*Pages 280*

- Author / Uploaded
- Radu Miron

*Table of contents : Preface......Page 7I Lagrange and Hamilton Spaces......Page 15Riemannian mechanical systems......Page 19The evolution semispray of the mechanical system TEXT......Page 22The nonlinear connection of TEXT......Page 28The canonical metrical connection TEXT......Page 34Semidefinite Finsler spaces......Page 39The notion of Finslerian mechanical system......Page 41The evolution semispray of the system TEXT......Page 44The canonical nonlinear connection of the Finslerian mechanical systems TEXT......Page 46The dynamical derivative determined by the evolution nonlinear connection TEXT......Page 48Metric N-linear connection of TEXT......Page 52The electromagnetism in the theory of the Finslerian mechanical systems TEXT......Page 55The almost Hermitian model on the tangent manifold TEXT of the Finslerian mechanical systems TEXT......Page 57Lagrange Spaces. Preliminaries......Page 61Lagrangian Mechanical systems, TEXT......Page 65The evolution semispray of TEXT......Page 67The evolution nonlinear connection of TEXT......Page 68Hamilton spaces. Preliminaries......Page 71The Hamiltonian mechanical systems......Page 75Canonical nonlinear connection of TEXT......Page 79The Cartan mechanical systems......Page 83Parallelism, paths and structure equations......Page 89Lagrangian Mechanical systems of order TEXT......Page 95Lagrangian mechanical system of order TEXT, TEXT......Page 98Canonical TEXTsemispray of mechanical system TEXT......Page 99Canonical nonlinear connection of mechanical system TEXT......Page 101Canonical metrical TEXTconnection......Page 102Notion of Cartan space......Page 105Canonical nonlinear connection of TEXT......Page 108Canonical metrical connection of TEXT......Page 110The duality between Lagrange and Hamilton spaces......Page 112II Lagrangian and Hamiltonian Spaces of higher order......Page 117The electromagnetism in the theory of the Riemannian mechanical systems TEXT......Page 138The almost Hermitian model of the RMS TEXT......Page 140III Analytical Mechanics of Lagrangian and Hamiltonian Mechanical Systems......Page 163Canonical TEXTmetrical connection of TEXT. Structure equations......Page 215The almost Hermitian model of the Lagrangian mechanical system TEXT......Page 218Generalized Lagrangian mechanical systems......Page 220The Riemannian TEXT almost contact model of the Lagrangian mechanical system of order TEXT, TEXT......Page 249Classical Riemannian mechanical system with external forces depending on higher order accelerations......Page 251Finslerian mechanical systems of order TEXT, TEXT......Page 254Hamiltonian mechanical systems of order TEXT......Page 257*

Lagrangian and Hamiltonian geometries. Applications to Analytical Mechanics Radu Miron

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Contents Preface

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I Lagrange and Hamilton Spaces

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1 The 1.1 1.2 1.3 1.4 1.5

Geometry of tangent manifold The manifold TM . . . . . . . . . . Semisprays on the manifold T M . . Nonlinear connections . . . . . . . N -linear connections . . . . . . . . Parallelism. Structure equations . .

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2 Lagrange spaces 2.1 The notion of Lagrange space . . . . . . . . . . . . . 2.2 Variational problem. Euler-Lagrange equations . . . 2.3 Canonical semispray. Nonlinear connection . . . . . . 2.4 Hamilton-Jacobi equations . . . . . . . . . . . . . . . 2.5 Metrical N -linear connections . . . . . . . . . . . . . 2.6 The electromagnetic and gravitational fields . . . . . 2.7 The almost Kählerian model of a Lagrange space Ln 2.8 Generalized Lagrange spaces . . . . . . . . . . . . . . 3 Finsler Spaces 3.1 Finsler metrics . . . . . . . 3.2 Geodesics . . . . . . . . . . 3.3 Cartan nonlinear connection 3.4 Cartan metrical connection

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4 The 4.1 4.2 4.3 4.4 4.5

Geometry of Cotangent Manifold Cotangent bundle . . . . . . . . . . . . . . . . . . Variational problem. Hamilton–Jacobi equations . Nonlinear connections . . . . . . . . . . . . . . . N −linear connections . . . . . . . . . . . . . . . Parallelism, paths and structure equations . . . .

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5 Hamilton spaces 5.1 Notion of Hamilton space . . . . . . . . . . . . . . . . . 5.2 Nonlinear connection of a Hamilton space . . . . . . . . 5.3 The canonical metrical connection of Hamilton space H n 5.4 Generalized Hamilton Spaces GH n . . . . . . . . . . . . 5.5 The almost Kählerian model of a Hamilton space . . . . 6 Cartan spaces 6.1 Notion of Cartan space . . . . . . . . . . . . 6.2 Canonical nonlinear connection of C n . . . . 6.3 Canonical metrical connection of C n . . . . . 6.4 The duality between Lagrange and Hamilton

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Lagrangian and Hamiltonian Spaces of higher order

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Geometry of the manifold T k M The bundle of acceleration of order k ≥ 1 . . . . . . . . The Liouville vector fields . . . . . . . . . . . . . . . . Variational Problem . . . . . . . . . . . . . . . . . . . Semisprays. Nonlinear connections . . . . . . . . . . . The dual coeﬀicients of a nonlinear connection . . . . . Prolongation to the manifold T k M of the Riemannian given on the base manifold M . . . . . . . . . . . . . . N −linear connections on T k M . . . . . . . . . . . . .

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1 The 1.1 1.2 1.3 1.4 1.5 1.6 1.7

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2 Lagrange Spaces of Higher–order 2.1 The spaces L(k)n = (M, L) . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples of spaces L(k)n . . . . . . . . . . . . . . . . . . . . . . . 2.3 Canonical metrical N −connection . . . . . . . . . . . . . . . . . .

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The Riemannian (k − 1)n−contact model of the space L(k)n . . . . 135 The generalized Lagrange spaces of order k . . . . . . . . . . . . . . 137

3 Higher-Order Finsler spaces 141 3.1 Notion of Finsler space of order k . . . . . . . . . . . . . . . . . . . 141 4 The Geometry of k−cotangent bundle 143 ∗k 4.1 Notion of k−cotangent bundle, T M . . . . . . . . . . . . . . . . . 143

III Analytical Mechanics of Lagrangian and Hamiltonian Mechanical Systems 149 1 Riemannian mechanical systems 1.1 Riemannian mechanical systems . . . . . . . . . . . . . . . . . . . 1.2 Examples of Riemannian mechanical systems . . . . . . . . . . . 1.3 The evolution semispray of the mechanical system ΣR . . . . . . . 1.4 The nonlinear connection of ΣR . . . . . . . . . . . . . . . . . . . 1.5 The canonical metrical connection CΓ(N ) . . . . . . . . . . . . . 1.6 The electromagnetism in the theory of the Riemannian mechanical systems ΣR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The almost Hermitian model of the RMS ΣR . . . . . . . . . . . .

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2 Finslerian Mechanical systems 2.1 Semidefinite Finsler spaces . . . . . . . . . . . . . . . . . . . . . . 2.2 The notion of Finslerian mechanical system . . . . . . . . . . . . 2.3 The evolution semispray of the system ΣF . . . . . . . . . . . . . 2.4 The canonical nonlinear connection of the Finslerian mechanical systems ΣF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The dynamical derivative determined by the evolution nonlinear connection N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Metric N-linear connection of ΣF . . . . . . . . . . . . . . . . . . 2.7 The electromagnetism in the theory of the Finslerian mechanical systems ΣF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The almost Hermitian model on the tangent manifold T M of the Finslerian mechanical systems ΣF . . . . . . . . . . . . . . . . . .

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3 Lagrangian Mechanical systems 3.1 Lagrange Spaces. Preliminaries . . . . . . . . . . . . . . . . . . . 3.2 Lagrangian Mechanical systems, ΣL . . . . . . . . . . . . . . . . . 3.3 The evolution semispray of ΣL . . . . . . . . . . . . . . . . . . . . 3.4 The evolution nonlinear connection of ΣL . . . . . . . . . . . . . . 3.5 Canonical N −metrical connection of ΣL . Structure equations . . 3.6 Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The almost Hermitian model of the Lagrangian mechanical system ΣL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Generalized Lagrangian mechanical systems . . . . . . . . . . . .

193 . 193 . 196 . 197 . 199 . 201 . 204

4 Hamiltonian and Cartanian mechanical systems 4.1 Hamilton spaces. Preliminaries . . . . . . . . . . 4.2 The Hamiltonian mechanical systems . . . . . . . 4.3 Canonical nonlinear connection of ΣH . . . . . . . 4.4 The Cartan mechanical systems . . . . . . . . . .

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5 Lagrangian, Finslerian and Hamiltonian mechanical systems of 223 order k ≥ 1 5.1 Lagrangian Mechanical systems of order k ≥ 1 . . . . . . . . . . . . 223 5.2 Lagrangian mechanical system of order k, ΣLk . . . . . . . . . . . . 227 5.3 Canonical k−semispray of mechanical system ΣLk . . . . . . . . . . 229 5.4 Canonical nonlinear connection of mechanical system ΣLk . . . . . . 232 5.5 Canonical metrical N −connection . . . . . . . . . . . . . . . . . . . 234 5.6 The Riemannian (k − 1)n almost contact model of the Lagrangian mechanical system of order k, ΣLk . . . . . . . . . . . . . . . . . . . 235 5.7 Classical Riemannian mechanical system with external forces depending on higher order accelerations . . . . . . . . . . . . . . . . . 237 5.8 Finslerian mechanical systems of order k, ΣF k . . . . . . . . . . . . 240 5.9 Hamiltonian mechanical systems of order k ≥ 1 . . . . . . . . . . . 243

Preface The aim of the present monograph is twofold: 1◦ to provide a Compendium of Lagrangian and Hamiltonian geometries and 2◦ to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. One knows (R. Abraham, J. Klein, R. Miron et al.) that the geometrical theory of nonconservative mechanical systems can not be rigourously constructed without the use of the geometry of the tangent bundle of the configuration space. The solution of this problem is based on the Lagrangian and Hamiltonian geometries. In fact, the construction of these geometries relies on the mechanical principles and on the notion of Legendre transformation. The whole edifice has as support the sequence of inclusions: {Rn } ⊂ {F n } ⊂ {Ln } ⊂ {GLn } formed by Riemannian, Finslerian, Lagrangian, and generalized Lagrangian spaces. The L−duality transforms this sequence into a similar one formed by Hamiltonian spaces. Of course, these sequences suggest the introduction of the correspondent Mechanics: Riemannian, Finslerian, Lagrangian, Hamiltonian etc. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus applied to the integral of action and these can be studied by using the methods of Lagrangian or Hamiltonian geometries. More general, the notions of higher order Lagrange or Hamilton spaces have been introduced by the present author [161], [162], [163] and developed is realized by means of two sequences of inclusions similarly with those of the geometry of order 1. The problems raised by the geometrical theory of Lagrange and Hamilton spaces of order k ≥ 1 have been investigated by Ch. Ehresmann, W. M. Tulczyjew, A. Kawaguchi, K. Yano, M. Crampin, Manuel de Léon, R. Miron, M. Anastasiei, I. Bucătaru et al. [175]. The applications lead to the notions of Lagrangian or Hamiltonian Analytical Mechanics of order k. For short, in this monograph we aim to solve some diﬀicult problems: i

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- The problem of geometrization of classical non conservative mechanical systems; - The foundations of geometrical theory of new mechanics: Finslerian, Lagrangian and Hamiltonian; - To determine the evolution equations of the classical mechanical systems for whose external forces depend on the higher order accelerations. This monograph is based on the terminology and important results taken from the books: Abraham, R., Marsden, J., Foundation of Mechanics, Benjamin, New York, 1978; Arnold, V.I., Mathematical Methods in Classical Mechanics, Graduate Texts in Math, Springer Verlag, 1989; Asanov, G.S., Finsler Geometry Relativity and Gauge theories, D. Reidel Publ. Co, Dordrecht, 1985; Bao, D., Chern, S.S., Shen, Z., An Introduction to Riemann-Finsler Geometry, Springer Verlag, Grad. Text in Math, 2000; Bucataru, I., Miron. R., Finsler-Lagrange geometry. Applications to dynamical systems, Ed. Academiei Romane, 2007; Miron, R., Anastasiei, M., The geometry of Lagrange spaces. Theory and applications to Relativity, Kluwer Acad. Publ. FTPH no. 59, 1994; Miron, R., The Geometry of HigherOrder Lagrange Spaces. Applications to Mechanics and Physics, Kluwer Acad. Publ. FTPH no. 82, 1997; Miron, R., The Geometry of Higher-Order Finsler Spaces, Hadronic Press Inc., SUA, 1998; Miron, R., Hrimiuc, D., Shimada, H., Sabau, S., The geometry of Hamilton and Lagrange Spaces, Kluwer Acad. Publ., FTPH no. 118, 2001. This book has been received a great support from Prof. Ovidiu Cârjă, the dean of the Faculty of Mathematics from “Alexandru Ioan Cuza” University of Iasi. My colleagues Professors M. Anastasiei, I. Bucătaru, I. Mihai, M. Postolache, K. Stepanovici made important remarks and suggestions and Mrs. Carmen Savin prepared an excellent print-form of the hand-written text. Many thanks to all of them. Iasi, June 2011

Radu Miron

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The purpose of the book is a short presentation of the geometrical theory of Lagrange and Hamilton spaces of order 1 or of order k ≥ 1, as well as the definition and investigation of some new Analytical Mechanics of Lagrangian and Hamiltonian type of order k ≥ 1. In the last thirty five years, geometers, mechanicians and physicists from all over the world worked in the field of Lagrange or Hamilton geometries and their applications. We mention only some important names: P.L. Antonelli [21], M. Anastasiei [9],[10], [11]. G. S. Asanov [27], A. Bejancu [38], I. Bucătaru [47], M. Crampin [64], R. S. Ingarden [114], S. Ikeda [116], M. de Leon [138], M. Matsumoto [144], R. Miron [164], [165], [166], H. Rund [218], H. Shimada [223], P. Stavrinos [237], L. Tamassy [247] and S. Vacaru [251]. The book is divided in three parts: I. Lagrange and Hamilton spaces; II. Lagrange and Hamilton spaces of higher order; III. Analytical Mechanics of Lagrangian and Hamiltonian mechanical systems. The part I starts with the geometry of tangent bundle (T M, π, M ) of a differentiable, real, n−dimensional manifold M . The main geometrical objects on T M , as Liouville vector field C, tangent structure J, semispray S, nonlinear connection N determined by S, N −metrical structure D are pointed out. It is continued with the notion of Lagrange space, defined by a pair Ln = (M, L(x, y)) with 1 L : T M → R as a regular Lagrangian and gij = ∂˙i ∂˙j L as fundamental tensor 2 field. Of course, L(x, y) is a regular Lagrangian if the Hessian matrix kgij (x, y)k is nonsingular. In the definition of Lagrange space Ln we assume that the fundamental tensor gij (x, y) has a constant signature. The known Lagrangian from Electrodynamics assures the existence of Lagrange spaces. The variational problem associated to the integral of action Z 1 I(c) = L(x, x)dt ˙ 0

allows us to determine the Euler–Lagrange equations, conservation law of energy, EL , as well as the canonical semispray S of Ln . But S determines the canonical nonlinear connection N and the metrical N −linear connection D, given by the generalized Christoffel symbols. The structure equations of D are derived. This theory is applied to the study of the electromagnetic and gravitational fields of the space Ln . An almost Kählerian model is constructed. This theory suggests to define the notion of generalized Lagrange space GLn = (M, g(x, y)), where g(x, y) is a metric tensor on the manifold T M . The space GLn is not reducible to a space

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PREFACE

Ln . A particular case of Lagrange space Ln leads to the known concept of Finsler space F n = (M, F (x, y)). It follows that the geometry of Finsler space F n can be constructed only by means of Analytical Mechanics principles. Since a Riemann space Rn = (M · g(x)) is a particular Finsler space F n = (M, F (x, y)) we get the following sequence of inclusions: (I)

{Rn } ⊂ {F n } ⊂ {Ln } ⊂ {GLn }.

The Lagrangian geometry is the geometrical study of the sequence (I). The geometrical theory of the Hamilton spaces can be constructed step by step following the theory of Lagrange spaces. The legitimacy of this theory is due to L−duality (Legendre duality) between a Lagrange space Ln = (M, L(x, y)) and a Hamilton space H n = (M, H(x, p)). Therefore, we begin with the geometrical theory of cotangent bundle (T ∗ M, π ∗ , M ), continue with the notion of Hamilton space H n (M, H(x, p)), where H : T ∗ M → R is a regular Hamiltonian, with the variational problem Z 1 dxi 1 I(c) = pi (t) − L(x(t), p(t)) dt, dt 2 0 from which the Hamilton–Jacobi equations are derived, Hamiltonian vector field of H n , nonlinear connection N , N ∗ −linear connection etc. The Legendre transformation L : Ln → H n transforms the fundamental geometrical object fields on Ln into the fundamental geometrical object fields on H n . The restriction of L to the Finsler spaces {F n } has as image a new class of spaces C n = (M, K(x, p)) called the Cartan spaces. They have the same beauty, symmetry and importance as the Finsler spaces. A pair GH n = (M, g ∗ (x, p)), where g ∗ is a metric tensor on T ∗ M is named a generalized Hamilton space. Remarking that R∗n = (M, g ∗ (x)), with g ∗ij (x) the contravariant Cartan space, we obtain a dual sequence of the sequence (I): (II)

{R∗n } ⊂ {C n } ⊂ {H n } ⊂ {GH n }.

The Hamiltonian geometry is geometrical study of the sequence II. The Lagrangian and Hamiltonian geometries are useful for applications in: Variational calculus, Mechanics, Physics, Biology etc. The part II of the book is devoted to the notions of Lagrange and Hamilton spaces of higher order. We study the geometrical theory of the total space of

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k−tangent bundle T k M , k ≥ 1, generalizing, step by step the theory from case k = 1. So, we introduce the Liouville vector fields, study the variational problem for a given integral of action on T k M , continue with the notions of k−semispray, nonlinear connection, the prolongation to T k M of the Riemannian structure defined on the base manifold M . The notion of N −metrical connections is pointed out, too. It follows the notion of Lagrange space of order k ≥ 1. It is defined as a pair L(k)n = (M, L(x, y (1) , y (2) , ..., y (k) )), where L is a Lagrangian depending on the 1 d k xi i (material point) x = (xi ) and on the accelerations y (k) = , k = 1, 2, ..., k; k! dt i = 1, 2, ..., n, n = dimM . Some examples prove the existence of the spaces L(k)n . The canonical nonlinear connection N and metrical N −linear connection are pointed out. The Riemannian almost contact model for L(k)n as well as the Generalized Lagrange space of order k end this theory. The methods used in the construction of the Lagrangian geometry of higherorder are the natural extensions of those used in the theory of Lagrangian geometries of order k = 1. Next chapter treats the geometry of Finsler spaces of order k ≥ 1, F (k)1 = (M, F (x, y (1) , ..., y (k) )), F : T k M → R being positive k−homogeneous on the fibres of T k M and fundamental tensor has a constant signature. Finally, one obtains the sequences (III)

{R(k)n } ⊂ {F (k)n } ⊂ {L(k)1 } ⊂ {GL(k)n }.

The Lagrangian geometry of order k ≥ 1 is the geometrical theory of the sequences of inclusions (III). The notion of Finsler spaces of order k, introduced by the present author, was investigated in the book The Geometry of Higher-Order Finsler Spaces, Hadronic k M of the theory of Press, 1998. Here it is a natural extension to the manifold T] Finsler spaces given in Section 3. It was developed by H. Shimada and S. Sabău [223]. The previous theory is continued with the geometry of k cotangent bundle T ∗k M , defined as the fibered product: T k−1 M XM T ∗ M , with the notion of Hamilton space of order k ≥ 1 and with the particular case of Cartan spaces of order k. An extension to L(k)n and H (k)n of the Legendre transformation is pointed out. The Hamilton–Jacobi equations of H (k)n , which are fundamental equations of these spaces, are presented, too.

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For the Hamilton spaces of order k, H (k)n = (M, H(x, y)(k) , ..., y (k−1) , p) a similar sequence of inclusions with (III) is introduced. These considerations allow to clearly define what means the Hamiltonian geometry of order k and what is the utility of this theory in applications. Part III of this book is devoted to applications in Analytical Mechanics. One studies the geometrical theory of scleronomic nonconservative classical mechanical systems ΣR = (M, T, F e), it is introduced and investigated the notion of Finslerian mechanical systems ΣF = (M, F, F e) and is defined the concept of Lagrangian mechanical system ΣL = (M, L, F e). In all these theories M is the configuration space, T is the kinetic energy of a Riemannian space Rn = (M, g), F (x, y) is the fundamental function of a Finsler space F n = (M, F (x, y)), L(x, y) is a regular Lagrangian and F e(x, y) are the external forces which depend on the material dxi i i points x ∈ M and their velocities y = x˙ = . dt In order to study these Mechanics we apply the methods from the Lagrangian geometries. The contents of these geometrical theory of mechanical systems ΣR , ΣF and ΣL is based on the geometrical theory of the velocity space T M . The base of these investigations are the Lagrange equations. They determine a canonical semispray, which is fundamental for all constructions. For every case of ΣR , ΣF and ΣL the law of conservation of energy is pointed out. We end with the corresponding almost Hermitian model on the velocity space T M . The dual theory via Legendre transformation, leads to the geometrical study of the Hamiltonian mechanical systems Σ∗R = (M, T ∗ , F e∗ ), Σ∗C = (M, K, F e∗ ) and Σ∗H = (M, H, F e∗ ) where T ∗ is energy, K(x, p) is the fundamental function of a given Cartan space and H(x, p) is a regular Hamiltonian on the cotangent bundle T ∗ M . The fundamental equations in these Hamiltonian mechanical systems are the Hamilton equations dxi ∂H dpi ∂H 1 1 =− ; = + Fi , H = H , dt ∂pi dt ∂xi 2 2 Fi (x, p) being the covariant components of external forces. The used methods are those given by the sequence of inclusions {R∗ } ⊂ {C n } ⊂ {H n } ⊂ {GH n }. More general, the notions of Lagrangian and Hamiltonian mechanical systems of order k ≥ 1 are introduced and studied. Therefore, to the sequences of inclusions III correspond the analytical mechanics of order k of the Riemannian, Finslerian, Lagrangian type. All these cases are the direct generalizations from case k = 1. This is the reason why we present here shortly the principal results. Especially,

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we paid attention to the following question: Considering a Riemannian (particularly Euclidian) mechanical system ΣR = 1 (M, T, F e), with T = gij (x)x˙ i x˙ j as energy, but with the external forces F e de2 i 2 i dx d x i pending on material point (x ) and on higher order accelerations , , .... 2 dt dt d k xi . What are the evolution equations of the system ΣR ? dtk Clearly, the classical Lagrange equations are not valid. In the last part of this book we present the solution of this problem. Finally, what is new in the present book? 1◦ A solution of the problem of geometrization of the classical nonconservative mechanical systems, whose external forces depend on velocities, based on the differential geometry of velocity space. 2◦ The introduction of the notion of Finslerian mechanical system. 3◦ The definition of Cartan mechanical system. 4◦ The study of theory of Lagrangian and Hamiltonian mechanical systems by means of the geometry of tangent and cotangent bundles. 5◦ The geometrization of the higher order Lagrangian and Hamiltonian mechanical systems. 6◦ The determination of the fundamental equations of the Riemannian mechanical systems whose external forces depend on the higher order accelerations.

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Part I Lagrange and Hamilton Spaces

1

3

The purpose of the part I is a short presentation of the geometrical theory of Lagrange space and of Hamilton spaces. These spaces are basic for applications to Mechanics, Physics etc. and have been introduced by the present author in 1980 and 1987, respectively. Therefore we present here the general framework of Lagrange and Hamilton geometries based on the books by R. Miron [161], [162], [163], R. Miron and M. Anastasiei [164], I. Bucătaru and R. Miron [49] and R. Miron, D. Hrimiuc, H. Shimada and S. Sabău [174]. Also, we use the papers R. Miron, M. Anastasiei and I. Bucătaru, The geometry of Lagrange spaces, Handbook of Finsler Geometry, P. L. Antonelli ed., Kluwer Academic; R. Miron, Compendium on the Geometry of Lagrange spaces, Handbook of Differential Geometry, vol. II, pp 438–512, Edited by F. J. E. Dillen and L. C. A. Verstraelen, 2006. The geometry of Lagrange space starts here with the study of geometrical theory of tangent bundle (T M, π, M ). It is continued with the notion of Lagrange space Ln = (M, L(x, y)) and with an important particular case the Finsler space F n = (M, F ). The Lagrangian geometry is the study of the sequence of inclusions (I) from the Preface. The geometry of Hamilton spaces follow the same pattern: the geometry of cotangent bundle (T ∗ M, π ∗ , M ), it continues with the notion of Hamilton space H n = (M, H(x, p)), with the concept of Cartan space C = (M, K(x, p)) and ends with the sequence of inclusions (II) from Introduction. The relation between the previous sequences are given by means of the Legendre transformation. In the following, we assume that all the geometrical object fields and mappings are C ∞ −differentiable and we express this by words “differentiable” or “smooth”.

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Chapter 1 The Geometry of tangent manifold The total space T M of tangent bundle (T M, π, M ) carries some natural object fields as Liouville vector field C, tangent structure J and vertical distribution V . An important object field is the semispray S defined as a vector field S on the manifold T M with the property J(S) = C. One can develop a consistent geometry of the pair (T M, S).

1.1 The manifold TM The differentiable structure on T M is induced by that of the base manifold M so that the natural projection π : T M → M is a differentiable submersion and the triple (T M, π, M ) is a differentiable vector bundle. Assuming that M is a real, n−dimensional differentiable manifold and (U, φ = (xi )) is a local chart at a point x ∈ M , then any curve σ : I → M , (Imσ ⊂ U ) that passes through x at t = 0 is analytically represented by xi = xi (t), t ∈ I, φ(x) = (xi (0)), (i, j, ... = 1, ..., n). The tangent vector [σ]x is determined by the real numbers xi = xi (0), y i =

dxi (0). dt

Then the pair (π −1 (U ), Φ), with Φ([σ]x ) = (xi , y i ) ∈ R2n , ∀[σ]x ∈ π −1 (U ) is a local chart on T M . It will be denoted by (π −1 (U ), ϕ = (xi , y i )). The set of these “induced” local charts determines a differentiable structure on T M such that π : T M → M is a differentiable manifold of dimension 2n and (T M, π, M ) is a differentiable vector bundle. A change on M , (U, φ = xi ) → (V, ψ = x˜i ) given by x˜i = x˜i (xj ), ofi coordinate ∂ x˜ with rank = n has the corresponding change of coordinates on T M : ∂xj 5

6

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

(π −1 (U ), Φ = (xi , y i )) → (π −1 (V ), Ψ = (˜ xi , y˜i )), (U ∩ V 6= ϕ), given by: i ∂ x˜ i i j = n, x ˜ = x ˜ (x ), rank ∂xj (1.1) i ∂ x ˜ y˜i = yj . j ∂x i 2 ∂ x˜ −1 The Jacobian of Ψ◦ Φ is det > 0. So the manifold T M is orientable. ∂xj The tangent space Tu T M at a point u ∈ TM to T M is a 2n−dimensional ∂ ∂ vector space, having the natural basis , at u. A change of coordinates ∂xi ∂y i (1.1) on T M implies the change of natural basis, at point u as follows: ∂ ∂ x˜j ∂ ∂ y˜j ∂ ∂xi = ∂xi ∂ x˜j + ∂xi ∂ y˜j , (1.2) j ∂ x ˜ ∂ ∂ = . ∂y i ∂xi ∂ y˜j ∂ ∂ i + Y (u) . Then a vector field ∂xi ∂y i X on T M is subsection X : T M → T T M of the projection π∗ : T T M → T M . This is π∗ (x, y, X, Y ) = (x, y). ∂ From the last formula (1.2) we can see that at point u ∈ T M span an ∂y i n−dimensional vector subspace V (u) of Tu T M . The mapping V : u ∈ T M → V (u) ⊂ Tu T[ M is an integrable distribution called the vertical distribution. Then V TM = V (u) is a subbundle of the tangent bundle (T T M, π∗ (u), T M ) to A vector Xu ∈ Tu T M is given by X = X i (u)

u∈T M

T M . As π : T M → M is a submersion it follows that π∗,u : Tu T M → Tπ(u) M is an epimorphism of linear spaces. The kernel of π∗,u is exactly the vertical subspaces V (u). We denote by X v (T M ) the set of all vertical vector field on T M . It is a real subalgebra of Lie algebra of vector fields on T M , X (T M ). Consider the cotangent space Tu∗ T M , u ∈ T M . It is the dual of space Tu T M and (dxi , dy i )u is the natural cobasis and with respect to (1.1) we have ∂ x˜i j ∂ y˜i j ∂ x˜i j i y = j dy + j dx . d˜ x = j dx , d˜ ∂x ∂x ∂x i

(1.3)

1.1. THE MANIFOLD TM

7

The almost tangent structure of tangent bundle is defined as J=

∂ ⊗ dxi i ∂y

(1.4)

By means of (1.2) and (1.3) we can prove that J is globally defined on T M and that we have ∂ ∂ ∂ = i, J = 0. (1.1.4’) J ∂xi ∂y ∂y i It follows that the following formulae hold: J 2 = J ◦ J = 0, KerJ = ImJ = V T M. The almost cotangent structure J ∗ is defined by J ∗ = dxi ⊗

∂ . ∂y i

Therefore, we obtain J ∗ (dxi ) = 0, J ∗ (dy i ) = dxi . The Liouville vector field on Tg M = T M \ {0} is defined by C = yi

∂ . ∂y i

(1.5)

It is globally defined on Tg M and C 6= 0. A smooth function f : T M → R is called r ∈ Z homogeneous with respect to the variables y i if, f (x, ay) = ar f (x, y), ∀a ∈ R+ . The Euler theorem holds: A function f ∈ F (T M ) differentiable on Tg M is r homogeneous with respect to y i if and only if ∂f (1.6) LC f = Cf = y i i = rf, ∂y LC being the Lie derivation with respect to C. A vector field X ∈ X (T M ) is r−homogeneous with respect to y i if LC X = (r − 1)X, where LC X = [C, X]. Finally an 1-form ω ∈ X ∗ (T M ) is r−homogeneous if LC ω = rω. Evidently, the notion of homogeneity can be extended to a tensor field T of type (r, s) on the manifold T M .

8

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

1.2 Semisprays on the manifold T M The notion of semispray on the total space T M of the tangent bundle is strongly related with the second order differential equations on the base manifold M : d 2 xi dx + 2Gi x, = 0. (1.1) 2 dt dt Writing the equation (1.1), on T M , in the equivalent form dxi dy i i i + 2G (x, y) = 0, y = dt2 dt

(1.2)

we remark that with respect to the changing of coordinates (1.1) on T M , the functions Gi (x, y) transform according to: yi j ∂e xi j ∂e f i 2G = j 2G − j y . ∂x ∂x

(1.3)

But (1.2) are the integral curve of the vector field: S = yi

∂ ∂ i − 2G (x, y) ∂xi ∂y i

(1.4)

By means of (1.3) one proves: S is a vector field globally defined on T M . It is called a semispray on T M and Gi are the coeﬀicients of S. S is homogeneous of degree 2 if and only if its coeﬀicients Gi are homogeneous functions of degree 2. If S is 2-homogeneous then we say that S is a spray. If the base manifold M is paracompact, then on T M there exist semisprays.

1.3 Nonlinear connections As we have seen in the first subsection of this chapter, the vertical distribution V is a regular, n-dimensional, integrable distribution on T M . Then it is naturally to look for a complementary distribution of V T M . It will be called a horizontal distribution. Such distribution is equivalent with a nonlinear connection. Consider the tangent bundle (T M, π, M ) of the base manifold M and the tangent bundle (T T M, π∗ , T M ) of the manifold T M . As we know the kernel of π∗ is the vertical subbundle (V T M, πV , T M ). Its fibres are the vertical spaces V (u), u ∈ TM.

1.3. NONLINEAR CONNECTIONS

9

For a vector field X ∈ X (T M ), in local natural basis, we can write: X = X i (x, y)

∂ ∂ i + Y (x, y) . ∂xi ∂y i

Shorter X = (xi , y i , X i , Y i ). The mapping π∗ : T T M → T M has the local form π∗ (x, y, X, Y ) = (x, X). The points of the submanifold V T M are of the form (x, y, 0, Y ). Let us consider the pull-back bundle π ∗ (T M ) = T M ×π T M = {(u, v) ∈ T M × T M |π(u) = π(v)}. The fibres of π ∗ (T M ), i.e., πu∗ (T M ) are isomorphic to Tπ(u) M . Then, we can define the following morphism π! : T T M −→ π ∗ (T M ) by π!(Xu ) = (u, π∗,u (Xu )). Therefore we have ker π! = ker π∗ = V T M. We can prove, without diﬀiculties that the following sequence of vector bundles over T M is exact: 0 −→ V T M −→T T M −→π ∗ (T M ) −→ 0 i

π!

(1.1)

Thus, we can give Definition 1.3.1. A nonlinear connection on the tangent manifold T M is a left splitting of the exact sequence (1.1). Consequently, a nonlinear connection on T M is a vector bundle morphism C : T T M → V T M , with the property that C ◦ i = 1V T M . The kernel of the morphism C is a vector subbundle of the tangent bundle (T T M, π∗ , T M ) denoted by (N T M, πN , T M ) and called the horizontal subbundle. Its fibres N (u) determine a regular n-dimensional distribution N : u ∈ T M → N (u) ⊂ Tu T M , complementary to the vertical distribution V : u ∈ T M → V (u) ⊂ Tu T M i.e. Tu T M = N (u) ⊕ V (u), ∀u ∈ T M. (1.2) Therefore, a nonlinear connection on T M induces the following Whitney sum: T T M = N T M ⊕ V T M. The reciprocal of this property is true, too.

(1.3.2’)

10

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

An adapted local basis to the direct decomposition (1.2) is of the form where

δ ∂ ∂ j = − N (x, y) i δxi ∂xi ∂y j

δ ∂ , δxi ∂y j

, u

(1.3)

δ and , (i = 1, ..., n) are vector fields that belong to N (u). δxi u They are n−linear independent vector fields and are independent from the ∂ vector fields (i = 1, ..., n) which belong to V (u). ∂y i u The functions Nij (x, y) are called the coeﬀicients of the nonlinear connection, denoted in the following by N . Remarking that π∗,u : Tu T M → Tπ(u) M is an epimorphism the restriction of π∗,u to N (u) is an isomorphism from N (u) so Tπ(u) M . So we can take the inverse map lh,u the horizontal lift determined by the nonlinear connection N . δ Consequently, the vector fields can be given in the form δxi u ∂ δ = lh,u . δxi u ∂xi π(u) With respect to a change of local coordinates on the base manifold M we have ∂ ∂e xj θ = . ∂xi ∂xi ∂e xj δ are changed, with respect to (1.1), in the form Consequently δxi u ∂e xj δ δ = . (1.4) δxi ∂xi δe xj It follows, from (1.3), that the coeﬀicients Nji (x, y) of the nonlinear connection N , with respect to a change of local coordinates on the manifold T M , (1.1) are transformed by the rule: xk ∂e yj ∂e xj k j ∂e e N = Nk i + i . (1.5) ∂xk i ∂x ∂x The reciprocal property is true, too. We can prove without diﬀiculties that there exists a nonlinear connection on T M if M is a paracompact manifold. The validity of this sentence is assured by the following theorem:

1.3. NONLINEAR CONNECTIONS

11

Theorem 1.3.1. If S is a semispray with the coeﬀicients Gi (x, y) then the system of functions ∂Gi i Nj (x, y) = (1.6) ∂y j is the system of coeﬀicients of a nonlinear connection N . Indeed, the formula (1.3) and

∂e xj ∂ ∂ = give the rule of transformation ∂y i ∂xi ∂e yj

(1.5) for Nji from (1.6).

The adapted dual basis {dxi , δy i } of the basis

δ ∂ , δxi ∂y i

has the 1-forms δy i

as follows: δy i = dy i + Nji dxj .

(1.7)

With respect to a change of coordinates, (1.1), we have ∂e xi j de x = j dx , ∂x i

∂e xi j δe y = j δy . ∂x i

(1.3.7’)

Now, we can consider the horizontal and vertical projectors h and v with respect to direct decomposition (1.2): h=

δ ∂ i ⊗ dx , v = ⊗ δy i . i i δx ∂y

(1.8)

Some remarkable geometric structures, as the almost product P and almost complex structure F, are determined by the nonlinear connection N : P=

δ ∂ i ⊗ dx − ⊗ δy i = h − v. i i δx ∂y F=

δ ∂ i ⊗ δy − ⊗ dxi . i i δx ∂y

(1.9) (1.3.9’)

It is not diﬀicult to see that P and F are globally defined on Tg M and P ◦ P = Id, F ◦ F = −Id.

(1.10)

With respect to (1.2) a vector field X ∈ X (T M ) can be uniquely written in the form X = hX + vX = X H + X V (1.11) with X H = hX and X V = vX.

12

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

An 1-form ω ∈ X ∗ (T M ) has the similar form ω = hω + vω, where hω(X) = ω(X H ), vω(X) = ω(X V ). A d−tensor field T on T M of type (r, s) is called a distinguished tensor field (shortly a d−tensor) if T (ω1 , ..., ωr , X1 , ..., Xs ) = T (ε1 ω1 , ..., εr ωr , ε1 X1 , ..., εs Xs ) where ε1 , ..., εr , ... are h or v. V H V Therefore hX = X H , vX = X , hω = ω , vω = ω are d−vectors or δ ∂ d−covectors. In the adapted basis , we have δxi ∂y i δ , δxi

∂ X V = X˙ i i ∂y

ω H = ωj (x, y)dxj ,

ω V = ω˙ j δy j .

X H = X i (x, y) and

A change of local coordinates on T M : (x, y) → (e x, ye) leads to the change of H V H V coordinates of X , X , ω , ω by the classical rules of transformation: ∂e xi j i e X = jX , ∂x

∂e xi ωj = j ω ei etc. ∂x

So, a d−tensor T of type (r, s) can be written as ...ir T = Tji11...j (x, y) s

∂ δ ⊗ ... ⊗ ⊗ dxj1 ⊗ ... ⊗ δy js . i i 1 r δx ∂y

(1.12)

A change of coordinates (1.1) implies the classical rule: xir ∂xk1 ∂xks ∂e xi1 ∂e ...h2 ...ir ... ... j = Tkh11...k . Teji11...j (e x , y e ) = s h h j s 1 r 1 s ∂x ∂x ∂e x ∂e y

(1.3.12’)

Next, we shall study the integrability of the nonlinear connection N and of the structuresP and F. δ i = 1, ..., n is an adapted basis to N , according to the Frobenius Since δxi δ δ theorem it follows that N is integrable if and only if the Lie brackets , , δxi δxj i, j = 1, ..., n are vector fields in the distribution N .

1.3. NONLINEAR CONNECTIONS

But we have

13

δ ∂ δ h , = R ij δxi δxj ∂y h

(1.13)

where

δNih δNjh = − . (1.3.13’) δxj δxi h It is not diﬀicult to prove that Rij are the components of a d−tensor field of type (1,2). It is called the curvature tensor of the nonlinear connection N . We deduce: Rhij

Theorem 1.3.2. The nonlinear connection N is integrable if and only if its curh vature tensor Rij vanishes. The weak torsion thij of N is defined by thij

∂N hi ∂N hj − . = ∂y i ∂y i

(1.14)

It is a d-tensor field of type (1, 2), too. We say that N is a symmetric nonlinear connection if its weak torsion thij vanishes. Now is not diﬀicult to prove: Theorem 1.3.3. 1◦ The almost product structure P is integrable if and only if the nonlinear connection N is integrable; 2◦ The almost complex structure F is integrable if and only if the nonlinear connection N is symmetric and integrable. The proof is simple using the Nijenhuis tensors NP and NF . For NP we have the expression NP (X, Y ) = P2 [X, Y ] + [PX, PY ] − P[PX, Y ] − P[X, PY ], ∀X, Y ∈ χ(T M ). Also we can see that each structure P or F characterizes the nonlinear connection N . Autoparallel curves of a nonlinear connection can be obtained considering the horizontal curves as follows. A curve c : t ∈ I ⊂ R → (xi (t), y i (t)) ∈ T M has the tangent vector c˙ given by δy i ∂ dxi δ + c˙ = c˙ + c˙ = dt δxi dt ∂y i H

V

(1.15)

14

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

where

δy i dy i dxj i = + Nj (x, y) . (1.3.15’) dt dt dt δy i The curve c is a horizontal curve if c˙V = 0 or = 0. dt Evidently, if the functions xi (t), t ∈ I are given and y i (t) are the solutions of this system of differential equations, then we have an horizontal curve xi = xi (t), y i = y i (t) in T M with respect to N . dxi In the case y i = , the horizontal curves are called the autoparallel curves of dt the nonlinear connection N . They are characterized by the system of differential equations dxj dxi dy i + Nji (x, y) = 0, y i = . (1.16) dt dt dt A theorem of existence and uniqueness can be easy formulated for the autoparallel curves of a nonlinear connection N given by its coeﬀicients Nji (x, y).

1.4 N -linear connections An N -linear connection on the manifold T M is a special linear connection D on T M that preserves by parallelism the horizontal distribution N and the vertical distribution V . We study such linear connections determining the curvature, torsion and structure equations. Throughout this subsection N is an a priori given nonlinear connection with the coeﬀicients Nji . Definition 1.4.1. A linear connection D on the manifold T M is called a distinguished connection (a d-connection for short) if it preserves by parallelism the horizontal distribution N . Thus, we have Dh = 0. It follows that we have also: Dv = 0 and DP = 0. If Y = Y H + Y V we get DX Y = (DX Y H )H + (DX Y V )V ,

∀X, Y ∈ χ(T M ).

We can easily prove: Proposition 1.4.1. For a d-connection the following conditions are equivalent: 1o DJ = 0; 2o DF = 0.

1.4. N -LINEAR CONNECTIONS

15

Definition 1.4.2. A d-connection D is called an N -linear connection if the structure J (or F) is absolute parallel with respect to D, i.e. DJ = 0. In an adapted basis an N -linear connection has the form: δ ∂ h δ h ∂ δ δ D = L ; D = L , ij ij δxj δxi δxj ∂y i δxh ∂y h δ ∂ h δ h ∂ D∂ ∂ = C ; D = C . ij ij ∂y j δxi ∂y j ∂y i δxh ∂y h

(1.1)

The set of functions DΓ = (Nji (x, y), Lhij (x, y), Cijh (x, y)) are called the local coeﬀicients of the N −linear connection D. Since N is fixed we denote sometimes by DΓ(N ) = (Lhij (x, y), Cijh (x, y)) the coeﬀicients of D. ∂Nih , 0 are the coeﬀicient of a special N −linear For instance the BΓ(N ) = ∂y j connection, derived only from the nonlinear connection N . It is called the Berwald connection of the nonlinear connection N . We can prove this showing that the h ∂Ni has the same rule of transformation, with respect to system of functions ∂y j (1.1), as the coeﬀicients Lhij . Indeed, under a change of coordinates (1.1) on T M the coeﬀicients (Lhij , Cijh ) are transformed by the rules: ∂e xh s ∂xp ∂xq ∂ 2x eh ∂xp ∂xq h e L − , Lij = ∂xs pq ∂e xi ∂e xj ∂xp ∂xq ∂e xi ∂e xj xh s ∂xp ∂xq C eh = ∂e C . ij ∂xs pq ∂e xi ∂e xj

(1.2)

So, the horizontal coeﬀicients Lhij of D have the same rule of transformation of the local coeﬀicients of a linear connection on the base manifold M . The vertical coeﬀicients Cijh are the components of a (1,2)-type d-tensor field. i But, conversely, if the set of functions (Lijk (x, y), Cjk (x, y)) are given, having the property (1.2), then the equalities (1.1) determine an N −linear connection D on T M . For an N −linear connection D on T M we shall associate two operators of h− and v− covariant derivation on the algebra of d−tensor fields. For each X ∈ χ(T M ) we set: H DX Y = DX H Y,

H DX f = X H f,

∀Y ∈ χ(T M ), ∀f ∈ F (T M ).

(1.3)

16

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

If ω ∈ χ∗ (T M ), we obtain H H (DX ω)(Y ) = X H (ω(Y )) − ω(DX Y ).

(1.4.3’)

H to any d-tensor field by asking So we may extend the action of the operator DX H preserves the type of d-tensor fields, is R-linear, satisfies the Leibniz rule that DX with respect to tensor product and commutes with the operation of contraction. H DX will be called the h-covariant derivation operator. In a similar way, we set: V V DX Y = DX V Y, DX f = X V (f ), ∀Y ∈ χ(T M ), ∀f ∈ F (T M ).

(1.4)

and V V DX ω = X V (ω(Y )) − ω(DX Y ), ∀ω ∈ χ(T M ). V Also, we extend the action of the operator DX to any d-tensor field in a similar H V way as we did for DX . DX is called the v-covariant of derivation operator. Consider now a d-tensor T given by (1.12). According to (1.1) its h−covariant derivation is given by ···ir H DX T = X k Tji11···j s |k

∂ δ ⊗ · · · ⊗ ⊗ dxj1 ⊗ · · · ⊗ dxjs i i 1 r δx ∂y

(1.5)

where ···ir Tji11···j s |k

···ir δTji11···j i ···i p s 2 ···ir = + Lipk1 Tjpi1 ···j + · · · + Lipkr Tj11···jsr−1 − s k δx p i1 ···ir r −Lpj1 k Tpji12···i ···js − · · · − Ljs k Tj1 ···js−1 p .

(1.4.5’)

V The v-covariant derivative DX T is as follows: ···ir V DX T = X k Tji11···j | s k

∂ δ ⊗ · · · ⊗ ⊗ dxj1 ⊗ · · · ⊗ dxjs , i i 1 r δx ∂y

(1.6)

where ···ir Tji11···j | s k

···ir ∂Tji11···j i1 pi2 ···ir ir i1 ···ir−1 p s + C T + · · · + C Tj1 ···js − = j ···j pk pk 1 s ∂y k p i1 ···ir r −Cjp1 k Tpji12···i ···js − · · · − Cjr k Tj1 ···js−1 p .

(1.4.6’)

1.4. N -LINEAR CONNECTIONS

17

For instance, if g is a d-tensor of type (0,2) having the components gij (x, y), we have δgij gij|k = k − Lpik gpj − Lpjk gip , δx (1.7) ∂gij p p − Cik gpj − Cjk gip . gij |k = ∂y k For the operators “| ” and “|” we preserve the same denomination of h− and v−covariant derivations. The torsion T of an N -linear is given by: T (X, Y ) = DX Y − DY X − [X, Y ], ∀X, Y ∈ χ(T M ).

(1.8)

The horizontal part hT (X, Y ) and the vertical one vT (X, Y ), for X ∈ {X H , X V } and Y ∈ {Y H , Y V } determine five d-tensor fields of torsion T : H H hT (X H , Y H ) = DX Y − DYH X H − [X H , Y H ]H , H H H H V vT (X , Y ) = −v[X , Y ] , hT (X H , Y V ) = −DYV X H − [X H , Y V ]V ,

H V vT (X H , Y V ) = DX Y − [X H , Y V ]V , V V vT (X V , Y V ) = DX Y − DYV X V − [X V , Y V ]V .

(1.9)

With respect to the adapted basis, the components of torsion are as follows: δ δ δ δ k δ k ∂ , ; vT , ; hT = T = R ji ji δxi δxj δxk δxi δxj ∂y k ∂ δ k δ hT , ; = C ji ∂y i δxj δxk (1.4.9’) ∂ ∂ δ , j = Pjik k ; vT i ∂y δx ∂y ∂ ∂ k ∂ vT , j = Sji , i ∂y ∂y ∂y k i i where Cjk are the v−coeﬀicients of D, Rjk is the curvature tensor of the nonlinear connection N and

T ijk

=

Lijk

−

Likj , S ijk

=

i Cjk

−

i Ckj , P ijk

∂Nji − Likj . = k ∂y

(1.10)

18

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

The N −linear connection D is said to be symmetric if T ijk = S ijk = 0. Next, we study the curvature of an N -linear connection D: R(X, Y )Z = DX DY Z − DY DX Z − D[X,Y ] Z, ∀X, Y, Z ∈ χ(T M ).

(1.11)

As D preserves by parallelism the distribution N and V it follows that the operator R(X, Y ) = DX DY − DY DX − D[X,Y ] carries the horizontal vector fields Z H into horizontal vector fields and vertical vector fields into verticals. Consequently we have the formula: R(X, Y )Z = hR(X, Y )Z H + vR(X, Y )Z V , ∀X, Y, Z ∈ χ(T M ). Since R(X, Y ) = −R(Y, X), we obtain: Theorem 1.4.1. The curvature R of a N -linear connection D on the tangent manifold T M is completely determined by the following six d-tensor fields: H H H H H R(X H , Y H )Z H = DX DY Z − DYH DX Z − D[X H ,Y H ] Z H , H H V H V R(X H , Y H )Z V = DX DY Z − DYH DX Z − D[X H ,Y H ] Z V , R(X V , Y H )Z H = DV DH Z H − DH DV Z H − D[X V ,Y H ] Z H , X Y Y X (1.12) V H V V H V H V V V R(X , Y )Z = DX DY Z − DY DX Z − D[X V ,Y H ] Z , V V H R(X V , Y V )Z H = DX DYV Z H − DYV DX Z − D[X V ,Y V ] Z H , V V V R(X V , Y V )Z V = DX DYV Z V − DYV DX Z − D[X V ,Y V ] Z V . As the tangent structure J is absolute parallel with respect to D we have that JR(X, Y )Z = R(X, Y )JZ. Then R(X, Y )Z has only three components R(X H , Y H )Z H , R(X V , Y H )Z H , R(X V , Y V )Z H . In the adapted basis these are: δ δ δ δ i R , = R ; h kj δxk δxj δxh δxi δ ∂ δ δ i R , = P ; h kj ∂y k δxj δxh δxi ∂ ∂ δ δ i R , = S . h kj ∂y k ∂y j δxh δxi

(1.13)

1.4. N -LINEAR CONNECTIONS

19

The other three componentsare obtained by applying the operator J to the ∂ δ δ ∂ i previous ones. So, we have R , = R etc. Therefore, the h jk δxh δxj ∂y h ∂y h i curvature of an N −linear connection DΓ = (Nji , Lijk , Cjk ) has only three local i i i components Rh jk , Ph jk and Sh jk . They are given by Rhi jk Ph ijk Shi jk

δLihj δLihk i s ; = − + Lshj Lisk − Lshk Lisj + Chs Rjk δxk δxj ∂Lihj i i s = − Chk|j + Chs Pjk ; k ∂y

(1.14)

i i ∂Chj ∂Chk s i s i − + Chj Csk − Chk Csj . = k j ∂y ∂y

If X i (x, y) are the components of a d-vector field on T M then from (1.12) we may derive the Ricci identities for X i with respect to an N -linear connection D. They are: s i i − X i |s Rsjk , = X s Rsi jk − X|si Tjk − X|k|j X|j|k s X|ji |k − X i |k|j = X s Ps ijk − X|si Cjk − X i |s P sjk ,

(1.15)

X i |j |k − X i |k |j = X s Ss ijk − X i |s S sjk . We deduce some fundamental identities for the N −linear connection D, apply∂ ing the Ricci identities to the Liouville vector field C = y i i . Considering the ∂y d-tensors Dji = y i |j , dij = y i |j (1.16) called h− and v−deflection tensors of D we obtain: Theorem 1.4.2. For any N -linear connection D the following identities hold: Di k|h − Di h|k = y s Rs i kh − Di s T s kh − di s Rs kh , Di k |h − di h|k = y s Ps ikh − Di s C s kh − di s P s kh ,

(1.17)

di k |h − di h |k = y s Ss ikh − di s S s kh . Others fundamental identities are the Bianchi identities, which are obtained by writing in the adapted basis the following Bianchi identities: Σ[DX T (Y, Z) − R(X, Y )Z + T (T (X, Y ), Z)] = 0, Σ[(DX R)(U, Y, Z) + R(T (X, Y ), Z)U ] = 0,

(1.18)

20

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

where Σ means cyclic summation over X, Y, Z.

1.5 Parallelism. Structure equations Let DΓ(N ) =

i ) (Lijk , Cjk

be an N -linear connection and the adapted basis

δ ∂ , , δxi ∂y i

i = 1, n. As we know, a curve c : t ∈ I → (xi (t), y i (t)) ∈ T M, has the tangent vector dxi δ δy i ∂ c˙ = c˙H + c˙V given by (3.15), i.e. c˙ = + . The curve c is horizontal if dt δxi dt ∂y i δy i = 0, and it is an autoparallel curve with respect to the nonlinear connection dt δy i dxi i N if = 0, y = . dt dt We set DX DX = Dc˙ X, DX = dt, ∀X ∈ χ(T M ). (1.1) dt dt DX Here is the covariant differential along the curve c with respect to the the dt N -linear connection D. δ ∂ Setting X = X H + X V , X H = X i i , X V = X˙ i i we have δx ∂y DX DX H DX V = + = dt dt dt (1.2) k k k k dx δy δ i i dx i δy i = X |k + X |k + X˙ |k + X˙ |k . dt dt δxi dt dt Consider the connection 1-forms of D: i ω i j = Lijk dxk + Cjk δy k .

Then

DX takes the form dt ( ) i i i i ˙ DX dX ∂ dX ωs δ ˙ sω s = + Xs + + X . dt dt dt δxi dt dt ∂y i

(1.3)

(1.4)

The vector field X on T M is said to be parallel along the curve c, with respect DX DX = 0. Using (1.2), the equation =0 to the N -linear connection D(N ) if dt dt DX H DX V is equivalent to = 0, = 0. According to (1.4) we obtain: dt dt

1.5. PARALLELISM. STRUCTURE EQUATIONS

21

δ ˙ i ∂ from χ(T M ) is parallel + X δxi ∂y i along the parametrized curve c in T M , with respect to the N -linear connection i ) if and only if its coeﬀicients X i (x(t), y(t)) and X˙ i (x(t), y(t)) DΓ(N ) = (Lijk , Cjk are solutions of the linear system of differential equations

Proposition 1.5.1. The vector field X = X i

dZ i ω i s (x(t), y(t)) s + Z (x(t), y(t)) = 0. dt dt A theorem of existence and uniqueness for the parallel vector fields along a curve c on the manifold T M can be formulated on the classical way. The horizontal geodesic of D are the horizontal curves c : I → T M with the dxi ˙ i δy i i property Dc˙ c˙ = 0. Taking X = ,X = = 0 we get: dt dt Theorem 1.5.1. The horizontal geodesics of an N -linear connection are characterized by the system of differential equations: d 2 xi dxj dxk dy i dxj i i + Ljk (x, y) = 0, + N j (x, y) = 0. dt2 dt dt dt dt by

(1.5)

Now we can consider a curve cVx0 on the fibre Tx0 M = π −1 (x0 ). It is represented xi = xi0 , y i = y i (t), t ∈ I,

cvx0 is called a vertical curve of T M at the point x0 ∈ M. The vertical geodesic of D are the vertical curves cvx0 with the property Dc˙vx0 c˙vx0 = 0. Theorem 1.5.2. The vertical geodesics at the point x0 ∈ M, of the N -linear i connection DΓ(N ) = (Lijk , Cjk ) are characterized by the following system of differential equations i

x =

xi0 ,

dy j dy k d2 y i i + Cjk (x0 , y) = 0. dt2 dt dt

(1.6)

Obviously, the local existence and uniqueness of horizontal or vertical geodesics are assured if initial conditions are given. Now, we determine the structure equations of an N -linear connection D, considering the connection 1-forms ωji , (1.3). First of all, we have:

22

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

Lemma 1.5.1. The exterior differential of 1-forms δy i = dy i + N i j dxj are given by: 1 d(δy i ) = Ri js dxs ∧ dxj + B i js δy s ∧ dxj (1.7) 2 where ∂N i j i . (1.5.7’) B jk = ∂y k Remark 1.5.2.1. B i jk are the coeﬀicients of the Berwald connection. Lemma 1.5.2. With respect to a change of local coordinate on the manifold T M , the following 2-forms d(dxi ) − dxs ∧ ω i s ; d(δy i ) − δy s ∧ ω i s transform as the components of a d-vector field. The 2-forms dω i j − ω s j ∧ ω i s transform as the components of a d-tensor field of type (1, 1). Theorem 1.5.3. The structure equations of an N -linear connection DΓ(N ) = i (Lijk , Cjk ) on the manifold T M are given by (0)

d(dxi ) − dxs ∧ ω i s = − Ω i (1)

d(δy i ) − δy s ∧ ω i s = − Ω i

(1.8)

dω i j − ω s j ∧ ω i s = −Ωi j (0)

i

(1)

where Ω and Ω i are the 2-forms of torsion: 1 i dxj ∧ δy k Ω i = T ijk dxj ∧ dxk + Cjk 2 (1) 1 1 Ω i = Ri jk dxj ∧ dxk + P ijk dxj ∧ δy k + S i jk δy j ∧ δy k 2 2

(0)

(1.9)

and the 2-forms of curvature Ωi j are given by 1 1 Ωi j = Rj i kh dxk ∧ dxh + Pj ikh dxk ∧ δy h + Sj ikh δy j ∧ δy h . 2 2

(1.10)

1.5. PARALLELISM. STRUCTURE EQUATIONS

23

Proof. By means of Lemma (1.5.2), the general structure equations of a linear connection on T M are particularized for an N -linear connection D in the form (1.8). Using the connection 1-forms ω i j (1.3) and the formula (1.7), we can calculate the (0)

(1)

forms Ω i , Ω i and Ωi j . Then it is very easy to determine the structure equations (1.9). Remark 1.5.3.1. The Bianchi identities of an N -linear connection D can be obtained from (1.8) by calculating the exterior differential of (1.8), modulo the (0) (1)

same system (1.8) and using the exterior differential of Ω i Ω i and Ωij .

24

CHAPTER 1. THE GEOMETRY OF TANGENT MANIFOLD

Chapter 2 Lagrange spaces The notion of Lagrange spaces was introduced and studied by the present author. The term “Lagrange geometry” is due to J. Kern, [121]. We study the geometry of Lagrange spaces as a subgeometry of the geometry of tangent bundle (T M, π, M ) of a manifold M , using the principles of Analytical Mechanics given by variational problem on the integral of action of a regular Lagrangian, the law of conservation, Nöther theorem etc. Remarking that the Euler - Lagrange equations determine a canonical semispray S on the manifold T M we study the geometry of a Lagrange space using this canonical semi-spray S and following the methods given in the first chapter. Beginning with the year 1987 there were published by author, alone or in collaborations, some books on the Lagrange spaces and the Hamilton spaces [174], and on the higher-order Lagrange and Hamilton spaces [161], as well.

2.1 The notion of Lagrange space First we shall define the notion of differentiable Lagrangian over the tangent manifold T M and Tg M = T M \ {0}, M being a real n−dimensional manifold. Definition 2.1.1. A differentiable Lagrangian is a mapping L : (x, y) ∈ T M → L(x, y) ∈ IR, of class C ∞ on Tg M and continuous on the null section 0 : M → T M of the projection π : T M → M . The Hessian of a differentiable Lagrangian L, with respect to y i , has the elements: 1 ∂ 2L . (2.1) gij = 2 ∂y i ∂y j 25

26

CHAPTER 2. LAGRANGE SPACES

Evidently, the set of functions gij (x, y) are the components of a d-tensor field, symmetric and covariant of order 2. Definition 2.1.2. A differentiable Lagrangian L is called regular if: rank(gij (x, y)) = n, onTg M.

(2.2)

Now we can give the definition of a Lagrange space: Definition 2.1.3. A Lagrange space is a pair Ln = (M, L(x, y)) formed by a smooth, real n-dimensional manifold M and a regular Lagrangian L(x, y) for which the d-tensor gij has a constant signature over the manifold Tg M. n For the Lagrange space L = (M, L(x, y)) we say that L(x, y) is the fundamental function and gij (x, y) is the fundamental (or metric) tensor. Examples. 1◦ Every Riemannian manifold (M, gij (x)) determines a Lagrange space Ln = (M, L(x, y)), where L(x, y) = gij (x)y i y j .

(2.3)

This example allows to say: If the manifold M is paracompact, then there exist Lagrangians L(x, y) such that Ln = (M, L(x, y)) is a Lagrange space. 2◦ The following Lagrangian from electrodynamics 2e Ai (x)y i + U (x), (2.4) m where γij (x) is a pseudo-Riemannian metric, Ai (x) a covector field and U (x) a smooth function, m, c, e are physical constants, is a Lagrange space Ln . This is called the Lagrange space of electrodynamics. We already have seen that gij (x, y) from (2.1.1) is a d-tensor field, i.e. with respect to (1.1.1), we have L(x, y) = mcγij (x)y i y j +

∂xh ∂xk geij (e x, ye) = ghk (x, y). ∂e xi ∂e xj Now we can prove without diﬀiculties:

2.2. VARIATIONAL PROBLEM. EULER-LAGRANGE EQUATIONS

27

Theorem 2.1.1. For a Lagrange space Ln the following properties hold: 1◦ The system of functions pi =

1 ∂L 2 ∂y i

determines a d-covector field. 2◦ The functions 1 1 ∂gij ∂ 3L Cijk = = 4 ∂y i ∂y j ∂y k 2 ∂y k are the components of a symmetric d−tensor field of type (0, 3). 3◦ The 1-form 1 ∂L i dx (2.5) 2 ∂y i depend on the Lagrangian L only and are globally defined on the manifold Tg M ω = pi dxi =

4◦ The 2-form θ = dω = dpi ∧ dxi

(2.6)

is globally defined on Tg M and defines a symplectic structure on Tg M.

2.2 Variational problem. Euler-Lagrange equations The variational problem can be formulated for differentiable Lagrangians and can be solved in the case when the integral of action is defined on the parametrized curves. Let L : T M → R be a differentiable Lagrangian and c : t ∈ [0, 1] → (xi (t)) ∈ U ⊂ M be a smooth curve, with a fixed parametrization, having Imc ⊂ U , where U is a domain of a local chart on the manifold M . The curve c can be extended to π −1 (U ) ⊂ Tg M by e c : t ∈ [0, 1] → (xi (t),

dxi (t)) ∈ π −1 (U ). dt

So, Ime c ⊂ π −1 (U ). The integral of action of the Lagrangian L on the curve c is given by the functional: Z 1 dx I(c) = L x, dt. (2.1) dt 0

28

CHAPTER 2. LAGRANGE SPACES

Consider the curves cε : t ∈ [0, 1] → (xi (t) + εV i (t)) ∈ M

(2.2)

which have the same end points xi (0) and xi (1) as the curve c, V i (t) = V i (xi (t)) being a regular vector field on the curve c, with the property V i (0) = V i (1) = 0 and ε is a real number, suﬀiciently small in absolute value, so that Imcε ⊂ U . The extension of a curve cε to Tg M is given by dxi dV i i i e cε : t ∈ [0, 1] 7→ x (t) + εV (t), +ε ∈ π −1 (U ). dt dt The integral of action of the Lagrangian L on the curve cε is Z 1 dx dV +ε I(cε ) = L x + εV, dt. dt dt 0

(2.2.1’)

A necessary condition for I(c) to be an extremal value of I(cε ) is dI(cε ) |ε=0 = 0. dε

(2.3)

d Under our condition of differentiability, the operator is permuting with the dε operator of integration. From (2.2.1’) we obtain Z 1 d dx dV dI(cε ) = L(x + εV, + ε )dt. (2.4) dε dε dt dt 0 But we have

d dx dV ∂L ∂L dV i |ε=0 = i V i + i L x + εV, +ε = dε dt dt ∂x ∂y dt dxi d ∂L d ∂L ∂L i i i − V + V , y = . = ∂xi dt ∂y i dt ∂y i dt

Substituting in (2.4) and taking into account the fact that V i (x(t)) is arbitrary, we obtain the following theorem. Theorem 2.2.1. In order for the functional I(c) to be an extremal value of I(cε ) it is necessary for the curve c(t) = (xi (t)) to satisfy the Euler-Lagrange equations: d ∂L dxi ∂L i = 0, y = . Ei (L) := i − ∂x dt ∂y i dt

(2.5)

2.2. VARIATIONAL PROBLEM. EULER-LAGRANGE EQUATIONS

For the Euler-Lagrange operator Ei =

29

∂ d ∂ − we have: ∂xi dt ∂y i

Theorem 2.2.2. The following properties hold true: 1◦ Ei (L) is a d-covector field. 2◦ Ei (L + L0 ) = Ei (L) + Ei (L0 ). 3◦ Ei (aL) aEi (L), a ∈ R. = dF ∂F = 0, ∀F ∈ F (T M ) with = 0. 4◦ Ei dt ∂y i The notion of energy of a Lagrangian L can be introduced as in the Theoretical Mechanics [17], [164], by ∂L EL = y i i − L. (2.6) ∂y We obtain, without diﬀiculties: Theorem 2.2.3. For every smooth curve c on the base manifold M the following formula holds: dxi dxi dEL i =− Ei (L), y = . (2.7) dt dt dt Consequently: Theorem 2.2.4. For any differentiable Lagrangian L(x, y) the energy EL is conserved along every solution curve c of the Euler-Lagrange equations Ei (L) = 0,

dxi = yi. dt

A Noether theorem can be proved: Theorem 2.2.5. For any infinitesimal symmetry on M × R of the Lagrangian L(x, y) and for any smooth function ϕ(x) the following function: F(L, ϕ) = V i

∂L − τ EL − ϕ(x) ∂y i

is conserved on every solution curve c of the Euler-Lagrange equations Ei (L) = 0, dxi i y = . dt Remark. An infinitesimal symmetry on M × R is given by x0i = xi + εV i (x, t), t0 = t + ετ (x, t).

30

CHAPTER 2. LAGRANGE SPACES

2.3 Canonical semispray. Nonlinear connection Now we can apply the previous theory in order to study the Lagrange space Ln = (M, L(x, y)). As we shall see that Ln determines a canonical semispray S and S gives a canonical nonlinear connection on the manifold Tg M. As we know, the fundamental tensor gij of the space Ln is nondegenerate, and Ei (L) is a d-covector field, so the equations g ij Ej (L) = 0 have a geometrical meaning. Theorem 2.3.1. If Ln = (M, L) is a Lagrange space, then the system of differential equations dxi g ij Ej (L) = 0, y i = (2.1) dt can be written in the form: d 2 xi dx dxi i i x, = 0, y = (2.3.1’) + 2G dt2 dt dt where 2 1 ij ∂ L k ∂L i 2G (x, y) = g y − j . (2.2) 2 ∂y j ∂xk ∂x Proof. We have 2 dy j ∂L ∂ L dxi i Ei (L) = i − + 2gij , y = . ∂x ∂y i ∂xk dt dt So, (2.1) implies (2.3.1’), (2.2). The previous theorem tells us that the Euler Lagrange equations for a Lagrange space are given by a system of n second order ordinary differential equations. According with theory from Section 1.2, Ch. 1, it follows that the equations (2.1) determine a semispray with the coeﬀicients Gi (x, y): ∂ ∂ S = y i i − 2Gi (x, y) i . (2.3) ∂x ∂y S is called the canonical semispray of the Lagrange space Ln . By means of Theorem 1.3.1, it follows: Theorem 2.3.2. Every Lagrange space Ln = (M, L) has a canonical nonlinear connection N which depends only on the fundamental function L. The local coefficients of N are given by 2 ∂Gi 1 ∂ ∂ L h ∂L i ik Nj= = g y − k . (2.4) ∂y j 4 ∂y j ∂y k ∂xh ∂x

2.3. CANONICAL SEMISPRAY. NONLINEAR CONNECTION

31

Evidently: Proposition 2.3.1. The canonical nonlinear connection N is symmetric, i.e. ∂Nji ∂N ik i tjk = − = 0. ∂y k ∂y j Proposition 2.3.2. The canonical nonlinear connection N is invariant with respect to the Carathéodory transformation L0 (x, y) = L(x, y) + Indeed, we have 0

Ei (L ) = Ei

∂φ(x) i y. ∂xi

dφ L(x, y) + dt

(2.5)

= Ei (L).

So, Ei (L0 (x, y)) = 0 determines the same canonical semispray as the one determined by Ei (L(x, y)) = 0. Thus, the Carathéodory transformation (2.5) preserves the nonlinear connection N . Example. The Lagrange space of electrodynamics, Ln = (M, L(x, y)), where L(x, y) is given by (2.4) with U (x) = 0 has the canonical semispray with the coeﬀicients: 1 Gi (x, y) = γ i jk (x)y j y k − g ij (x)Fjk (x)y k , (2.6) 2 where γ i jk (x) are the Christoffel symbols of the metric tensor gij (x) = mcγij (x) of the space Ln and Fjk is the electromagnetic tensor e ∂Ak ∂Aj Fjk (x, y) = − k . (2.7) 2m ∂xj ∂x Therefore, the integral curves of the Euler-Lagrange equation are given by the solution curves of the Lorentz equations: dxj dxk dxk d 2 xi i ij + γ jk (x) = g (x)Fjk (x) . (2.8) dt2 dt dt dt According to (2.4), the canonical nonlinear connection of the Lagrange space of electrodynamics Ln has the local coeﬀicients given by i N i j (x, y) = γjk (x)y k − g ik (x)Fkj (x).

(2.9)

It is remarkable that the coeﬀicients N i j from (2.9) are linear with respect to yi.

32

CHAPTER 2. LAGRANGE SPACES

Proposition 2.3.3. The Berwald connection of the canonical nonlinear connection i N has the coeﬀicients BΓ(N ) = (γjk (x), 0). Proposition 2.3.4. The solution curves of the Euler-Lagrange equations and the autoparallel curves of the canonical nonlinear connection N are given by the Lorentz equations (2.8). In the last part of this section, we underline the following theorem: Theorem 2.3.3. The autoparallel curves of the canonical nonlinear connection N are given by the following system: d 2 xi dx dxj i + N j x, = 0, dt2 dt dt where N i j are given by (2.4).

2.4 Hamilton-Jacobi equations Consider a Lagrange space Ln =(M, L(x,y)) and N (N i j ) its canonical nonlinear δ ∂ connection. The adapted basis , to the horizontal distribution N and δxi ∂y i the vertical distribution V has the horizontal vector fields: δ ∂ j ∂ = − N . i δxi ∂xi ∂y j

(2.1)

δy i = dy i + N i j dxj .

(2.2)

Its dual is (dxi , δy i ), with Theorem 1.1.1 give us the momenta 1 ∂L , 2 ∂y i

(2.3)

ω = pi dxi

(2.4)

θ = dω = dpi ∧ dxi .

(2.5)

pi = the 1-form and the 2-form

These geometrical object fields are globally defined on Tg M . θ is a symplectic structure on the manifold Tg M.

2.4. HAMILTON-JACOBI EQUATIONS

33

Proposition 2.4.1. In the adapted basis the 2-form θ is given by θ = gij δy i ∧ dxj .

(2.6)

H = pi y i − L(x, y).

(2.7)

δ ∂L ∂ ∂L 1 δ ∂L δ ∂L 1 dxs + s i δy s ∧dxi = − i s dxs ∧ Indeed, θ = dpi ∧dxi = s i s i 2 δx ∂y ∂y ∂y 4 δx ∂y δx ∂y i s i dx + gis δy ∧ dx . But is not diﬀicult to see that the coeﬀicient of dxs ∧ dxi vanishes. The triple (Tg M , θ, L) is called a Lagrangian system. 1 1 The energy EL of the space Ln is given by (2.6). Denoting H = EL , L = L, 2 2 then (2.6) can be written as:

But, along the integral curve of the Euler-Lagrange equations (2.5) we have ∂L dpi ∂H = − = − . ∂xi ∂xi dt And from (2.7), we get ∂H dxi i =y = . ∂pi dt So, we obtain: Theorem 2.4.1. Along to integral curves of the Euler-Lagrange equations the Hamilton-Jacobi equations: ∂H dpi ∂H dxi = ; = − i, dt ∂pi dt ∂x where H is given by (2.7) and pi =

(2.8)

1 ∂L , are satisfied. 2 ∂y i

These equations are important in applications. Example. For the Lagrange space of Electrodynamics with the fundamental function L(x, y) from (2.4) and U (x) = 0 we obtain e i e2 1 ij γ (x)pi pj − A (x)pi + Ai (x)Ai (x) H= 2 3 2mc mc 2mc (Ai = γ ij Aj ).

34

CHAPTER 2. LAGRANGE SPACES

Then, the Hamilton - Jacobi equations can be written without diﬀiculties. Now we remark that θ being a symplectic structure on Tg M , exterior differential dθ vanishes. But in adapted basis dθ = dgij ∧ δy i ∧ dxj + gij dδy i ∧ dxj = 0 reduces to: 1 δgij δgik 1 ∂g ∂g ij kj − j δy i ∧ dxj ∧ dxk + − δy k ∧ δy i ∧ dxj + k k i 2 δx δx 2 ∂x ∂y 1 i R dxs ∧ dxr + B i rs δy s ∧ dxr ∧ dxj = 0. +gij 2 rs We obtain Theorem 2.4.2. For any Lagrange space Ln the following identities hold gijkk − gikkj = 0,

gij kk − gik kj = 0.

(2.9)

Indeed, taking into account the h− and v−covariant derivations of the metric ! ∂Nji gij with respect to Berwald connection BΓ(N ) = , 0 , i.e. ∂y k δgij r r − Bik gkj − Bjk gir δxk ∂gij ∂gij i i and gij kk = , according with the properties = 2Cijk , Bjk = Bkj , we k k ∂y ∂y obtain (2.9). gijkk =

2.5 Metrical N -linear connections Let N (Nji ) be the canonical nonlinear connection of the Lagrange space Ln = i (M, L) and D an N −linear connection with the coeﬀicients DΓ(N ) = (Lijk , Cjk ). Then, the h− and v− covariant derivations of the fundamental tensor gij , gij|k and gij |k are given by (1.4.7). Applying the theory of N −linear connection from Chapter 1, one proves without diﬀiculties, the following theorem: Theorem 2.5.1. 1◦ On the manifold Tg M there exist only one N −linear connection D which verifies the following axioms:

2.5. METRICAL N -LINEAR CONNECTIONS

A1 A2 A3 A4 A5 2◦

35

N is canonical nonlinear connection of the space Ln . gij|k = 0 (D is h-metrical); gij |k = 0 (D is v-metrical); T ijk = 0 (D is h-torsion free); S i jk = 0 (D is v-torsion free).

i The coeﬀicients DΓ(N ) = (Lijk , Cjk ) of D are expressed by the following generalized Christoffel symbols: δg δg 1 δg rj jk rk + k − r Lijk = g ir j 2 δx δx δx (2.1) 1 ∂g ∂g ∂g rk rj jk i Cjk = g ir + − 2 ∂y j ∂y k ∂y r

3◦ This connection depends only on the fundamental function L(x, y) of the Lagrange space Ln . The N -linear connection D given by the previous theorem is called the canonical i metric connection and denoted by CΓ(N ) = (Lijk , Cjk ). By means of §1.5, Ch. 1, the connection 1-forms ω i j of the CΓ(N ) are i ω i j = Lijk dxk + Cjk δy k ,

(2.2)

Theorem 2.5.2. The canonical metrical connection CΓ(N ) satisfies the following structure equations: d(dx ) − dx ∧ ω i

k

i

(0)

k

= − Ω i, (1)

d(δy i ) − δy k ∧ ω i k = − Ω i ,

(2.3)

dωji − ωjk ∧ ωki = −Ωij (0)

(1)

where 2-forms of torsion Ω i and Ω i are as follows (0)

i dxj ∧ δy k , Ω i = Cjk

1 Ω i = Ri jk dxj ∧ dxk + P i jk dxj ∧ δy k 2 and the 2-forms of curvature Ωi j are 1 1 Ωi j = Rj i kh dxk ∧ dxh + Pj ikh dxk ∧ δy h + Sj ikh δy k ∧ δy h . 2 2 (1)

(2.4)

(2.5)

36

CHAPTER 2. LAGRANGE SPACES

i i The d-tensors of torsion R jk , P jk are given by (1.3.13’) and (1.4.10), and the i i i d-tensors of curvature Rj kh , Pj kh , Sj kh have the expressions (1.4.14). i ) we can deStarting from the canonical metrical connection CΓ(N ) = (Lijk , Cjk n rive other N connections depend only on the space L : Berwald connection -linear i ∂N j BΓ(N ) = , 0 ; Chern-Rund connection RΓ(N ) = (Lijk , 0) and Hashiguchi ∂y k ∂N i j i , Cjk . For special transformations of these connecconnection HΓ(N ) = ∂y k tions, the following commutative diagram holds:

RΓ(N ) %

−→

CΓ(N ) &

& BΓ(N ) %

HΓ(N ) Some properties of the canonical metrical connection CΓ(N ) are given by: Proposition 2.5.1. We have: P h 1◦ (ijk) Rijk = 0, (Rijk = gih R jk ). 2◦ Pijk = gih P h jk is totally symmetric. 3◦ The covariant curvature d-tensors Rijkh = gjr Ri r kh , Pijkh = gjr Pi r kh and Sijkh = gir Si r kh are skew-symmetric with respect to the first two indices. 4◦ Sijkh = Ciks C s jh − Cihs C s jk . s 5◦ Cikh = gis Cjh .

These properties can be proved using the property dθ = 0, with θ = gij δy i ∧dxj , the Ricci identities applied to the fundamental tensor gij and the equations gij|k = 0, gij |k = 0. By the same method we can study the metrical connections with a priori given h− and v− torsions. ¯ ¯ i , C¯ i ) Theorem 2.5.3. 1◦ There exists only one N -linear connection DΓ(N ) = (L jk jk which satisfies the following axioms: A01

N is canonical nonlinear connection of the space Ln ,

2.5. METRICAL N -LINEAR CONNECTIONS

A02 A03 A04 A05 2◦

37

¯ is h-metrical), gij|k = 0 (D ¯ is v-metrical), gij |k = 0 (D i is a priori given. The h-tensor of torsion T¯jk i The v-tensor of torsion S¯jk is a priori given.

¯ i , C¯ i ) of D ¯ are given by The coeﬀicients (L jk jk ¯ i = Li + 1 g ih (gjr T¯r kh + gkr T¯r jh − ghr T¯r kj ), L jk jk 2 1 i i C¯jk = Cjk + g ih (gjr S¯r kh + gkr S¯r jh − ghr S¯r kj ) 2

(2.6)

i where (Lijk , Cjk ) are the coeﬀicients of the canonical metrical connection.

From now on T¯ijk , S¯ijk will be denoted by T ijk , S ijk and the N −linear connection given by the previous theorem will be called metrical N −connection of the Lagrange space Ln . Some particular cases can be studied using the expressions of the coeﬀicients ¯ i and C¯ i . For instance the semi-symmetric case will be obtained taking T i = L jk jk jk δji σk − δki σj , S ijk = δji τk − δki τj . Proposition 2.5.2. The Ricci identities of the metrical N -linear connection DΓ(N ) are given by: X i |j|k − X i |k|j = X r Rr i jk − X i |r T r jk − X i |r Rr jk , X i |j |k − X i |k|j = X r Pr ijk − X i |r C r jk − X i |r P r jk ,

(2.7)

X i |j |k − X i |k |j = X r Sr i jk − X i |r S r jk . Of course these identities can be extended to a d-tensor field of type (r, s). Denoting Di j = y i |j , di j = y i |j . (2.8) we have the h− and v− deflection tensors. They have the known expressions: Di j = y s Li sj − N i j ; di j = δ i j + y s C i sj . According to Ricci identities (2.7) we obtain:

(2.5.7’)

38

CHAPTER 2. LAGRANGE SPACES

Theorem 2.5.4. For any metrical N -linear connection the following identities hold: Di j|k − Di k|j = y s Rs i jk − Di s T sjk − di s Rsjk , Di j |k − di k|j = y s Ps ijk − Di s C s jk − di s P s jk ,

(2.9)

di j |k − di k |j = y s Ss ijk − di s S sjk . We will apply this theory in a next section taking into account the canonical metrical connection CΓ(N ) and taking T ijk = 0, S ijk = 0. Of course the theory of parallelism of vector fields and the h−geodesics or v−geodesics for the metrical connection N −linear connections can be obtained as a consequence of the corresponding theory from Ch. 1.

2.6 The electromagnetic and gravitational fields Let us consider a Lagrange spaces Ln = (M, L) endowed with the canonical nonlinear connection N and with the canonical metrical N −connection CΓ(N ) = i (Lijk , Cjk ). The covariant deflection tensors Dji and dji are given by Dij = gis Ds j , dij = gis ds j . We have: Dij|k = gis Ds j|k , dij|k = gis ds j|k etc. So, we have Proposition 2.6.1. The covariant deflection tensors Dij and dij of the canonical metrical N -connection CΓ(N ) satisfy the identities: Dij|k − Dik|j = y s Rsijk − dis Rs jk , Dij |k − dik|j = y s Psijk − Dis C s jk − dis P s jk ,

(2.1)

dij |k − dik |j = y s Ssijk . The Lagrangian theory of electrodynamics lead us to introduce [182], [184], [183], [175]: Definition 2.6.1. The d-tensor fields: 1 1 Fij = (Dij − Dji ), fij = (dij − dji ) 2 2 are the h- and v-electromagnetic tensor of the Lagrange space Ln = (M, L).

(2.2)

2.6. THE ELECTROMAGNETIC AND GRAVITATIONAL FIELDS

39

The Bianchi identities for CΓ(N ) and the identities (2.1) lead to the following important result: Theorem 2.6.1. The following generalized Maxwell equations hold: X Fij|k + Fjk|i + Fki|j = − Cios Rs jk , (ijk)

where Cios

(2.3)

Fij |k + Fjk |i + Fki |j = 0, X j = Cijs y , and means cyclic sum. (ijk)

Corollary 2.6.1.1. If the canonical nonlinear connection N of the space Ln is integrable then the equations (2.3) reduce to: X X Fij|k = 0, Fij |k = 0. (2.6.3’) (ijk)

(ijk)

If we put F ij = g is g jr Fsr

(2.4)

hJ i = F ij |j , vJ i = F ij |j ,

(2.5)

and then one can prove: Theorem 2.6.2. The following laws of conservation hold: 1 hJ i |i = {F ij (Rij − Rji ) + F ij |r Rr ij }, 2 vJ i |i = 0,

(2.6)

where Rij is the Ricci tensor Ri hjh . Remark 2.6.2.1. In the Lagrange space of electrodynamics the tensor Fjk is given by (2.3.6). The previous theoryX one reduces to the classical theory. Namely Fij (x) Fij|k = 0 and Fij |k = 0, hj|ii = 0, vj i = 0. satisfy the Maxwell equations (ijk)

Now, considering the lift to Tg M of the fundamental tensor gij (x, y) of the space n L , given by G = gij dxi ⊗ dxj + gij δy i ⊗ δy j

40

CHAPTER 2. LAGRANGE SPACES

we can obtain the Einstein equations of the canonical metric connection CΓ(N ). The curvature Ricci and scalar curvatures: ′′

Rij = Ri h jh , Sij = Si h jh ,0 Pij = Pi hjh , Pi hhj R = g ij Rij , S = g ij Sij . H

V

1

(2.7)

2

Let us denote by T ij , T ij , T ij and T ij the components in adapted basis of the energy momentum tensor on the manifold Tg M. Thus we obtain:

δ ∂ , δxi ∂y i

Theorem 2.6.3. 1◦ The Einstein equations of the Lagrange space Ln = (M, L(x, y)) with respect to the canonical metrical connection i CΓ(N ) = (Lijk , Cjk ) are as follows: H 1 1 Rij − Rgij = κ T ij ,0 Pij = κ T ij 2 V 2 1 Sij − Sgij = κ T (i)(j) ,00 Pij = κ T ij , 2 where κ is a real constant. H

(2.8)

V

2◦ The energy momentum tensors T ij and T ij satisfy the following laws of conservation H Vi 1 ih s s i κ T ij = − (Pjs Rhi + 2Rij Ps ), κ T j |i = 0. (2.9) 2 The physical background of the previous theory is discussed by Satoshy Ikeda in the last chapter of the book [116]. The previous theory is very simple in the particular Lagrange spaces Ln having Pi hjk = 0. We have: Corollary 2.6.3.1. 1◦ If the canonical metrical connection CΓ(N ) has the property Pj i kh = 0, then the Einstein equations are H V 1 1 Rij − Rgij = κ T ij , Sij − Sgij = κ T (i)(j) 2 2

2◦ The following laws of conservation hold: Hi

Vi

T j|i = 0, T j |i = 0.

(2.10)

2.7. THE ALMOST KÄHLERIAN MODEL OF A LAGRANGE SPACE LN

41

Remark 2.6.3.1. The Lagrange space of Electrodynamics, Ln , has i CΓ(N ) = (γjk (x), 0), Pj ikh = 0, Sj ikh = 0. The Einstein equations (2.10) reduce to the classical Einstein equations of the space Ln .

2.7 The almost Kählerian model of a Lagrange space Ln A Lagrange space Ln = (M, L) can be thought as an almost Kähler space on the manifold Tg M = T M \ {0}, called the geometrical model of the space Ln . As we know from section 3, Ch. 1 the canonical nonlinear connection N determines an almost complex structure F(Tg M ), expressed in (1.3.9’). This is F=

δ ∂ i ⊗ δy − ⊗ dxi . i i δx ∂y

(2.1)

i F is integrable if and only if Rjk = 0. F is globally defined on Tg M and it can be considered as a F(Tg M )− linear mapping from χ(Tg M ) to χ(Tg M ): ∂ δ ∂ δ = − , F = , (i = 1, ..., n). (2.7.1’) F δxi ∂y i ∂y i δxi

The lift of the fundamental tensor gij of the space Ln with respect to N is defined by G = gij dxi ⊗ dxj + gij δy i ⊗ δy j . (2.2) Evidently G is a (pseudo-)Riemannian metric on the manifold Tg M. The following result can be proved without diﬀiculties: Theorem 2.7.1. 1◦ The pair (G, F) is an almost Hermitian structure on Tg M, n determined only by the fundamental function L(x, y) of L . 2◦ The almost symplectic structure associated to the structure (G, F) is given by θ = gij δy i ∧ dxj . 3◦ The space (Tg M , G, F) is almost Kählerian. Indeed: 1◦ N, G, F are determined only by L(x, y). We have G(FX, FY ) = G(X, Y ), ∀X, Y ∈ χ(Tg M ).

(2.3)

42

CHAPTER 2. LAGRANGE SPACES

2◦ In the adapted basis θ(X, Y ) = G(FX, Y ) is (2.3). 3◦ Taking into account (2.1.1), it follows that θ is a symplectic structure (i.e. dθ = 0). The space H 2n = (Tg M , G, F) is called the almost Kählerian model of the Lan grange space L . It has a remarkable property: Theorem 2.7.2. The canonical metrical connection D with coeﬀicients CΓ(N ) = i (Lijk , Cjk ) of the Lagrange space Ln is an almost Kählerian connection, i.e. DG = 0,

DF = 0.

(2.4)

Indeed, (2.1), (2.2), in the adapted basis imply (2.4). We can use this geometrical model to study the geometry of Lagrange space Ln . For instance, the Einstein equations of the (pseudo) Riemannian space (Tg M , G) equipped with the metrical canonical connection CΓ(N ) are the Einstein equations of the Lagrange space studied in the previous section of this chapter. G.S. Asanov showed [27] that the metric G given by the lift (2.2) does not satisfies the principle of the Post-Newtonian calculus because the two terms of G have not the same physical dimensions. This is the reason to introduce a new lift which can be used in a gauge theory of physical fields. Let us consider the scalar field: E = ||y||2 = gij (x, y)y i y j .

(2.5)

ε is called the absolute energy of the Lagrange space Ln . We assume ||y||2 > 0 and consider the following lift of the fundamental tensor gij : 0 a2 i j G= gij dx ⊗ dx + gij δy i ⊗ δy j (2.6) 2 ||y|| where a > 0 is a constant. Let us consider also the tensor field on Tg M: 0 ||y|| ∂ a δ j F= − ⊗ dx + ⊗ δy i , (2.7) i i a ∂y ||y|| δx and 2-form

0

θ= where θ is from (2.3). We can prove:

a θ, ||y||

(2.8)

2.8. GENERALIZED LAGRANGE SPACES 0

43

0

Theorem 2.7.3. 1◦ The pair (G, F) is an almost Hermitian structure on the manifold Tg M , depending only on the the fundamental function L(x, y) of the n space L . 0

0

0

2◦ The almost symplectic structure θ associated to the structure (G, F) is given by (2.8). 0

0

0

3◦ θ being conformal to symplectic structure θ, the pair (G, F) is conformal to the almost Kählerian structure (G, F).

2.8 Generalized Lagrange spaces A first natural generalization of the notion of Lagrange space is provided by a notion which we call a generalized Lagrange space. This notion was introduced by author in the paper [183]. Definition 2.8.1. A generalized Lagrange space is a pair GLn = (M, gij (x, y)), where gij (x, y) is a d-tensor field on the manifold Tg M , of type (0,2), symmetric, of rank n and having a constant signature on Tg M. We continue to call gij (x, y) the fundamental tensor on GLn . One easily sees that any Lagrange space Ln = (M, L(x, y)) is a generalized Lagrange space with the fundamental tensor 1 ∂ 2 L(x, y) . gij (x, y) = 2 ∂y i ∂y j

(2.1)

But not any space GLn is a Lagrange space Ln . Indeed, if gij (x, y) is given, it may happen that the system of partial differential equations (2.1) does not admits solutions in L(x, y). 1◦ A necessary condition in order that the system (2.1) ∂gij admit a solution L(x, y) is that the d-tensor field = 2Cijk be completely ∂y k symmetric.

Proposition 2.8.1.

2◦ If the condition 1◦ is verified and the functions gij (x, y) are 0-homogeneous with respect to y i ; then the function L(x, y) = gij (x, y)y i y j + Ai (x)y i + U (x)

(2.2)

44

CHAPTER 2. LAGRANGE SPACES

is a solution of the system of partial differential equations (2.1) for any arbitrary d-covector field Ai (x) and any arbitrary function U (x), on the base manifold M . The proof of previous statement is not complicated. In the case when the system (2.1) does not admit solutions in the functions L(x, y) we say that the generalized Lagrange space GL2 = (M, gij(x,y) ) is not reducible to a Lagrange space. Remark 2.8.0.1. The Lagrange spaces Ln with the fundamental function (2.2) give us an important class of Lagrange spaces which includes the Lagrange space of electrodynamics. Examples. 1◦ The pair GLn = (M, gij ) with the fundamental tensor field gij (x, y) = e2σ(x,y) γij (x)

(2.3)

where σ is a function on (Tg M ) and γij (x) is a pseudo-Riemannian metric on ∂σ the manifold M is a generalized Lagrange space if the d-covector field i no ∂y vanishes. It is not reducible to a Lagrange space. R. Miron and R. Tavakol [183] proved that GLn = (M, gij (x, y)) defined by (2.3) satisfies the Ehlers - Pirani - Schild’s axioms of General Relativity. 2◦ The pair GLn = (M, gij (x, y)), with gij (x, y) = γij (x) + 1 −

1 yi yj , n2 (x, y)

yi = γij (x)y j

(2.4)

where γij (x) is a pseudo-Riemannian metric and n(x, y) > 1 is a smooth function (n is a refractive index) give us a generalized Lagrange space GLn which is not reducible to a Lagrange space. This metric has been called by R. G. Beil, [40] the Miron’s metric from Relativistic Optics. The restriction of the fundamental tensor gij (x, y) (2.4) to a section SV : xi = xi , y i = V i (x), (V i being a vector field) of the projection π : T M → M , is given by gij (x, V (x)). It gives us the known Synge’s metric tensor of the Relativistic Optics [243].

2.8. GENERALIZED LAGRANGE SPACES

45

For a generalized Lagrange space GLn = (M, gij (x, y)) an important problem is to determine a nonlinear connection obtained from the fundamental tensor gij (x, y). In the particular cases, given by the previous two examples this is possible. But, generally no. We point out a method of determining a nonlinear connection N , strongly connected to the fundamental tensor gij of the space GLn , if such kind of nonlinear connection exists. Consider the absolute energy E(x, y) of space GLn : ε(x, y) = gij (x, y)y i y j

(2.5)

ε(x, y) is a Lagrangian. The Euler - Lagrange equations of ε(x, y) are d ∂ε ∂ε − = 0, ∂xi dt ∂y i

dxi y = . dt i

(2.6)

Of course, according the general theory, the energy Eε of the Lagrangian E(x, y), ∂ε is Eε = y i i − E and it is preserved along the integral curves of the differential ∂y equations (2.6). If E(x, y) is a regular Lagrangian - we say that the space GLn is weakly regular - it follows that the Euler-Lagrange equations determine a semispray with the coeﬀicients 2 2 is ∂ε 1 ∂ ε ∂ ε ∨ ∨ 2Gi (x, y) =g y j − s , g ij = . (2.7) s j ∂y ∂x ∂x 2 ∂y i ∂y j Consequently, the nonlinear connection N with the coeﬀicients Nji =

∂Gi is ∂y j

determined only by the fundamental tensor gij (x, y) of the space GLn . In the case when we can not derive a nonlinear connection from the fundamental tensor gij , we give a priori a nonlinear connection N and study the geometry of pair (GLn , N ) by the methods of the geometry of Lagrange space Ln . δ ∂ For instance, using the adapted basis , to the distributions N and δxi ∂y i V , respectively and its dual (dxi , δy i ) we can lift gij (x, y) to Tg M: G(x, y) = gij (x, y)dxi ⊗ dxj +

a gij (x, y)δy i ⊗ δy j 2 kyk

46

CHAPTER 2. LAGRANGE SPACES

and can consider the almost complex structure F=−

kyk ∂ a δ i ⊗ dx + ⊗ δy i i i a ∂y kyk δx

with ε(x, y) > 0 and kyk = ε1/2 (x, y). The space (Tg M , F) is an almost Hermitian space geometrical associated to the n pair (GL , N ). J. Silagy, [241], make an exhaustive and interesting study of this diﬀicult problem.

Chapter 3 Finsler Spaces An important class of Lagrange Spaces is provided by the so-called Finsler spaces. The notion of Finsler space was introduced by Paul Finsler in 1918 and was developed by remarkable mathematicians as L. Berwald [42], E. Cartan [56], H. Buseman [52], H. Rund [218], S.S. Chern [60], M. Matsumoto [144], and many others. This notion is a generalization of Riemann space, which gives an important geometrical framework in Physics, especially in the geometrical theory of physical fields, [18], [116], [129], [246], [252]. In the last 40 years, some remarkable books on Finsler geometry and its applications were published by H. Rund, M. Matsumoto, R. Miron and M. Anastasiei, A. Bejancu, Abate-Patrizio, D. Bao, S.S. Chern and Z. Shen, P. Antonelli, R. Ingarden and M. Matsumoto, R. Miron, D. Hrimiuc, H. Shimada and S. Sabău, G.S. Asanov, M. Crampin, P.L. Antonelli, S. Vacaru, S. Ikeda. In the present chapter we will study the Finsler spaces considered as Lagrange spaces and applying the mechanical principles. This method simplifies the theory of Finsler spaces. So, we will treat: Finsler metric, Cartan nonlinear connection derived from the canonical spray, Cartan metrical connection and its structure equations. Some examples: Randers spaces, Kropina spaces and some new classes of spaces more general as Finsler spaces: the almost Finsler Lagrange spaces and the Ingarden spaces will close this chapter.

3.1 Finsler metrics Definition 3.1.1. A Finsler space is a pair F n = (M, F (x, y)) where M is a real n−dimensional differentiable manifold and F : T M → R is a scalar function 47

48

CHAPTER 3. FINSLER SPACES

which satisfies the following axioms: 1◦ F is a differentiable function on Tg M and F is continuous on the null section of the projection π : T M → M . 2◦ F is a positive function. 3◦ F is positively 1-homogeneous with respect to the variables y i 4◦ The Hessian of F 2 with the elements: gij (x, y) =

1 ∂F 2 2 ∂y i ∂y j

(3.1)

is positively defined on the manifold Tg M. Of course, the axiom 4◦ is equivalent with the following: 4◦0 The pair (M, F 2 (x, y)) = LnF is a Lagrange space with positively defined fundamental tensor, gij . LnF will be called the Lagrange space associated to the Finsler space F n . It follows that all properties of the Finsler space F n derived from the fundamental function F 2 and the fundamental tensor gij of the associated Lagrange space LnF . Remarks 1◦ Sometimes we will ask for gij to be of constant signature and rank(gij (x, y)) = n on Tg M. 2◦ Any Finsler space F n = (M, F (x, y)) is a Lagrange space LnF = (M, F 2 (x, y)), but not conversely. Examples 1◦ A Riemannian manifold (M, γij (x)) determines a Finsler space F n = (M, F (x, y)), where q F (x, y) = γij (x)y i y j . (3.2) The fundamental tensor is gij (x, y) = γij (x). 2◦ Let us consider, in a preferential local system of coordinates, the following function: p F (x, y) = 4 (y 1 )4 + ... + (y n )4 . (3.3) Then F satisfy the axioms 1◦ -4◦ .

3.1. FINSLER METRICS

49

Remark 3.1.0.1. This example was given by B. Riemann. 3◦ Antonelli-Shimada’s ecological metric is given, in a preferential local system of coordinates, on Tg M , by F (x, y) = eϕ L, ϕ = αi xi , (αi are positive constants), and where L = {(y 1 )m + (y 2 )m + · · · + (y n )m }1/m , m ≥ 3,

(3.4)

m being even. 4◦ Randers metric is defined by F (x, y) = α(x, y) + β(x, y),

(3.5)

where α2 (x, y) := aij (x)y i y j , (M, aij (x)) being a Riemannian manifold and β(x, y) := bi (x)y i . The fundamental tensor gij is expressed by: gij =

00 α+β hij + di dj , hij := aij − l i l j , α 0

0

di = bi + l i , l i :=

(3.1.5’)

∂α . ∂y i

One can prove that gij is positively defined under the condition b2 = aij bi bj < 1. In this case the pair F n = (M, α + β) is a Finsler space. The first example motivates the following theorem: Theorem 3.1.1. If the base manifold M is paracompact, then there exist functions F : T M → R such that the pair (M, F ) is a Finsler spaces. Regarding the axioms 1◦ − 4◦ we can see without diﬀiculties: Theorem 3.1.2. The system of axioms of a Finsler space is minimal. Some properties of Finsler space F n :

50

CHAPTER 3. FINSLER SPACES

1◦ The components of the fundamental tensor gij (x, y) are 0-homogeneous with respect to y i . 2◦ The components of 1-form 1 ∂F 2 pi = 2 ∂y i are 1-homogeneous with respect to y i .

(3.6)

3◦ The components of the Cartan tensor Cijk

1 ∂ 3F 2 1 ∂gij = = 4 ∂y i ∂y j ∂y k 2 ∂y k

(3.7)

are −1−homogeneous with respect to y i . Consequently we have Coij = y s Csij = 0.

(3.8)

If X i and Y i are d-vector fields, then kXk2 := gij (x, y)X i Y i is a scalar field. < X, Y >:= gij (x, y)X i Y j is a scalar field. Assuming kXku 6= 0, kY ku 6= 0 the angle φ = ∠(X, Y ) at a point u ∈ Tg M is given by < X, Y > (u) cos φ = . kXku · kY ku The vectors Xu , Yu are orthogonal if hX, Y i(u) = 0. Proposition 3.1.1. In a Finsler space F n the following identities hold: 1◦ F 2 (x, y) = gij (x, y)y i y j 2 ◦ pi y i = F 2 . Proposition 3.1.2.

1◦ The 1-form ω = pi dxi

(3.9)

θ = dω = dpi ∧ dxi

(3.10)

is globally defined on Tg M. 2◦ The 2-form is globally defined on Tg M.

3.2. GEODESICS

51

3◦ θ is a symplectic structure on Tg M. Definition 3.1.2. A Finsler space F n = (M, F ) is called reducible to a Riemann space if its fundamental tensor gij (x, y) does not depend on the variable y i . Proposition 3.1.3. A Finsler space F n is reducible to a Riemann space if and only if the tensor Cijk is vanishing on Tg M.

3.2 Geodesics In a Finsler space F n = (M, F (x, y)) one can define the notion of arc length of a smooth curve. Let c be a parametrized curve in the manifold M : c : t ∈ [0, 1] → (xi (t)) ∈ U ⊂ M

(3.1)

U being a domain of a local chart in M . The extension e c of c to Tg M has the equations xi = xi (t), y i =

dxi (t), t ∈ [0, 1]. dt

Thus the restriction of the fundamental function F (x, y) to e c is F (x(t),

(3.2.1’) dx (t)), dt

t ∈ [0, 1]. We define the “length” of curve c with extremities c(0), c(1) by the number Z 1 dx ℓ(c) = F (x(t), (t))dt. (3.2) dt 0 The number ℓ(c) does not depend on a changing of coordinates on Tg M and, by means of 1-homogeneity of function F , ℓ(c) does not depend on the parametrization of the curve c. ℓ(c) depends on the curve c, only. We can choose a canonical parameter on c, considering the following function s = s(t), t ∈ [0, 1]: Z t dx s(t) = F (x(τ ), (τ ))dτ. dt 0 This function is derivable and its derivative is ds dx = F (x(t), (t)) > 0, t ∈ (0, 1). dt dt

52

CHAPTER 3. FINSLER SPACES

So the function s = s(t), t ∈ [0, 1], is invertible. Let t = t(s), s ∈ [s0 , s1 ] be its inverse. The change of parameter t → s has the property dx F x(s), (s) = 1. (3.3) ds Variational problem on the functional ℓ will gives the curves on Tg M which extremize the arc length. These curves are geodesics of the Finsler space F n . So, the solution curves of the Euler-Lagrange equations: ∂F dxi d ∂F i − i = 0, y = (3.4) dt ∂y i ∂x dt are the geodesics of the space F n . Definition 3.2.1. The curves c = (xi (t)), t ∈ [0, 1], solutions of the EulerLagrange equations (3.4) are called the geodesics of the Finsler space F n . The system of differential equations (3.4) is equivalent to the following system dF ∂F d ∂F 2 ∂F 2 − =2 , i i dt ∂y ∂x dt ∂y i

dxi y = . dt i

In the canonical parametrization, according to (3.3) we have: Theorem 3.2.1. In the canonical parametrization the geodesics of the Finsler space F n are given by the system of differential equations dxi d ∂F 2 ∂F 2 i Ei (F ) := − = 0, y = . ds ∂y i ∂xi ds 2

(3.5)

Now, remarking that F 2 = gij y i y j , the previous equations can be written in the form: j k d 2 xi dx dx dx dxi i i + γjk x, = 0, y = , (3.6) ds2 ds ds ds ds i where γjk are the Christoffel symbols of the fundamental tensor gij : ∂g ∂g 1 ∂g jr jk rk + − . γ i jk = g ir 2 ∂xj ∂xk ∂xr

(3.7)

A theorem of existence and uniqueness of the solution of differential equations (3.6) can be formulated in the classical manner.

3.3. CARTAN NONLINEAR CONNECTION

53

3.3 Cartan nonlinear connection Considering the Lagrange space LnF = (M, F 2 ) associated to the Finsler space F n = (M, F ) we can obtain some main geometrical object field of F n . So, Theorem 2.3.1, aﬀirms: Theorem 3.3.1. For the Finsler space F n the equations d ∂F 2 ∂F 2 dxi ij 2 ij i g Ej (F ) := g − = 0, y = dt ∂y j ∂xj dt can be written in the form

d 2 xi dx dxi i i + 2G x, = 0, y = dt2 dt dt

(3.1)

where i 2Gi (x, y) = γjk (x, y)y j y k

(3.3.1’)

Consequently the equations (3.1) give the integral curves of the semispray: S = yi

∂ ∂ i − 2G (x, y) . ∂xi ∂y i

(3.2)

Since Gi are 2-homogeneous functions with respect to y i it follows that S is a spray. S determine a canonical nonlinear connection N with the coeﬀicients 1 ∂ ∂Gi i Nj= = {γ i rs (x, y)y r y s }. (3.3) j ∂y 2 ∂yj N is called the Cartan nonlinear connection of the space F n . The tangent bundle T (T M ), the horizontal distribution N and the vertical distribution V give us the direct decomposition of vectorial spaces Tu (Tg M ) = N (u) ⊕ V (u), ∀u ∈ Tg M. (3.4) δ ∂ The adapted basis to N and V is , and the dual adapted basis is δxi ∂y i (dx, δy i ), where δ = ∂ − Nij (x, y) ∂ δxi ∂xi ∂y j (3.3.4’) δy i = dy i + N i (x, y)dxj . j

One obtains:

54

CHAPTER 3. FINSLER SPACES

Theorem 3.3.2.

1◦ The horizontal curves in F n are given by i

i

x = x (t),

δy i = 0. dt

2◦ The autoparallel curves of the Cartan nonlinear connection N coincide to the integral curves of the spray S, (3.2).

3.4 Cartan metrical connection Let N (Nji ) be the Cartan nonlinear connection of the Finsler space F n . According to section 5 of chapter 2 one introduces the canonical metrical N −linear connection of the space F n . But, for these spaces the system of axioms, from Theorem 2.5.1 can be given in the Matsumoto’s form [144], [145]. Theorem 3.4.1. 1◦ On the manifold Tg M , for any Finsler space F n = (M, F ) i i , Cjk ) there exists only one linear connection D, with the coeﬀicients CΓ = (Nji , Fjk which verifies the following axioms: A1 . The deflection tensor field Dji = y|ji vanishes. A2 . gij|k = 0, (D is h−metrical). A3 . gij |k = 0, (D is v−metrical). A4 . T ijk = 0, (D is h−torsion free). A5 . S i jk = 0, (D is v−torsion free). i i 2◦ The coeﬀicients (Nji , Fjk , Cjk ) are as follows:

a. Nji are the coeﬀicients (3.3.3) of the Cartan nonlinear connection. i i b. Fjk , Cjk are expressed by the generalized Christoffel symbols: δg δg δg 1 sk js jk i + k − s Fjk = g is 2 δxj δx δx 1 ∂g ∂g ∂g sk js jk i Cjk = g is + − 2 ∂y j ∂y k ∂y s

3◦ This connection depends only on the fundamental function F .

(3.1)

3.4. CARTAN METRICAL CONNECTION

55

A proof can be find in the books [21], [164]. The previous connection is named the Cartan metrical connection of the Finsler space F n . Now we can develop the geometry of Finsler spaces, exactly as the geometry of the associated Lagrange space LnF = (M, F 2 ). Also, in the case of Finsler space the geometrical model H 2n = (Tg M , G, F) is an almost Kählerian space. A very good example is provided by the Randers space, introduced by R.S. Ingarden. A Randers space is the Finsler space F n = (M, α + β) equipped with Cartan nonlinear connection N . It is denoted by RF n = (M, α + β, N ). The geometry of these spaces was much studied by many geometers. A good monograph in this respect is the D. Bao, S.S. Chern and Z. Shen’s book [35]. The Randers spaces RF n can be generalized considering the Finsler spaces F n = (M, α + β), where α(x, y) is the fundamental function of a Finsler space F 0n = (M, α). The Finsler space F n = (M, α + β) equipped with the Cartan nonlinear connection N of the space F 0n = (M, α) is a generalized Randers space [152], [35]. Evidently, this notion has some advantages, since we can take some remarkable Finsler spaces F 0n , (M. Anastasiei β−transformation of a Finsler space [14]). As an application of the previous notions, we define the notion of Ingarden space IF n , [115], [49]. This is the Finsler space F n = (M, α + β) equipped i i (x) being the Christoffel with the nonlinear connection N = γjk (x)y k − Fji (x), γjk symbols of the Riemannian metric aij (x), which defines α2 = aij (x)y i y j and the electromagnetic tensor Fji (x) determined by β = bi (x)y i . While the spaces RF n 1 have not the electromagnetic field F = (Dij − Dji ), the Ingarden spaces have 2 such kind of tensor fields and they give us the remarkable Maxwell equations, [49]. Also, the autoparallel curve of the nonlinear connection N are given by the known Lorentz equations. An example of a special Lagrange space derived from a Finsler one, [164] is as follows: Let us consider the Lagrange space Ln = (M, L(x, y)) with the fundamental function L(x, y) = F 2 (x, y) + β,

56

CHAPTER 3. FINSLER SPACES

where F is the fundamental function of a priori given Finsler space F n = (M, F ) and β = bi (x)y i . These spaces have been called the almost Finsler Lagrange Spaces (shortly AFLspaces), [164], [49]. They generalize the Lagrange space from Electrodynamics. Indeed, the Euler - Lagrange equations of AFL-spaces are exactly the Lorentz equations d 2 xi dxj dxk 1 i dxj i + γjk (x, y) = Fj (x) . dt2 dt dt 2 dt To the end of these three chapter we can do a general remark: The class of Riemann spaces {Rn } is a subclass of the class of Finsler spaces {F n }, the class {F n } is a subclass of class of Lagrange spaces {Ln } and this is a subclass of class of generalized Lagrange spaces {GLn }. So, we have the following sequence of inclusions: (I)

{Rn } ⊂ {F n } ⊂ {Ln } ⊂ {GLn }.

Therefore, we can say: The Lagrange geometry is the geometric study of the terms of the sequence of inclusions (I).

Chapter 4 The Geometry of Cotangent Manifold The geometrical theory of cotangent bundle (T ∗ M , π ∗ , M ) on a real, finite dimensional manifold M is important in differential geometry. Correlated with that of tangent bundle (T M, π, M ), introduced in Ch. 1, one obtains a framework for construction of Lagrangian and Hamiltonian mechanical systems, as well as, for the duality between them, via Legendre transformation. We study here the fundamental geometric objects on T ∗ M , as Liouville vector field C ∗ , Liouville 1-form ω, symplectic structure θ = dω, Poisson structure, N −linear connection etc.

4.1 Cotangent bundle Let M be a real n−dimensional differentiable manifold. Its cotangent bundle (T ∗ M , π ∗ , M ) can be constructed as the dual of the tangent bundle (T M, π, M ), [154]. If (xi ) is a local coordinate system on a domain U of a chart on M , the induced system of coordinates on π ∗−1 (U ) is (xi , pi ), (i, j, k, .. = 1, 2, ..., n), p1 , ..., pn are called “momentum variables”. We denote (xi , pi ) = (x, p) = u. A change of coordinates on T ∗ M is given by i ∂e x x ei = x ei (x1 , ..., xn ), rank =n ∂xj ∂xj pei = pj . ∂e xi 57

(4.1)

58

CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

The natural frame

∂ ∂ , ∂xi ∂pi

= (∂˙i , ∂˙ i ) is transformed by (4.1) in the form

∂e xj e ∂ pej ˙ j ˙ i ∂e xi ˙ j ∂i = ∂j + i ∂ , ∂ = j ∂ . ∂xi ∂x ∂x

(4.2)

The natural coframe (dxi , dpi ) is changed by the rule de xi =

∂xj ∂ 2 xj ∂e xi j dx ; de p = dp + pj de xk . i j j i i k ∂x ∂e x ∂e x ∂e x

(4.1.2’)

The Jacobian matrix of change of coordinates (4.1) is ∂e xi ∂xj 0 J(u) = . j ∂ pej ∂x ∂xi ∂e xi u It follows

det J(u) = 1 for every u ∈ T ∗ M.

So we get: Theorem 4.1.1. The differentiable manifold T ∗ M is orientable. One can prove that if M is a paracompact manifold, then T ∗ M is paracompact, too. The kernel of differential dπ ∗ : T T ∗ M → T ∗ M is the vertical subbundle V T ∗ M of the tangent bundle T T ∗ M . The fibres Vu of V T ∗ M , ∀u ∈ T ∗ M determine a distribution V on T ∗ M , called the vertical distribution. It is locally generated by the tangent vector fields (∂˙ 1 , ..., ∂˙ n ). Consequently, V is an integrable distribution of local dimension n. Noticing the formulae (4.2), (4.1.2’) one can introduce the following geometrical object fields: C ∗ = pi ∂˙ i , (4.3) called the Liouville–Hamilton vector field on T ∗ M , ω = pi dxi ,

(4.4)

called the Liouville 1-form on T ∗ M , θ = dω = dpi ∧ dxi ,

(4.5)

4.1. COTANGENT BUNDLE

59

θ is a symplectic structure on T ∗ M . All these geometrical object fields do not depend on the change of coordinates (4.1). The Poisson brackets {, } on the manifold T ∗ M are defined by {f, g} =

∂f ∂g ∂g ∂f − , ∀f, g ∈ F (T ∗ M ). i i ∂pi ∂x ∂pi ∂x

(4.6)

Of course, {f, g} ∈ F (T ∗ M ) and {f, g} does not depend on the change of coordinates (4.1). Also, the following properties hold: 1◦ {f, g} = −{g, f }, 2◦ {f, g} is R−linear in every argument, 3◦ {{f, g}, h} + {{g, h}, f } + {{h, f }, g} = 0 (Jacobi identity), 4◦ {·, gh} = {·, g}h + {·, h}g. The pair {F(T ∗ M ), {, }} is a Lie algebra, called the Poisson–Lie algebra. The relation between the structures θ and {, } can be given by means of the notion of Hamiltonian system. Definition 4.1.1. A differentiable Hamiltonian is a real function H : T ∗ M → R which is of C ∞ class on T] M ∗ = T ∗ M \ {0} and continuous on T ∗ M . ∞ structure on M . Then H = p An example: Let gij (x) be a C −Riemannian ∗ ij g (x)pi pj is a differentiable Hamiltonian on T M .

Definition 4.1.2. A Hamiltonian system is a triple (T ∗ M, θ, H). Let us consider the F(T ∗ M )−modules χ(T ∗ M ) (tangent vector fields on T ∗ M ), χ∗ (T ∗ M ) (cotangent vector fields on T ∗ M ). The following F(T ∗ M )−linear mapping Sθ = χ(T ∗ M ) → χ∗ (T M ) can be defined by Sθ (X) = iX θ, ∀X ∈ χ(T ∗ M ). One proves, without diﬀiculties:

(4.7)

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

Proposition 4.1.1. Sθ is an isomorphism. We have: ∂ ∂ = dxi Sθ = −dpi , Sθ i ∂x ∂pi Sθ (C ∗ ) = ω.

(4.1.6’) (4.1.7’)

Theorem 4.1.2. The following properties of the Hamiltonian system (T ∗ M, θ, H) hold: 1◦ There exists a unique vector field XH ∈ X (T ∗ M ) for which: iH θ = −dH.

(4.8)

2◦ The integral curves of the vector field XH are given by the Hamilton–Jacobi equations: dxi ∂H dpi ∂H , (4.9) = = − i. dt ∂pi dt ∂x Proof. 1◦ The existence and uniqueness of the vector field XH is assured by the isomorphism Sθ : XH = Sθ−1 (−dH). (4.10) XH is called the Hamiltonian vector field. 2◦ The local expression of XH is given (by (4.6)): XH =

∂H ∂ ∂H ∂ − . ∂pi ∂xi ∂xi ∂pi

(4.11)

Consequently: the integral curves of the vector field XH are given by the Hamilton–Jacobi equations (4.8). Along the integral curves of XH , we have dH = {H, H} = 0. dt Thus: The differentiable Hamiltonian H(x, p) is constant along the integral curves of the Hamilton vector field XH . The structures θ and {, } have a fundamental property: Theorem 4.1.3. The following formula holds: {f, g} = θ(Xf , Xg ), ∀f, g ∈ X ∗ (T ∗ M ), ∀X ∈ X (T ∗ M ).

(4.12)

4.2. VARIATIONAL PROBLEM. HAMILTON–JACOBI EQUATIONS

61

Proof. From (4.10): {f, g} = Xf g = −Xg f = −df (Xg ) = (iXf θ)(Xg ) = θ(Xf , Xg ).

As a consequence, we obtain: dpi dxi = {H, xi }, = {H, pi }. dt dt

(4.13)

It is remarkable the Jacobi method for integration of Hamilton–Jacobi equations (4.9). Namely, we look for a solution curves γ(t) in T ∗ M of the form xi = xi (t), pi =

∂S (x(t)) ∂xi

(4.14)

where S ∈ F (M ). Substituting in (4.9), we have dxi ∂H ∂S dpi ∂ 2 S ∂H ∂H = (x(t)) i (x(t)); = i j = − i. dt ∂pi ∂x dt ∂x ∂x ∂pj ∂x It follows

(4.15)

∂S ∂H ∂H ∂H ∂H dH x, = − dt = 0. ∂x ∂xi ∂pi ∂pi ∂xi ∂S =const, which is called the Hamilton–Jacobi equation Consequently, H x, ∂x of Mechanics. This equation determines the function S and the first equation (4.14) gives us the curves γ(t). The Jacobi method suggests to obtain the Hamilton–Jacobi equation by the variational principle.

4.2 Variational problem. Hamilton–Jacobi equations The variational problem for the Hamiltonian systems (T ∗ M, θ, H) is defined as follows: Let us consider a smooth curve c defined on a local chart π ∗−1 (U ) of the cotangent manifold T ∗ M by: c : t ∈ [0, 1] → (x(t); p(t)) ∈ π ∗−1 (U )

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

analytically expressed by xi = xi (t), pi = pi (t), t ∈ [0, 1].

(4.1)

Consider a vector field V i (t) and a covector field ηi (t) on the domain U of the local chart (U, φ) on M , and assume that we have V i (0) = V i (1) = 0, ηi (0) = ηi (1) = 0 (4.2)

dV i dV i (0) = (1) = 0. dt dt

The variations of the curve c determined by the pair (V i (t), ηi (t)) are defined by the curves c(ε1 , ε2 ): xi = xi (t) + ε1 V i (t), (4.3) pi = pi (t) + ε2 ηi (t), t ∈ [0, 1], when ε1 and ε2 are constants, small in absolute value such that the image of any curve c belongs to the open set π ∗−1 (U ) in T ∗ M . The integral of action of Hamiltonian H(x, p) along the curve c is defined by Z 1 dxi I(c) = pi (t) − H(x(t), p(t)) dt. (4.4) dt 0 The integral of action I(c(ε1 , ε2 )) is: i R1 dx dV i I(c(ε1 , ε2 )) = 0 (pi + ε2 ηi (t)) + ε1 − dt dt

(4.5)

−H(x + ε1 V, p + ε2 η)] dt. The necessary conditions in order that I(c) is an extremal value of I(c(ε1 , ε2 )) are ∂I(c(ε1 , ε2 )) ∂I(c(ε1 , ε2 )) = 0, = 0. (4.6) ∂ε1 ∂ε2 ε1 =ε2 =0 ε1 =ε2 =0 Under our conditions of differentiability, the operators

∂ ∂ , and the oper∂ε1 ∂ε2

4.2. VARIATIONAL PROBLEM. HAMILTON–JACOBI EQUATIONS

63

ator of integration commute. Therefore, from (4.5) we deduce: R1 dV i ∂H i dt = 0 0 pi (t) dt (t) − ∂xi V R1 0

i

dx ∂H − dt ∂pi

(4.7)

ηi dt = 0.

Denoting: ◦

dpi ∂H + i (4.8) dt ∂x and noticing the conditions (4.2) one obtain that the equations (4.7) are equivalent to: Z 1 ◦ Z 1 i ∂H dx − ηi dt = 0. (4.9) E i (H)V i dt = 0; dt ∂pi 0 0 E i (H) =

But (V i , ηi ) are arbitrary. Thus, (4.9) lead to the following result: Theorem 4.2.1. In order to the integral of action I(c) to be an extremal value for the functionals I(c) is necessary that the curve c to satisfy the Hamilton–Jacobi equations: dxi ∂H dpi ∂H = , = − i. (4.10) dt ∂pi dt ∂x ◦

The operator E i (H) has a geometrical meaning: ◦

Theorem 4.2.2. E i (H) is a covector field. Proof. With respect to a change of local coordinates (4.1.1) on the manifold T ∗ M , we have Z 1 ◦ Z 1f ◦ e Ve i dt − E i (H) E i (H)V i dt (4.11) 0 # 0 Z 1 "f i ◦ ◦ x j e ∂e E i (H) = − E (4.12) j (H) V dt = 0. j ∂x 0 Since the vector V i is arbitrary, we obtain: ◦ ◦ f ∂e xi e E i (H) j = E j (H). ∂x

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

A consequence of the last theorem: The Hamilton–Jacobi equation have a geometrical meaning on the cotangent manifold T ∗ M . The notion of homogeneity for the Hamiltonian systems is defined in the classical manner [174]. A smooth function f : T ∗ M → R is called r−homogeneous r ∈ Z with respect to the momenta variables pi if we have: LC∗ f = C∗ f = p : ∂˙ i f = rf.

(4.13)

A vector field X ∈ χ(T ∗ M ) is r−homogeneous if LC∗ X = (r − 1)X,

(4.14)

where LC∗ X = [C∗ ; X]. So, we have 1◦

∂ ∂ , = ∂˙ i are 1- and 0-homogeneous. i ∂x ∂pi

2◦ If f ∈ F (T ∗ M ) is s−homogeneous and X ∈ χ(T ∗ M ) is r−homogeneous then f X is r + s−homogeneous. 3◦ C∗ = pi ∂˙ i is 1-homogeneous. 4◦ If X is r−homogeneous and f is s−homogeneous then Xf is r+s−1−homogeneous. ˙ is s − 1−homogeneous. 5◦ If f is s−homogeneous, then ∂˙ i ∂f Analogously, for q−forms ω ∈ Λq (T ∗ M ). The q−form ω is r−homogeneous if LC∗ ω = rω.

(4.15)

It follows: 10 If ω, ω 0 are s− respectively s0 −homogeneous, then ω∧ω 0 is s+s0 −homogeneous. 20 dxi , dpi are 0− respectively 1−homogeneous. 30 The Liouville 1−form ω = pi dxi is 1-homogeneous. 40 The canonical symplectic structure θ is 1-homogeneous.

4.3. NONLINEAR CONNECTIONS

65

4.3 Nonlinear connections On the manifold T ∗ M there exists a remarkable distribution. It is the vertical distribution V . As we know, V is integrable, having (∂˙ 1 , ∂˙ 2 , ..., ∂˙ n ) as a local adapted basis and dimV = n =dimM . Definition 4.3.1. A nonlinear connection N on the manifold T ∗ M is a differentiable distribution N on T ∗ M supplementary to the vertical distribution V : Tu T ∗ M = Nu ⊕ Vu , ∀u ∈ T ∗ M.

(4.1)

N is called the horizontal distribution on T ∗ M . It follows that dimN = n. connection N is given we can consider an adapted basis When a nonlinear δ δ , ..., n expressed in the form δx1 δx δ ∂ ∂ = + N (i = 1, 2, ..., n). ij δxi ∂xi ∂pj

(4.2)

The system of function Nij (x, p) is well determined by (4.1). It is called system of coeﬀicients of the nonlinear connection N . This system defines a geometrical object field on T ∗ M . δ ˙i , ∂ give us an adapted basis to the direct decomThe set of vector fields δxi position (4.1). Its dual adapted basis is (dxi , δpi ), (i = 1, ..., n), where δpi = dpi − Nji dxj .

(4.3)

It is not diﬀicult to prove that if M is a paracompact manifold, then on the cotangent manifold T ∗ M there exists a nonlinear connection. Let N be a nonlinear connection with the coeﬀicients Nij (x, p) and define the set of function τij (x, y) by 1 τij = (Nij − Nji ). (4.4) 2 It is not diﬀicult to see that τij is transformed, with respect to (4.1) as a covariant tensor field on the base manifold M . So, it is a distinguished tensor field, shortly a d−tensor of torsion of the nonlinear connection. τij = 0 has a geometrical meaning. In this case N is a symmetric nonlinear connection.

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

With respect to a symmetric nonlinear connection N the symplectic structure θ can be written in an invariant form: θ = δpi ∧ dxi ,

(4.5)

and the Poisson structure {, } can be expressed in the invariant form: ∂f δg ∂g δf {f, g} = − . (4.6) ∂pi δxi ∂pi δxi Of course, we can consider the curvature tensor of N as the d−tensor field δNji δNhi Rijh = − . (4.7) δxh δxj It is given by δ δ , h = Rijh ∂˙ i . (4.8) j δx δx Therefore Rijh (x, p) is the integrability tensor of the horizontal distribution N . Thus N is an integrable distribution if and only if the curvature tensor Rijh (x, p) vanishes. A curve c : I ⊂ R → c(t) ∈ T ∗ M . Imc ⊂ π ∗ (U ) expressed by (xi = xi (t), dc pi = pi (t), t ∈ I). The tangent vector can be written in the adapted basis as: dt dc dxi δ δpi ∂ = + , dt dt δxi dt ∂pi where dpi dxj δpi = − Nji (x(t), p(t)) . dt dt dt δpi The curve c is horizontal if = 0. We obtain the system of differential equations dt which characterize the horizontal curves: dpi dxj xi = xi (t), − Nji (x, p) = 0. (4.9) dt dt Let gij (x, y) be a d−tensor by (d−means distinguished) with the properties gij = gji and det(gij ) 6= 0 on T ∗ M . Its contravariant g ij (x, y) can be considered, ∂ δ = δi , gij g ts = δis . As usually, we put = ∂˙ and ∂˙i = gij ∂˙ j . So, one can i δx ∂pi ˇ : X (T ∗ M ) → X (T ∗ M ): consider the following F(T ∗ M )−linear mapping F ˇ i ) = −∂˙i , F( ˇ ∂˙i ) = δi , (i = 1, ..., n). F(δ (4.10) It is not diﬀicult to prove, [174]:

4.3. NONLINEAR CONNECTIONS

Theorem 4.3.1.

67

ˇ is globally defined on T ∗ M . 1◦ The mapping F

ˇ is a tensor field of type (1, 1) on T ∗ M . 2◦ F ˇ in the adapted basis (δi , ∂˙ i ) is 3◦ The local expression of F ˇ = −gij ∂˙ i ⊗ dxj + g ij δi ⊗ δpj . F

(4.11)

ˇ is an almost complex structure on T ∗ M determined by N and by gij (x, p), 4◦ F i.e. ˇ◦F ˇ = −I. F (4.12) Also, nonlinear symmetric connection N being considered, we can define the tensor: G(x, p) = gij (x, p)dxi ⊗ dxj + g ij δpi ⊗ δpj . (4.13) If d−tensor gij (x, p) has a constant signature on T ∗ M (for instance it is positively defined), then it follows: Theorem 4.3.2. gij .

1◦ G is a Riemannian structure on T ∗ M determined by N and

2◦ The distributions N and V are orthogonal with respect to G. ˇ is an almost Hermitian structure on T ∗ M . 3◦ The pair (G, F) ˇ is the canonical symplec4◦ The associated almost symplectic structure to (G, F) tic structure θ = δpi ∧ dxi . Remarking that the vector fields (δi , ∂˙ i ) are transformed by (4.1) in the form i ∂xj ∂e xi ˙ j e ˙ e δj , ∂ = j ∂ , δi = ∂e xi ∂x

(4.14)

and the 1-forms (dxi , δpi ) are transformed by ∂xj ∂e xi j pi = δpj , de x = j dx , δe ∂x ∂e xi i

(4.3.13’)

we can consider the horizontal and vertical projectors with respect to the direct decomposition (4.1): δ (4.15) h = i ⊗ dxi , v = ∂˙ i ⊗ δpi . δx

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

They have the properties: h + v = I, h2 = h, v 2 = v, h ◦ v + v ◦ h = 0. We set

hX = X H ,

vX = X V , ∀X ∈ X (T ∗ M )

ω H = ω ◦ h, ω v = ω ◦ v, ∀ω ∈ X ∗ (T ∗ M ). Therefore, for every vector field X on T ∗ M , represented in adapted basis in the form X = X i δi + X˙ i ∂˙ i we have

δ , X V = vX = X˙ i ∂˙ i i δx where the coeﬀicients X i (x, p) and X˙ i (x, p) are transformed by (4.1): X H = hX = X i

∂xj ˙ ∂e xi j f i ˙ e X = j X , Xi = Xj . ∂x ∂e xi For this reason, X i (x, p) are the coeﬀicients of a distinguished vector field and X˙ i (x, p) are the coeﬀicients of a distinguished covector field, shortly denoted by d−vector (covector) fields. Analogously, for the 1-form ω: ω = ωi dxi + ω˙ i δpi . Therefore, ω H = ωi dxi , ω V = ω˙ i δpi . Then ωi (x, p) are component of an one d−form on T ∗ M and ω˙ i (x, p) are the components of a d−vector field on T ∗ M . On the same way, we can define a distinguished (d−) tensor field. A d−tensor field can be represented in adapted basis in the form ...ir T = Tji11...j (x, p)δi1 ⊗ ... ⊗ ∂˙ js ⊗ dxj1 ⊗ ... ⊗ δpr . s

(4.16)

Its coeﬀicients are transformed by (4.1) in the form: ∂e xi1 ∂e xir ∂xk1 ∂xk2 h1 ...hr i1 ...ir e Tj1 ...js (e x, pe) = h ... h ... j Tk1 ...ks (x, p). ∂x 1 ∂x r ∂e xj1 ∂e xs

(4.3.15’)

4.4. N −LINEAR CONNECTIONS

69

...ir So, a d−tensor field T can be given by the coeﬀicients Tji11...j (x, p) whose rule s of transformation, with respect to (4.1) is the same with that of a tensor of the same type on the base manifold M of cotangent bundle (T ∗ M, M, π ∗ ). One can speak of the d−tensor algebra on T ∗ M , which is not diﬀicult to be defined.

4.4 N −linear connections As we know, a nonlinear connection N determines a direct decomposition (4.1) in respect to which we have X = X H + X V , ω = ω H + ω V , ∀X ∈ X (T ∗ M ), ∀ω ∈ X ∗ (T ∗ M ).

(4.1)

Assuming that N is a symmetric nonlinear connection we can give: Definition 4.4.1. A linear connection D on the manifold T ∗ M is called an N −linear connection if: 1◦ D preserves by parallelism the distributions N and V . 2◦ The canonical symplectic structure θ = δpi ∧ dxi has the associate tensor θ = δpi ⊗ dxi parallel with respect to D: Dθ = 0.

(4.2)

It follows that: DX h = DX v = 0 (4.3) DX Y = DX H Y + DX V Y. We obtain new operators of derivations in the algebras of d−tensors, defined by H V DX = DX H , DX = DX V , ∀X ∈ X (T ∗ M ).

(4.4)

We have V H V H f = XV X f = X F f, DX ; DX + DX DX = DX V V H H Y (f Y ) = X V f Y + f DX Y ; DX (f Y ) = X H f Y + f DX DX

(4.5) V H , DfVX = f DX DfHX = f DX V H θ = 0. θ = 0, DX DX

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

The operators DH , DV have the property of localization. They have the similarly properties with covariant derivative but they are not covariant derivative. DH is called h−covariant derivative and DV is called v−covariant derivative. They act on the 1-forms ω on T ∗ M by the rules: H H Y) ω)(Y ) = X H ω(Y ) − ω(DX (DX

(4.6) V ω)(Y (DX

) = X ω(Y ) − V

V ω(DX Y

).

The extension of these operators to the d−tensor fields is immediate. The torsion T of an N −linear connection has the form Π(X, Y ) = DX Y − DY X − [X, Y ].

(4.7)

It is characterized by the following vector fields: T(X H , Y H ), T(X H , Y V ), T (X V , X V ). Taking the h− and v−components we obtain the d−tensor of torsion T (X H , Y H ) = hT(X H , Y H ) + vT(X H , Y H ), etc. Proposition 4.4.1. The d−tensors of torsion of an N −linear connection D are: H H hT(X H , Y H ) = DX Y − DYH X H − [X H , Y H ]H

hT(X H , Y V ) = −DYV X H + [X H , Y V ]H vT(X H , Y H ) = −[X H , Y H ]V

(4.8)

H V vT(X H , Y V ) = DX Y − [X H , Y V ]V V V Y − DYV X V − [X V , Y V ]V = 0. vT(X V , Y V ) = DX

The curvature R of an N −linear connection D is given by R(X, Y )Z = (DX DY − DY DX )Z − D[X,Y ] Z.

(4.9)

Remarking that the vector field R(X, Y )Z H is horizontal one and R(X, Y )Z V is a vertical one, we have v(R(X, Y )Z H ) = 0, h(R(X, Y )Z V ) = 0.

(4.10)

4.4. N −LINEAR CONNECTIONS

71

We will see that the d−tensors of curvature of D are: R(X H , Y H )Z H , R(X H , Y V )Z H , R(X V , Y V )Z H .

(4.11)

By means of (4.9) one obtains: Proposition 4.4.2.

1◦ The Ricci identities of D are [DX , DY ]Z = R(X, Y )Z − D[X,Y ] Z.

(4.12)

2◦ The Bianchi identities are given by X {(DX T)(Y, Z) − R(X, Y )Z + T(T(X, Y ), Z)} = 0, (XY Z)

(4.13)

X

{(DX R)(U, Y, Z) − R(T(X, Y ), Z)U } = 0,

(X,Y,Z)

where

X

means the cyclic sum.

(XY Z)

The previous formulae can be expressed in local coordinates, taking δi , ∂˙ i , as the vectors X, Y, Z, U . But, first of all, we must introduce the local coeﬀicients of an N −linear connection D. In the adapted basis (δi , ∂˙ i ) we take into account the properties: Dδj = DδHj , D∂˙ j = D∂V˙ j . (4.14) Then, the following theorem holds: Theorem 4.4.1. 1◦ An N −linear connection D on T ∗ M can be uniquely represented in adapted basis (δi , ∂˙ i ) in the form: i ˙h Dδj δi = Hijh δh , Dδj ∂˙ i = −Hhj ∂ ,

(4.15) D∂˙ j δi =

Cihj δh ,

˙i

D∂˙ j ∂ =

−Chij ∂˙ h .

i (x, p) transform by the 2◦ With respect to (4.4.1) on T ∗ M the coeﬀicients Hjk rule r s ∂e xi r ∂ 2x ei i ∂x ∂x e Hrs j k = r Hjk − j k (4.16) ∂e x ∂e x ∂x ∂x ∂x and Cijk (x, p) is a d−tensor of type (2, 1).

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

The proof of this theorem is not diﬀicult. Conversely, if N is an a priori given nonlinear connection and a set of functions i Hjk (x, p), Cijk (x, p) on T ∗ M , verifying 2◦ is given, then there exists an unique N −linear connection D with the property (4.16). The action of D on the adapted cobasis (dxi , δpi ) is given by i Dδj dxi = −Hkj dxk , Dδj δpi = Hijk δpk ,

(4.17) i

D∂˙ j dx =

−Ckij dxk ,

D∂˙ j δpi =

Cikj δpk .

i The pair DΓ(N ) = (Hjk , Cijk ) is called the system of coeﬀicients of D. r Let us consider a d−tensor field T with the local coeﬀicients Tji,...,i (x, p), (see 1 ,...,js H i (4.3.15)) and a horizontal vector field X = X = X δi . H By means of previous theorem we obtain for h−covariant derivation DX of tensor T : ...ir H DX T = X k Tji11...j δi1 ⊗ ... ⊗ ∂˙ js ⊗ dxj1 ⊗ ... ⊗ δpir , (4.18) s ,|k

where ...ir ...ir i1 ir 2 ...ir 1 ...m Tji11...j = δk Tji11...j + Tjmi Hmk + ... + Tjir1 ...j Hmk − s 1 ...js s s |k i1 ...ir m −Tm...j Hj 1 k s

− ... −

(4.4.16’)

...ir m Tji11...m Hj s k .

i The operator “|” is called h−covariant derivative with respect to DΓ(N ) = (Hjk , Cijk ). V Now, taking X = X V = Xi ∂˙ i , the v−covariant derivative DX T has the following form ...ir k V DX T = X˙ k Tji11...j δi ⊗ ... ⊗ ∂˙ js ⊗ dxj1 ⊗ ... ⊗ δpir , (4.19) s

where

...ir k ...ir ...m ir k i1 k r Tji11...j = ∂˙ k Tji11...j + Tjm...i Cm + ... + Tji11...j Cm − s s 1 ...js s i1 ...ir mk −Tm...j Cj1 s

− ... −

...ir mk Tji11...m Cjs .

The operator “|” is called the v−covariant derivative. Proposition 4.4.3. The following properties hold: ...ir 1◦ Tji11...j is a d−tensor of type (r, s + 1). s |k

...ir k is a d−tensor of type (r + 1, s). 2◦ Tji11...j s

(4.4.17’)

4.4. N −LINEAR CONNECTIONS

3◦ f|m =

73

δf , f |m = ∂˙ m f . m δx i 4◦ X|ki = δk X i + X m Hmk ; m ωi|k = δk ωi − ωm Hik ;

ik X i |k = ∂˙ k X i + X m Cm ,

(4.20) ω1 |k = ∂˙ k ωi − ωm Cimk .

5◦ The operators “| ” and “|” are distributive with respect to addition and verify the rule of Leibnitz with respect to tensor product of d−tensors. Let us consider the deflection tensors of DΓ(N ): ∆ij = pi|j , δˇij = pi |j .

(4.21)

One gets mj i i ∆ij = Nij − pm Hijm ;δ− j = δ j − pm Ci .

(4.4.19’)

i Proposition 4.4.4. If Hjk (x, p) is a system of differentiable functions, verifying (4.16), then Nij (x, p) given by

Nij = pm Hijm

(4.22)

determine a nonlinear connection in T ∗ M . As in the case of tangent bundle, one can prove that if the base manifold M is paracompact, then on T ∗ M there exist the N −linear connections. Indeed, on M there exists a Riemannian metric gij (x). Then the pair DΓ(N ) = i i (γjk , 0) is an N −linear connection on T ∗ M , γjk (x) being the Christoffel coeﬀicients m and Nij = pm γij , are the coeﬀicients of a nonlinear connection on T ∗ M . In the adapted basis (δi , ∂˙ i ), the Ricci identitites (4.12) can be written in the form i i i m X|j|h − X|h|j = X m Rmi jh − X|m Tjh − X i |m Rmjh h X|ji |h − X i |h|j = X m Pmi jh − X i|m Cjmh − X i |m Pmj

(4.23)

X i |j |h − X i |h |j = X m Smijh − X i |m Smjh , where i i i i i Tjh = Hjh − Hhj , Si jh = Cijh − Cihj , Pjh = Hjh − ∂˙ i Nhj

(4.24)

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

and Rmjh , Cimh are the d−tensors of torsion and Rki jh , Pk ij h , Skijh are the d−tensors of curvature: i m i i m i i − Ck im Rmjh , Hmh − Hkh Hkh Hmj + Hkj Rki jh = δh Hkj − δj Hkh i Pk ij h = ∂˙ h Hkj − Ckih|j + Ckim P hmj , Skijh = ∂˙ h Ckij − ∂˙ j Ckih +

(4.25)

+Ck mj Cmih − Ck mh Cmij . Evidently, these d−tensors of curvature verify: R(δj δh )δk = Rki hj δi ; R(δj , ∂˙ h )δk = Pk hj i δi , (4.26) R(∂˙ j , ∂˙ h )δk = Skihj δi , and

R(δj δh )∂˙ k = Ri khj ∂˙ i ; R(δj , ∂˙ h )∂˙ k = −Pi kj h ∂˙ i , (4.4.23’) ˙j

˙h

˙k

R(∂ , ∂ )∂ = −Si

hkj ˙ j

∂ .

The Bianchi identities (4.13), in adapted basis (δi , ∂˙ i ) can be written without diﬀiculties. Applications: 1◦ For a d− tensor g ij (x, p) the Ricci identities are g ij|k|h − g ij|h|k = g mj Rmi

kh + g

im

j Rmkh − g ij|m T mkh − g ij |m Rmkh ,

h , g ij|k |h − g ij |h|k = g mj Pmi kh + g im Pmj kh − g ij|m Ck mh − g ij |m Pmk

(4.27)

g ij |k |h − g ij |h |k = g mj Smikh + g im Smjkh − g ij |m Smkh . In particular, if g ij is covariant constant with respect to N −connection D, i.e. (4.28) g ij|k = 0, g ij |k = 0, then, from (4.27) we have: g mj Rmi kh + g im Rmj kh = 0, g mj Pmi kh + g im Pmj kh = 0, g mj Smikh + g im Smjkh = 0.

(4.4.25’)

4.5. PARALLELISM, PATHS AND STRUCTURE EQUATIONS

75

Such kind of equations will be used for the N −linear connections compatible with a metric structure G in (4.13). The Ricci identities applied to the Liouville–Hamilton vector field C ∗ = pi ∂˙ i i lead to the important identities, which imply the deflection tensors ∆i , andδ− j. Theorem 4.4.2. Any N −linear connection D on T ∗ M satisfies the following identities: m ∆ij|k − ∆ik|j = −pm Ri mjk − ∆im T mjk −δ− i Rmjk , k m k mk − m i ∆ij |k −δ− (4.29) i |j = −pm Pi j − ∆i mCj −δi P mj , j k mjk − − k j − m jk δi | −δj | = −pm Si −δi Sm . i i One says that are N −linear connection D is of Cartan type if ∆ij = 0,δ− j = δj .

Proposition 4.4.5. Any N −linear connection D of Cartan type satisfies the identities: k pm Rimjk + Rijk = 0, pm Pi m + P kij = 0, pm Si mjk + Si jk = 0. j

(4.30)

Finally, we remark that we can explicitly write, in adapted basis, the Bianchi identities (4.13).

4.5 Parallelism, paths and structure equations Consider the Hamiltonian systems (T ∗ M, H, θ), an N −linear connection DΓ(N ) = i (Hjk , Ci jk ). A curve c : I → T ∗ M , locally represented by xi = xi (t), pi = pi (t), has the tangent vector

(4.1)

dc in adapted basis (δi , ∂˙ i ): dt δpi ˙ i dc dxi = δi + ∂, dt dt dt

(4.2)

δpi dpi dxj with = − Nji . The operator of covariant derivative D along the curve dt dt dt c is dxi δpi H V Dδi + D ˙i . Dc˙ = Dc˙ + Dc˙ = dt dt ∂

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

If X = X i δi + X˙ i ∂˙ i is a vector field then the operator X of covariant differential acts on the vector fields D by the rule: i DX = (dxi + X m ωm )δi + (dX˙ i − X˙ m ωim )∂˙ i ,

(4.3)

i ωji = Hjk dxk + Cjik δpk

(4.4)

where is called the connection 1−form.

DX is: dt ! i i m ˙ DX dX ω ω dXi = + X m m δi + − X˙ m i ∂˙ i , dt dt dt dt dt

Therefore, the covariant differential

(4.5)

dX i dX i where is (x(t), p(t)), t ∈ I and dt dt ωji dxk δpk i = H jk (x(t), p(t)) + Cj ik (x(t), p(t)) . dt dt dt As usually, we introduce:

(4.6)

Definition 4.5.1. The vector field X = X i (x, p)δi + X˙ m (x, p)∂˙ i is parallel, with DX respect to DΓ(N ), along smooth curve c : I → T ∗ M if = 0, ∀t ∈ I. dt From (4.5) one obtains: Theorem 4.5.1. The vector field X = X i δi + X˙ i ∂˙ i is parallel, with respect to the i N −linear connection DΓ(Hjk , Cijk ), along of curve c if, and only if, the functions X i , X˙ i , (i = 1, ..., n) are solutions of the differential equations: i dX i dX˙ i ωim m ωm ˙ +X = 0, − Xm = 0. dt dt dt dt

(4.7)

The proof is immediate, by means of (4.5). A theorem of existence and uniqueness for the parallel vector fields along a given parametrized curve c in the manifold T ∗ M can be formulated in the classical manner. Let us consider the case when a vector field X is absolute parallel on T ∗ M , with respect to an N −linear connection D, i.e. DX = 0 on T ∗ M , [174]. Using the formula (4.3), DX = 0 if and only if i dX i + X m ωm = 0, dX˙ i − X˙ m ωim = 0.

4.5. PARALLELISM, PATHS AND STRUCTURE EQUATIONS

77

Remarking (4.4), and that we have dX i = X|ki dxk + X i |k δpk dX˙ i = X˙ i |k dxk + X˙ i |k δpk it follows that the equation DX = 0 along any curve c on T ∗ M is equivalent to X|ji = 0, X i |j = 0

(4.8)

X˙ i|j = 0, X˙ i |j = 0. The differential consequence of the previous system are given by the Ricci identities (4.26), taken modulo (4.8). They are X h Rhi jk = 0, X h Ph ijk = 0, X h Shijk = 0 (4.9) X˙ h Ri hjk

= 0,

X˙ h Pi hj k

= 0, X˙ h Si

hjk

= 0.

But, {X i , X˙ i } being arbitrary, it follows Theorem 4.5.2. The N −linear connection DΓ(N ) is with the absolute parallelism of vectors if, and only if, the curvature tensor R of D vanishes. Definition 4.5.2. The curve c : I → T ∗ M is called autoparallel for N −linear connection D if Dc˙ · c˙ = 0. Taking into account the fact that c˙ =

dc dxi δpi ˙ i = δi + ∂ , one obtains dt dt dt

Theorem 4.5.3. A curve c : I → T ∗ M is autoparallel with respect to DΓ(N ) if, and only if, the functions xi (t), pi (t), t ∈ I, are solutions of the system of differential equations d2 xi dxs ωsi d δpi δps ωis + = 0, − = 0. dt2 dt dt dt dt dt dt

(4.10)

Of course, a theorem of existence and uniqueness for the autoparallel curves can be formulated. Definition 4.5.3. An horizontal path of DΓ(N ) is an horizontal autoparallel curve.

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

Theorem 4.5.4. The horizontal paths of DΓ(N ) are characterized by the differential equations d 2 xi dxj dxk i = 0. (4.11) + Hjk (x, p) dt2 dt dt The curve c : I → T ∗ M is vertical at the point xi0 = xi (t0 ), t0 ∈ I of M if c(t ˙ 0 ) ∈ V (vertical distribution). A vertical path at the point (xi0 ) ∈ M is a vertical autoparallel curve at point (xi0 ). Theorem 4.5.5. The vertical paths at a point x0 = (xi0 ) ∈ M with respect to DΓ(N ) are solutions of the differential equations d 2 pi dpj dpk jk − C (x , p) = 0, xi = xi0 . 0 i 2 dt dt dt

(4.12)

i Now, we can write the structure equations of DΓ(N ) = (Hjk , Ci jk ), remarking that the following geometric objects are a d−vector field and a d−covector field, respectively: i d(dxi ) − dxm ∧ ωm ; dδpi + δpm ∧ ωim i is a d−tensor of type (1,1). Here d is the operator of exterior and dωji − ωjm ∧ ωm differential. By a straightforward calculus we can prove:

Theorem 4.5.6. For any N −linear connection D with the coeﬀicients DΓ(N ) = i (Hjk , Cijk ) the following structure equations hold: i d(dxi ) − dxm ∧ ωm = −Ωi

d(δpi ) + δpm ∧ ωim = −Ω˙ i

(4.13)

i dωji − ωjm ∧ ωm = −Ωij ,

where Ωi , Ω˙ i are 2−forms of torsion: 1 Ωi = T ijk dxj ∧ dxk + Cjik dxj ∧ δpk , 2 1 1 Ω˙ i = Rijk dxj ∧ dxk + P kij dxj ∧ δpk + Si jk δpj ∧ δpk , 2 2

(4.14)

4.5. PARALLELISM, PATHS AND STRUCTURE EQUATIONS

79

and when Ωij is 2-forms of curvature: 1 1 Ωij = Rj i km dxk ∧ dxm + Pj ik m dxk ∧ δpm + Si jkm δpk ∧ δpm . 2 2

(4.15)

Remarks. 1◦ The torsion and curvature d−tensor forms (4.14) and (4.15) are expressed in Section 4 of this chapter. 2◦ The Bianchi identities can be obtained exterior differentiating the structure equations modulo the same equations.

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CHAPTER 4. THE GEOMETRY OF COTANGENT MANIFOLD

Chapter 5 Hamilton spaces The notion of Hamilton space was defined and investigated by R. Miron in the papers [154], [174], [175]. It was studied by D. Hrimiuc [109], H. Shimada [223] et al. It was applied by P.L. Antonelli, S. Vacaru, D. Bao et al. [251]. The geometry of these spaces is the geometry of a Hamiltonian system (T ∗ M, θ, H), where H(x, p) is a fundamental function of an Hamilton space H n = (M, H(x, p)). Consequently, we can apply the geometric theory of cotangent manifold T ∗ M presented in the previous chapter, establishing the all fundamental geometric objects of the spaces H n. As we see, the geometry of Hamilton spaces can be studied as dual geometry, via Legendre transformation of the geometry of Lagrange spaces. In this chapter we study the geometry of Hamilton spaces, combining these two methods. So, we defined the notion of Hamilton space H n , the canonical nonlinear connection N of H n , the metrical structure and the canonical metrical linear connection of H n . The fundamental equations of H n are the Hamilton–Jacobi equations. The notion of parallelism with respect to a N −metrical connection and its consequences are studied. As applications we study the notion of Hamilton spaces of the electrodynamics. Also the almost Kählerian model of the spaces H n is pointed out.

5.1 Notion of Hamilton space Let M be a real differential manifold of dimension n and (T ∗ M, π ∗ , M ) its cotangent bundle. A differentiable Hamiltonian is a mapping H : T ∗ M → R differen∗ M = T ∗ M \ {0} and continuous on the null section. tiable on T] The Hessian of H, with respect to the momenta variable pi of differential Hamil81

82

CHAPTER 5. HAMILTON SPACES

tonian H(x, p) has the components 1 ∂ 2H . g = 2 ∂pi ∂pj ij

(5.1)

Of course, the Latin indices i, j, ... run on the set {1, ..., n}. The set of functions g ij (x, p) determine a symmetric contravariant of order 2 ∗ tensor field on Tg M . H(x, p) is called regular if ∗M . det(g ij (x, p)) 6= 0 on T]

(5.2)

Now, we can give Definition 5.1.1. A Hamilton space is a pair H n = (M, H(x, p)) where M is a smooth real, n−dimensional manifold, and H is a function on T ∗ M having the properties ∗ M and it 1◦ H : (x, p) ∈ T ∗ M → H(x, p) ∈ R is a differentiable function on T] is continuous on the null section of the natural projection π ∗ T ∗ M → M .

2◦ The Hessian of H with respect to momenta pi given by kg ij (x, p)k, (5.1) is ∗M . nondegenerate i.e. (5.2) is verified on T] ∗M . 3◦ The d−tensor field g ij (x, p) has constant signature on the manifold T]

One can say that the Hamilton space H n = (M, H(x, p)) has as fundamental function a differentiable regular Hamiltonian for which its fundamental or metric tensor g ij (x, p) is nonsingular and has constant signature. Examples 1◦ If Rn = (M, gij (x)) is a Riemannian space then H n = (M, H(x, p)) with H(x, p) =

1 ij g (x)pi pj mc

(5.3)

is an Hamilton space. 2◦ The Hamilton space of electrodynamics is defined by the fundamental function e2 i 1 ij 2e i (5.4) H(x, p) = g (x)pi pj − A (x)pi − A (x)Ai (x), mc mc2 mc2 where m, c, e are the known physical constants, gij (x) is a pseudo-Riemannian tensor and Ai (x) are a d−covector (electromagnetic potentials) and Ai (x) = g ij Aj .

5.1. NOTION OF HAMILTON SPACE

83

Remark 5.1.0.1. The kinetic energy for a Riemannian metric Rn = (M, gij (x)) is given by 1 T (x, x) ˙ = gij (x)x˙ i x˙ j 2 and for the Hamilton space H n = (M, H(x, p)) with H(x, p) = g ij (x)pi pj it is 1 T ∗ (x, p) = g ij pi pj . 2 Therefore, generally, for an Hamilton space H n = (M, H(x, p)) is convenient to introduce the energy given by the Hamiltonian 1 H(x, p) = H(x, p). 2

(5.5)

∂˙ i ∂˙ j H = g ij (x, p).

(5.6)

One obtain: Consider the Hamiltonian system (T ∗ M, θ, H), where θ is the natural symplectic structure on T ∗ M, θ = dpi ∧ dxi (Ch. 4). The isomorphism Sθ , defined by (4.1.7) can be used and Theorem 4.2 can be applied. One obtain Theorem 5.1.1. For any Hamilton space H n = (M, H(x, p)) the following properties hold: 1◦ There exists a unique vector field XH ∈ X (T ∗ M ) for which iH θ = −dH.

(5.7)

∂H ∂ ∂H ∂ − . ∂pi ∂xi ∂xi ∂pi

(5.8)

2◦ XH is expressed by XH =

3◦ The integral curves of XH are given by the Hamilton–Jacobi equations ∂H dpi ∂H dxi = ; = − i. dt ∂pi dt ∂x Since

dH = {H, H} = 0, dt

(5.9)

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CHAPTER 5. HAMILTON SPACES

dH = 0. Then: The fundamental function H(x, p) of a Hamilton space dt H n is constant along the integral curves of the Hamilton - Jacobi equations (5.9). The Jacobi method of integration of (5.9) expound in chapter 4 is applicable and the variational problem, in §4.2, ch. 4, can be formulated again in the case of Hamiltonian systems (T ∗ M, θ, H). it follows

5.2 Nonlinear connection of a Hamilton space Let us consider a Hamilton space H n ⊂ (M, H(x, p)). The theory of nonlinear connection on the manifold T ∗ M , in ch. 4, can be applied in the case of spaces H n . We must determine a nonlinear connection N which depend only by the Hamilton space H n , i.e. N must be canonical related to H n , as in the case of canonical nonlinear connection from the Lagrange spaces. To do this, direct method is given by using the Legendre transformation, suggested by Mechanics. We will present, in the end of this chapter, this method. Now, following a result of R. Miron we enounce without demonstration (which can be found in ch. 2, §2.3) the following result: Theorem 5.2.1.

1◦ The following set of functions ∂ 2H ∂ 2H 1 1 Nij = {gij , H} − gik + gjk 4 4 ∂pk ∂xj ∂pk ∂xi

(5.1)

are the coeﬀicients of a nonlinear connection of the Hamilton space H n = (M, H(x, p)). 2◦ The nonlinear connection N with coeﬀicients Nij , (5.1) depends only on the fundamental function H(x, p). The brackets {, } from (5.1) are the Poisson brackets (, ), §5.1. Indeed, by a straightforward computation, it follows that, under a coordinate change ( ), Nij (x, p) from (5.1) obeys the rule of transformation of the coeﬀicients of a nonlinear connection N . Evidently Nij depend on the fundamental function H(x, p) only. N − will be called the canonical nonlinear connection of the Hamilton space H n . Remark 5.2.1.1. If the fundamental function H(x, p) of H n is globally defined, on T ∗ M then the horizontal distribution determined by the canonical nonlinear connection N has the same property.

5.3. THE CANONICAL METRICAL CONNECTION OF HAMILTON SPACE H N

85

It is not diﬀicult to prove: Proposition 5.2.1. The canonical nonlinear connection N has the properties 1 τij = (Nij − Nji ) = 0 (it is symmetric) 2

(5.2)

Rijk + Rjki + Rkij = 0

(5.3)

Taking into account that N is a horizontal distribution, we have the known direct decomposition: ∗ M = N ⊕ V , ∀u ∈ T ∗M , ] Tu T] u u

(5.4)

it follows that (δi , ∂˙ i ) is an adapted basis to the previous splitting, where δi = ∂i − Nij ∂˙ j

(5.5)

and the dual basis (dxi , δpi ) is: δpi = dpi − Nij dxj .

(5.2.5’)

Therefore we apply the theory to investigate the notion of metric N −linear connection determined only by the Hamilton space H n .

5.3 The canonical metrical connection of Hamilton space Hn i Let us consider the N −linear connection DΓ(N ) = (Hjk , Cijk ) for which N is the canonical nonlinear connection. By means of theory from chapter 4 we can prove:

Theorem 5.3.1. In a Hamilton space H n = (M, H(x, p)) there exists a i unique N −linear connection DΓ(N ) = (Hjk , Cijk ) verifying the axioms: 1◦ N is the canonical nonlinear connection. 2◦ The fundamental tensor g ij is h−covariant constant ij j i = 0. g|k = δk g ij + g sj Hsk + g is Hsk

(5.1)

3◦ The fundamental tensor g ij is v−covariant constant g ij |k = ∂˙ k C ij + g sj Csik + g is Csjk = 0.

(5.2)

86

CHAPTER 5. HAMILTON SPACES

4 DΓ(N ) is h−torsion free: i i i Tjk = Hjk − Hkj =0

(5.3)

5◦ DΓ(N ) is v−torsion free: Sijk = Cijk − Cikj = 0. 2) The N −connectiom DΓ(N ) which verify the previous axioms has the coefi i ficients Hjk and Cijk given by the coeﬀicients Hjk and Cijk given by the following generalized Christoffel symbols: 1 i Hjk = g is (δj gsk + δk gjs − δs gjk ), 2

(5.4)

1 Cijk = − gis (∂˙ j g sk + ∂˙ k g js − ∂ s g jk ). 2 Clearly CΓ(N ) with the coeﬀicients (5.4) is determined only by means of the Hamilton space H n . It is called canonical metrical N −linear connection. Now, applying theorems from ch. 4 we have: Theorem 5.3.2. With respect to CΓ(N ) we have the Ricci identities: i i X|j|k − X|k|j = X m Rmi

jk

− X i |m Rmjk ,

i Cj mk − X i |m P X|ji |k − X i |k|j = X m Pmi jk − X|m

k mj

(5.5)

X i |j |k − X i |k |j = X m Smijk , where the d−tensors of torsion Rijk , P ijk and d tensors of curvature Rj i kh , Pi jh k , Sijkh are expressed in chapter 4. The structure equations, parallelism, autoparallel curves of the Hamilton spaces are studied exactly as in chapter 4. Example. The Hamilton space of electrodynamics H n = (M, H(x, p)) where the 1 ij fundamental function is expressed in (5.4). The fundamental tensor is g (x). mc The canonical nonlinear connection has the coeﬀicients e (5.6) Nij = γijh (x)ph + (Ai|k + Ak|i ) c

5.4. GENERALIZED HAMILTON SPACES GH N

87

where γijh (x) are the Christoffel symbols of the covariant tensor of metric tensor s g ij (x). Evidently Ai|k = ∂k Ai − As γik . δgij ∂gij Remarking that k = we deduce that: the coeﬀicients of canonical metδx ∂xk rical N −connection HΓ(N ) are: i i Hjk (x) = γjk (x), Cijk = 0. i These geometrical object fields H, gij , Hjk , Cijk allow to develop the geometry of the Hamilton spaces of electrodynamics.

5.4 Generalized Hamilton Spaces GH n A straightforward generalization of the notion of Hamilton space is that of generalized Hamilton space. Definition 5.4.1. Ageneralized Hamilton space is a pair GH n = (M, g ij (x, p)) where M is a smooth real n−dimensional manifold and g ij (x, p) is a d−tensor ∗ M of type (0, 2) symmetric, nondegenerate and of constant signature. field on T] The tensor g ij (x, p) is called fundamental. If M a paracompact on T ∗ M there exist the tensors g ij (x, p) which determine a generalized Hamilton space. From the definition 5.1.1 it follows that: Any Hamilton space H n = (M, H) is a generalized Hamilton space. ◦ ij

The contrary aﬀirmation is not true. Indeed, if g (x) is a Riemannian tensor metric on M , then ◦ ij

g ij (x, p) = e−2σ(x,p) g (x), σ ∈ F (T ∗ M ) determine a generalized Hamilton space and g ij (x, p) does not the fundamental tensor of an Hamilton space. So, it is legitime the following definition: Definition 5.4.2. A generalized Hamilton space GH n = (M, g ij (x, p)) is called ∗M reducible to a Hamilton space if there exists a Hamilton function H(x, p) on T] such that 1 (5.1) g ij (x, p) = ∂˙ i ∂˙ j H. 2

88

CHAPTER 5. HAMILTON SPACES

Let us consider the Cartan tensor: 1 C ijk = ∂˙ i g jk . 2

(5.2)

In a similar manner with the case of generalized Lagrange space (ch. 2, §2.8) we have: Proposition 5.4.1. A necessary condition that a generalized Hamilton space GH n = (M, g ij (x, p)) be reducible to a Hamilton one is that the Cartan tensor C ijk (x, p) be totally symmetric. There exists a particular case when the previous condition is suﬀicient, too. Theorem 5.4.1. A generalized Hamilton space GH n = (M, g ij (x, p)) for which the fundamental tensor g ij (x, p) is 0-homogeneous is reducible to a Hamilton space, if and only if the Cartan tensor C ijk (x, p) is totally symmetric. Indeed, in this case H(x, p) = g ij (x, p)pi pj is a solution of the equation (5.1). Let gij (x, p) be the covariant of fundamental tensor g ij (x, p), then the following tensor field 1 Cijk = − gis (∂˙ j g sk + ∂˙ k g js − ∂˙ s g sk ) (5.3) 2 determine the coeﬀicients of a v−covariant derivative, which is metrical, i.e. g ij |k = ∂˙ k g ij + Csik g sj + Csjk g is = 0.

(5.4)

The proof is very simple. We use this v−derivation in the theory of canonical metrical connection of the spaces GH n . In general, we cannot determine a nonlinear connection from the fundamental tensor g ij of GH n . Therefore we study the geometry of spaces GH n when a nonlinear connection N is a priori given. In this case we can apply the methods used in the construction of geometry of spaces H n . Finally we remark that the class of spaces GH n include the class of spaces H n : {GH n } ⊃ {H n }.

(5.5)

5.5 The almost Kählerian model of a Hamilton space Let H n = (M, H(x, p)) be a Hamilton space and g ij (x, p) its fundamental tensor.

5.5. THE ALMOST KÄHLERIAN MODEL OF A HAMILTON SPACE

89

The canonical nonlinear connection N has the coeﬀicients (5.1). The adapted δ j i ˙ ˙ basis to the distributions N and V is = δi = ∂i + Nij ∂ , ∂ and its dual δxi basis (dxi , δpi = dpi − Nji dxj ). ∗ M = T ∗ M \ {0} can Thus, the following tensor on the cotangent manifold T] be considered: G = gij (x, p)dxi ⊗ dxj + g ij δpi ⊗ δpj . (5.1) ∗ M . If the fundamental tensor G determine a pseudo-Riemannian structure on T] g ij (x, p) is positive defined, then G is a Riemannian structure on T ∗ M . G is called the N −lift of the fundamental tensor g ij . Clearly, G is determined by Hamilton space H N , only. Some properties of G:

1◦ G is uniquely determined by g ij and Nij . 2◦ The distributions N and V are orthogonal. ∗ M ) → X (T ∗ M ) defined in (4.3.10) ˇ : X (T] Taking into account the mapping F or, equivalently by: ˇ = −gij ∂˙ i ⊗ dxj + g ij δi ⊗ δpj , F (5.2)

one obtain: Theorem 5.5.1.

∗M . ˇ is globally defined on the manifold T] 1◦ F

∗M : ˇ is an almost complex structure on T] 2◦ F

ˇ·F ˇ = −I. F

(5.3)

ˇ depends on the Hamilton spaces H n only. 3◦ F Finally, one obtain a particular form of the Theorem 4.3.2: Theorem 5.5.2. The following properties hold: ∗M . ˇ is an almost Hermitian structure on the manifold T] 1◦ The pair (G, F)

2◦ The structure (G, F) is determined only by the fundamental function H(x, p) of the Hamilton space H n . ˇ is the canonical symplec3◦ The associated almost symplectic structure to (G, F) tic structure θ = dpi ∧ dxi = δpi ∧ dxi .

90

CHAPTER 5. HAMILTON SPACES

∗ M , G, F) ˇ is almost Kählerian. 4◦ The space (T]

The proof is similar with that for Lagrange space (cf. Ch. 3). ∗ M , G, F) ˇ is called the almost Kählerian model of the The space H2 n = (T] Hamilton space H n . By means of H2n we can realize the study of gravitational ∗M . and electromagnetic fields [176], [182], [183], [185], on T]

Chapter 6 Cartan spaces A particular class of Hamilton space is given by the class of Cartan spaces. It is formed by the spaces H n = (M, H(x, p)) for which the fundamental function H is 2−homogeneous with respect to momenta pi . It is remarkable that these spaces appear as dual of the Finsler spaces, via Legendre transformations. Using this duality, several important results in Cartan spaces can be obtained: the canonical nonlinear connection, the canonical metrical connection etc. Therefore, the theory of Cartan spaces has the same symmetry and beauty like Finsler geometry. Moreover, it gives a geometrical framework for the Hamiltonian Mechanics or Physics fields. The modern formulation of the notion of Cartan space is due of R. Miron, but its geometry is based on the investigations of E. Cartan, A. Kawaguchi, H. Rund, R. Miron, D. Hrimiuc, and H. Shimada, P.L. Antonelli, S. Vacaru et. al. This concept is different from the notion of areal space defined by E. Cartan. In the final part of this chapter we shortly present the notion of duality between Lagrange and Hamilton spaces.

6.1 Notion of Cartan space Definition 6.1.1. A Cartan space is a pair C n = (M, K(x, p)) where M is a real n−dimensional smooth manifold and K : T ∗ M → R is a scalar function which satisfies the following axioms: ∗ M and continuous on the null section of π ∗ : T ∗ M → 1. K is differentiable on T] M.

2. K is positive on the manifold T ∗ M . 91

92

CHAPTER 6. CARTAN SPACES

3. K is positive 1−homogeneous with respect to the momenta pi . 4. The Hessian of K 2 having the components 1 g ij (x, p) = ∂˙ i ∂˙ j K 2 2

(6.1)

∗M . is positive defined on the manifold T]

It follows that g ij (x, p) is a symmetric and nonsingular d−tensor field of type (0, 2). So, we have

∗M . rankkg ij (y, p)k = n on T]

(6.2)

The functions g ij (x, p) are 0−homogeneous with respect to momenta pi . For a Cartan space C n = (M, K(x, p)) the function K is called fundamental function and g ij the fundamental or metric tensor. If the base M is paracompact, then on the manifold T ∗ M there exists real function K such that the pair (M, K) is a Cartan space. Indeed, on M there exists a Riemann structure gij (x), x ∈ M . Then K(x, p) = {g ij (x)pi pj }1/2

(6.3)

determine a Cartan space. Other examples are given by K = α∗ + β ∗

(6.4)

(α∗ )2 K= β

(6.5)

α∗ = {g ij (x)pi pj }1/2 , β ∗ = bi (x)pi .

(6.6)

where K from (6.4) is called a Randers metric and K from (6.5) is called a Kropina metric. A first and immediate result: Theorem 6.1.1. If C n = (M, K) is a Cartan space then the pair HCn = (M, K 2 ) is an Hamilton space.

6.1. NOTION OF CARTAN SPACE

93

HCn is called the associate Hamilton space with C n . This is the reason that the geometry of Cartan space include the geometry of associate Hamilton space. So, we have the sequence of inclusions {Rn } ⊂ {C n } ⊂ {H n } ⊂ {GH n }

(6.7)

where {Rn } is the class of Riemann spaces Rn = (M, g ij (x)) which give the Cartan spaces with the metric (6.3). Now we can apply the theory from previous chapter. The canonical symplectic structure θ on T ∗ M : θ = dpi ∧ dxi

(6.8)

and C n determine the Hamiltonian system (T ∗ M, θ, K 2 ). Then, setting 1 K = K 2, 2

(6.9)

we have: Theorem 6.1.2. For any Cartan space C n = (M, K(x, p)), the following properties hold: ∗ M for which 1◦ There exists a unique vector field XK 2 on T]

iXK θ = −dK.

(6.10)

2◦ The vector field XK is expressed by XK =

∂K ∂ ∂K ∂ − . ∂pi ∂xi ∂xi ∂pi

(6.11)

3◦ The integral curves of the vector field XK are given by the Hamilton–Jacobi equations: dxi ∂K dpi ∂K = , = − i. (6.12) dt ∂pi dt ∂x One deduce

dK = {K, K} = 0. So: dt

1 2 K of a Cartan space is 2 constant along the integral curves of the Hamilton–Jacobi equations (6.12).

Proposition 6.1.1. The fundamental function K =

94

CHAPTER 6. CARTAN SPACES

The Jacobi method of integration of (6.12) can be applied. The Hamilton–Jacobi equations of a Cartan space C n are fundamental for the geometry of C n . Therefore, the integral curves of the system of differential equations (6.12) are called the geodesics of Cartan space C n . Other properties of the space C n : 1 1◦ pi = ∂˙ i K 2 is a 1-homogeneous d−vector field. 2 1 2◦ g ij = ∂˙ j pi = ∂˙ i ∂˙ j K 2 is 0-homogeneous tensor field (the fundamental tensor 2 of C n ). 1 3◦ C ijh = − ∂˙ i ∂˙ j ∂˙ h K 2 is (−1)−homogeneous with respect to pi . 4 Proposition 6.1.2. We have the following identities: pi = g ij pj , pi = gij pj ,

(6.13)

K 2 = g ij pi pj = pi pi ,

(6.14)

C ijh ph = C ihj ph = C hij ph = 0.

(6.15)

Proposition 6.1.3. The Cartan space C n = (M, K(x, p)) is Riemannian if and only if the d−tensor C ijh vanishes. Consider d−tensor: 1 Cijh = − gis ∂˙ s g jh = gis C sjh . 2

(6.16)

Thus Cijh are the coeﬀicients of a v−metric covariant derivation: g ij |h = 0, Sijh = 0 pi | = k

(6.17)

δik .

6.2 Canonical nonlinear connection of C n The canonical nonlinear connection of a Cartan space C n = (M, K) is the canonical nonlinear connection of the associate Hamilton space HCn = (M, K 2 ). Its coeﬀicients Nij are given by (5.2.1).

6.2. CANONICAL NONLINEAR CONNECTION OF C N

95

Setting 1 i γjh = g is (∂j gsh + ∂h gjs − ∂s gjh ) (6.1) 2 for the Christoffel symbols of the covariant of fundamental tensor of C n and using the notations 0 i 0 i γjh = γjh pi , γj0 = γjh pi ph , (6.2) we obtain: Theorem 6.2.1 (Miron). The canonical nonlinear connection of the Cartan space C n = (M, K(x, p)) has the coeﬀicients 1 0 ˙h Nij = γij0 − γh0 ∂ gij . 2

(6.3)

The proof is based on the formula (5.2.1). Evidently, the canonical nonlinear connection is symmetric Nij = Nji .

(6.4)

Let us consider the adapted basis (δi , ∂˙ i ) to the distributions N (determined by canonical nonlinear connection) and V (vertical distribution). We have δi = ∂i + Nij ∂˙ j .

(6.5)

The adapted dual basis (dxi , δpi ) has the forms δpi : δpi = dpi − Nij dxj .

(6.2.5’)

The d−tensor of integrability of horizontal distribution N is Rijh = δh Nji − δj Nhi .

(6.6)

By a direct calculus we obtain Rijh + Rjki + Rkij = 0.

(6.7)

Proposition 6.2.1. The horizontal distribution N determined by the canonical nonlinear connection of a Cartan space C n is integrable if and only if the d−tensor field Rijk vanishes. Proposition 6.2.2. The canonical nonlinear connection of a Cartan space C n (M, K(x, p)) depends only on the fundamental function K(x, p).

96

CHAPTER 6. CARTAN SPACES

6.3 Canonical metrical connection of C n Consider the canonical metrical N −linear connection of the Hamilton space HeN = (M, K 2 ). It is the canonical metrical N −linear connection of the Cartan space i C n = (M, K). It is denoted by CΓ(N ) = (Hjk , Cijk ) and shortly is named canonical metrical connection of C n . Then, theorem 5.3.1. implies: Theorem 6.3.1. 1) In a Cartan space C n = (M, K(x, p)) there exists a unique i N −linear connection CΓ(N ) = (Hjk , Cijk ) verifying the axioms: 1◦ N is the canonical nonlinear connection of C n . 2◦ CΓ(N ) is h−metrical ij g|h = 0.

(6.1)

g ij |h = 0.

(6.2)

3◦ CΓ(N ) is v−metrical: i i 4◦ CΓ(N ) is h−torsion free: T ijh = Hjh − Hjh = 0.

5◦ CΓ(N ) is v−torsion free: Si jh = Cijh − Cihj = 0. 2) The connection CΓ(N ) has the coeﬀicients given by the generalized Christoffel symbols: 1 i Hjh = g is (δj gsh + δh gjs − δs gjh ) 2 (6.3) 1 Cijh = − gis (∂˙ j g sh + ∂˙ h g js − ∂˙ s g jh ). 2 3) CΓ(N ) depends only on the fundamental function K of the Cartan space C n . The connection CΓ(N ) is called canonical metrical connection of Cartan space C n . i We denote CΓ = (Nij , Hjk , Cijk ). Evidently, the d−tensor Cijh has the properties (6.15) and the coeﬀicients CΓ have 1, 0, −1 homogeneity degrees.

Proposition 6.3.1. The canonical metrical connections has the tensor of deflection: i i i ∆ij = pi|j = 0, δ− (6.4) j = pj | = δj .

6.3. CANONICAL METRICAL CONNECTION OF C N

97

But, the previous result allows to give a characterization of CΓ by a system of axioms of Matsumoto type: Theorem 6.3.2. 1◦ For any space C n = (M, K(x, p)) there exists a unique lini ear connection CΓ = (Nij , Hjk , Cijk ) on the manifold Tg M verifying the following axioms: ij i 10 . ∆ij = 0; 20 . g|h = 0; 30 . g ij |h = 0; 40 . Tjh = 0; 50 . Sijh = 0.

2◦ The previous metrical connection is exactly the canonical metrical connection CΓ. The following properties of CΓ(N ) are immediately ph 1 K|h = 0, K| = ; K ◦

h

2 2◦ K|h = 2ph ;

3◦ pi|j = 0, pi |j = δij ; 4◦ pi|j = 0, pi |j = g ij . And the d−tensors of torsion of CΓ(N ) are i i i Rijh = δh Nij − δj Nih , Cijh , Tjh = 0, Sijh = 0, Pjh = Hjh − ∂˙ i Njh .

(6.5)

Of course, we have i i Rijh = −Rihj , Pjh = Phj , Cijh = Cihj .

(6.6)

Proposition 6.3.2. The d−tensor of curvature Skijh is given by: ij ih Skijh = Ckmh Cm − Ckmj Cm .

(6.7)

Denoting by Rijkh = g is Rsj kh , etc. and applying the Ricci identities (5.3.5), one obtains: Theorem 6.3.3. The canonical metrical connection CΓ(N ) of the Cartan space C n satisfies the identities: Rijkh + Rjikh = 0, P ijk h + P jik

h

= 0, S ijkh + S jikh = 0.

(6.8)

Ri ojk + Rijk = 0, Pi oj k + P kij = 0, Siojk = 0.

(6.9)

Rojk = 0, P koj = 0.

(6.10)

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CHAPTER 6. CARTAN SPACES

Of course, the index “o” means the contraction by pi or pi . The 1-form connections of CΓ(N ) are: ω i j = H ijh dxh + Cjih δph .

(6.11)

Taking into account (6.2.5’), one obtains Theorem 6.3.4. The structure equations of the canonical metrical connection CΓ(N ) of the Cartan space C n = (M, K(x, p)) are i d(dxi ) − dxm ∧ ωm = −Ωi ; ◦

d(δpi ) + δpm ∧ ωim = −Ωi ;

(6.12)

i dωji − ωjm ∧ ωm = −Ωij , ◦

Ωi , Ωi being the 2-forms of torsion: Ωi = Cjik dxj ∧ δpk ◦

1 Ωi = Rijk dxj ∧ dxk + P kij dxj ∧ δpk 2

(6.13)

and Ωij is 2-form of curvature: 1 Ωij = Rji 2

km dx

k

1 ∧ dxm + Pj ik m dxk ∧ δpm + Sj ikm δpk ∧ δpm . 2

(6.14)

Applying Proposition 4.4.2, we determine the Bianchi identities of CΓ(N ). Now, we can develop the geometry of the associated Hamilton space HCn = (M, K 2 (x, p)). Also, in the case of Cartan space, the geometrical model K2n = ∗ M , G, F) is an almost Kählerian one. (T] Before finish this chapter is opportune to say some words on the Legendre transformation.

6.4 The duality between Lagrange and Hamilton spaces The duality, via Legendre transformation, between Lagrange and Hamilton space was formulated by R. Miron in the papers and it was developed by D. Hrimiuc, P.L. Antonelli, D. Bao, et al. Of course, it was suggested by Theoretical Mechanics.

6.4. THE DUALITY BETWEEN LAGRANGE AND HAMILTON SPACES

99

The theory of Legendre duality is presented here follows the Chapter 7 of the book [174]. 1 Let L be a regular Lagrangian, L = L on a domain D ⊂ T M and let H be a 2 1 regular Hamiltonian H = H, on a domain D∗ ⊂ T ∗ M . 2 ∂ ∂ Hence for ∂˙i = i , ∂˙ i = the matrices with entries: ∂y ∂pi gij (x, y) = ∂˙i ∂˙j L(x, y), (6.1) g ∗ij (x, p) = ∂˙ i ∂˙ j H(x, p) are nondegenerate on D and on D∗ , (i, j, ... = 1, 2, ..., n). Since L ∈ F (D) is a differentiable map consider the fiber derivative of L locally given by φ(x, y) = (xi , ∂˙j L(x, y)) (6.2) which is called the Legendre transformation. It is easy to see that: L is a regular Lagrangian if and only if φ is a local diffeomorphism, [174]. In the same manner, for H ∈ FD∗ , the fiber derivative is locally given by ψ(x, y) = (xi , ∂˙ i H(x, y)),

(6.3)

which is a local diffeomorphism if and only if H is regular. 1 Now, let us consider L(x, y) = L(x, y) the fundamental function of a Lagrange 2 space. Then φ defined by (6.2) is a diffeomorphism between two open set U ⊂ D and U ∗ ⊂ D∗ . In this case, we can define H∗ (x, p) = pi y i − L(x, y),

(6.4)

where y = (y i ) is solution of the equation pi = ∂˙i L(x, y).

(6.4.4’)

Remark 6.4.0.1. In the theory of Lagrange space the function E(x, p) = y i ∂˙i L − L(x, y) is the energy of Lagrange space Ln . It follows without diﬀiculties that H ∗n = (M, H ∗ (x, p)), H ∗ = 2H∗ is a Hamilton space. Its fundamental tensor g ∗ij (x, p) is given by g ij (x, φ−1 (x, p)). We set H ∗n = Leg Ln and say that H ∗n is the dual of Ln via Legendre transformation determined by Ln .

100

CHAPTER 6. CARTAN SPACES

Remark 6.4.0.2. The mapping Leg was used in chapter 2 to transform the Euler– Lagrange equations (ch. 2, part I) into the Hamilton–Jacobi equations (ch. 2, part I). 1 Analogously, for a Hamilton space H n = (M, H(x, p)), H = H, consider the 2 function L∗ (x, y) = pi y j − H(x, p), where p = (pi ) is the solution of the equation (6.4). Thus L∗n = (M, L∗ (x, y)), L∗ = 2L∗ , is a Lagrange space, dual, via Legendre transformation of the Hamilton space H n . So, L∗n = Leg H n . One proves: Leg(Leg Ln ) = Ln , Leg(Leg H n ) = H n . The diffeomorphisms φ and ψ have the property φ = ψ −1 . And they transform the fundamental object fields from Ln in the fundamental object fields of H ∗n = Leg Ln , and conversely. For instance, the Euler–Lagrange equation of Ln are transformed in the Hamilton–Jacobi equations of H n . The canonical nonlinear connection of Ln is transformed in the canonical nonlinear connection of H ∗n etc. Examples. 1◦ The Lagrange space of Electrodynamics Ln0 = (M, L0 (x, y)): L0 (x, y) = mcγij (x)y i y j +

2e Ai (x)y i m

where γij (x) is a pseudo-Riemannian metric, has the Legendre transformation: φ : xi = xi , p i =

1 ∂L0 e = mcγ (x)y + Ai (x) ij 2 ∂y i m

and

1 ij e φ =ψ: x =x, y = γ (x) pj − Aj (x) . mc m ∗ The Hamilton space H0 = Leg L0 has the fundamental function: −1

i

H0∗ =

i

i

1 ij 2e i e2 i γ (x)pi pj − A (x)p + A (x)Aj (x). i mc mc2 mc3

(6.5)

2◦ The Lagrange space of Electrodynamics Ln = (M, L(x, y)) in which L = L0 + U (x), ((2.1.4, Ch. 2)), i.e. L(x, y) = mcγij (x)y i y j +

2e Ai (x)y i + U (x), m

6.4. THE DUALITY BETWEEN LAGRANGE AND HAMILTON SPACES

101

has the Legendre transformation: φi : xi = xi , pi = mcγij (x)y j +

1 ij e φ =ψ: x =x, y = γ (x) pj − Aj . mc m n ∗ = Leg L has the fundamental function H (x, p) given by −1

And H ∗n

e Ai (x) m

i

H ∗ (x, p) =

i

1 c(m +

U c2 )

e2 + 3 c (m +

i

γ ij (x)pi pj −

U c2 )

2e c2 (m +

Ai (x)Ai (x) −

pi A U ) c2

i

(x)+

1 U (x) . c c

3◦ The class of Finsler space {F n } is inclosed in the calss of Lagrange space {Ln }, that is {F n } ⊂ {LN }, one can consider the restriction of the Legendre transformation Leg Ln to the class of Finsler spaces. In this case, the mapping φ : (x, y) ∈ D → (x, p) ∈ D∗ is given by φ(x, y) = (x, p) with 1 pi = ∂˙ i F 2 2 n ∗n ∗n and one obtains: Leg(F ) = C , C = (M, K ∗ (x, p)) with

(6.6)

K ∗2 = 2pi y i − F 2 (x, y) = y i ∂˙i F 2 (x, y) = F 2 (x, y) and y i is solution of the equation (6.4.6). Theorem 6.4.1. The dual, via Legendre transformation, of a Finsler space F n = (M, F (x, y)) is a Cartan space C ∗n = (M, F (x, φ−1 (x, p))).

102

CHAPTER 6. CARTAN SPACES

Part II Lagrangian and Hamiltonian Spaces of higher order

103

105

In this part of the book we study, the notions of Lagrange and Hamilton spaces of order k. They was introduced and investigated by the author [161]. Without explicitly formulated a clear definition of these spaces, a major contributions to edifice of these geometrics have been done by M. Crampin and colab. [64], M. de Leone and colab. [138], A. Kawaguchi [120], I. Vaisman [254] etc. For details, we refer to the books: The Geometry of Higher Order Lagrange Spaces. Applications to Mechanics and Physics, Kluwer Acad. Publ. FTPH, 82, 1997, [161]; The Geometry of Higher–Order Finsler Spaces, Hadronic Press, Inc. USA, 1998, [162]; The Geometry of Higher Order Hamilton Spaces. Applications to Hamiltonian Mechanics, Kluwer Acad. Publ., FTPH 132, 2003 [163]; as well as the papers [167]–[172]. This part contents: The geometry of the manifold of accelerations T k M , Lagrange spaces of order k, L(k)n , Finsler spaces of order k, F (k)n and dual, via Legendre transformation Hamilton spaces H (k)n and Cartan spaces C (k)n .

106

Chapter 1 The Geometry of the manifold T k M The importance of Lagrange geometries consists of the fact that the variational problems for Lagrangians have numerous applications: Mathematics, Physics, Theory of Dynamical Systems, Optimal Control, Biology, Economy etc. But, all of the above mentioned applications have imposed also the introduction of the notions of higher order Lagrange spaces. The base manifold of this space is the bundle of accelerations of superior order. The methods used in the construction of the geometry of higher order Lagrange spaces are the natural extensions of those used in the edification of the Lagrangian geometries exposed in chapters 1, 2 and 3. The concept of higher order Lagrange space was given by author in the books [161], [155]. The problems raised by the geometrization of Lagrangians systems of order k > 1 were been investigated by many scholars: Ch. Ehresmann [82], P. Libermann [143], J. Pommaret [207], J. T. Synge [243], M. Crampin [64], P. Saunders [230], G.S. Asanov [27], D. Krupka [132], M. de Léon [141], H. Rund [218], A. Kawaguchi [119], K. Yano [256], K. Kondo [128], D. Grigore [92], R. Miron [155], [156] et al. In this chapter we shall present, briefly, the following problems: 1◦ The geometry of total space of the bundle of higher order accelerations. 2◦ The definition of higher order Lagrange space, based on the nondegenerate Lagrangians of order k ≥ 1. 3◦ The solving of the old problem of prolongation of the Riemannian structures, given on the base manifold M , to the Riemannian structures on the total space of the bundle of accelerations of order k ≥ 1, we prove for the first time the existence of Lagrange spaces of order k ≥ 1. 107

108

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

4◦ The elaboration of the geometrical ground for variational calculus involving Lagrangians which depend on higher order accelerations. 5◦ The introduction of the notion of higher order energies and proof of the law of conservation. 6◦ The notion of k−semispray. Nonlinear connection the canonical metrical connection and the structure equations. 7◦ The Riemannian (k −1)n−almost contact model of a Lagrange space of order k. Evidently, we can not suﬀiciently develop these subjects. For much more informations one can see the books [161], [162]. Throughout in this chapter the differentiability of manifolds and of mappings means the class C ∞ .

1.1 The bundle of acceleration of order k ≥ 1 In Analytical Mechanics a real n−dimensional differentiable manifold M is considered as space of configurations of a physical system. A point (xi ) ∈ M is called a material point. A mapping c : t ∈ I → (xi (t)) ∈ U ⊂ M is a law ofi moving (ak law 1 d xi dx ,··· , of evolution), t is time, a pair (t, x) is an event and the k-uple dt k! dtk gives the velocity and generalized accelerations of order 2, ..., k − 1. The factors 1 (h = 1, ..., k) are introduced here for the simplicity of calculus. In this chapter h! we omit the word “generalized” and say shortly, the acceleration of order h, for 1 d h xi . A law of moving c : t ∈ I → c(t) ∈ U will be called a curve parametrized h! dth by time t. In order to obtain the differentiable bundle of accelerations of order k, we use the accelerations of order k, by means of geometrical concept of contact of order k between two curves in the manifold M . Two curves ρ, σ : I → M in M have at the point x0 ∈ M , ρ(0) = σ(0) = x0 ∈ U (and U a domain of local chart in M ) have a contact of order k if we have dα (f ◦ ρ)(t) dα (f ◦ σ)(t) |t=0 = |t=0 , (α = 1, ..., k). dtα dtα

(1.1)

1.1. THE BUNDLE OF ACCELERATION OF ORDER K ≥ 1

109

It follows that: the curves ρ and σ have at the point x0 = ρ(0) = σ(0) a contact of order k if and only if the accelerations of order 1, 2, ..., k on the curve ρ at x0 have the same values with the corresponding accelerations on the curve σ at point x0 . The relation “to have a contact of order k” is an equivalence. Let [ρ]x0 be a class of equivalence and T kx0 M the set of equivalence classes. Consider the set T kM =

[

T kx0 M

(1.2)

x0 ∈M

and the mapping: π k : [ρ]x0 ∈ T k M → x0 ∈ M,

∀[ρ]x0 .

(1.1.2’)

Thus the triple (T k M, π k , M ) can be endowed with a natural differentiable structure exactly as in the cases k = 1, when (T 1 M, π 1 , M ) is the tangent bundle. If U ⊂ M is a coordinate neighborhood on the manifold M , x0 ∈ U and the curve ρ : I → U , ρ0 = x0 is analytical represented on U by the equations xi = xi (t), t ∈ I, then Txk0 M can be represented by: xi0

i

= x (0),

(1)i y0

1 d k xi dxi (k)i (0), ..., y0 = (0). = dt k! dtk

(1.3)

Setting (1)i

(k)i

ϕ : ([ρ]x0 ) ∈ T k M → ϕ([ρ]x0 ) = (xi0 , y0 , ..., y0 ) ∈ R(k+1)n ,

(1.4)

it follows that the pair (π k )−1 (U ), ϕ) is a local chart on T k M induced by the local chart (U, φ) on the manifold M . So a differentiable atlas of the manifold M determine a differentiable atlas on k T M and the triple (T k M, π k , M ) is a differentiable bundle. Of course the mapping π k : T k M → M is a submersion. (T k M, π k , M ) is called the k accelerations bundle or tangent bundle of order k or k−osculator bundle. A change of local coordinates (xi , y (1)i , ..., y (k)i ) →

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

110

(e xi , ye(1)i , ..., ye(k)i ) on the manifold T k M , according with (1.1.3), is given by: i ∂e x i i 1 n =n x e = x e (x , ..., x ), rank ∂xj ∂e xi (1)j (1)i ye = j y ∂x ∂e y (1)i (1)j ∂e y (1)i (2)j (2)i 2e y = y y + 2 j (1)j ∂x ∂y ................................................. ∂e y (k−1)i (1)j ∂e y (k−1)i (2)j ∂e y (k−1)i (k)j (k)i ke y = y +2 y +...+ k (k−1)j y . ∂xj ∂y (1)j ∂y

(1.5)

And remark that we have the following identities: ∂e y (α)i ∂e y (α+1)i ∂e y (k)i = = ... = (k−α)j , ∂xj ∂y (1)j ∂y

(1.1.5’)

(α = 0, ..., k − 1; y (0) = x). We denote a point u ∈ T k M by u = (x, y (1) , ..., y (k) ) and its coordinates by (xi , y (1)i , ..., y (k)i ). A section of the projection π k is a mapping S : M → T k M with the property π k ◦ S = 1M . And a local section S has the property π k ◦ S|U = 1U . If c : I → M is a smooth curve, locally represented by xi = xi (t), t ∈ I, then the mapping e c : I → T k M given by: i

i

x = x (t), y

(1)i

1 d(k) xi 1 dxi (k)i (t), ..., y = (t), t ∈ I = 1! dt k! dtk

(1.6)

is the extension of order k to T k M of c. We have π k ◦ e c = c. The following property holds: If the differentiable manifold M is paracompact, then T k M is a paracompact manifold. We shall use the manifold Tg M = T k M \ {0}, where 0 is the null section of π k .

1.2. THE LIOUVILLE VECTOR FIELDS

111

1.2 The Liouville vector fields The natural basis at point u ∈ T k M of Tu (T k M ) is given by ∂ ∂ ∂ , , ..., (k)i . ∂xi ∂y (1)i ∂y u A local coordinate changing (1.1.5) transform the natural basis by the following rule: ∂e y (k)j ∂ ∂e xj ∂ ∂e y (1)j ∂ ∂ + ... + = + ∂xi ∂xi ∂e xj ∂xi ∂e ∂xi ∂e y (1)j y (k)j ∂e y (1)j ∂ ∂e y (k)j ∂ ∂ = + ... + (1)i (k)j ∂y (1)i ∂y (1)i ∂e y (1)j ∂y ∂e y ............................................................... ∂ ∂e y (k)j ∂ = , ∂y (k)i ∂y (k)i ∂e y (k)j

(1.1)

calculated at the point u ∈ T k M . The natural cobasis (dxi , dy (1)i , ..., dy (k)i )u is transformed by (1.1.5) as follows: de xi =

∂e xi j dx , ∂xj

∂e y (1)i j ∂e y (1)i (1)j de y = dx + (1)j dy , ∂xj ∂y ............................................................... ∂e y (k)i j ∂e y (k)i (1)j ∂e y (k)i (k)j (k)i de y = dx + (1)j dy + ... + (k)j dy . ∂xj ∂y ∂y (1)i

(1.2.1’)

The formulae (1.2.1) and (1.2.1’) allow to determine some important geometric object fields on the total space of accelerations bundle T k M . ∂ , The vertical distribution V1 is local generated by the vector fields (1)i ∂y ∂ ..., (k)i , i = 1, ..., n. V1 is integrable and of dimension kn. The distribution ∂y ∂ ∂ V2 local generated by , ..., (k)i is also integrable, of dimension (k − 1)n ∂y (2)i ∂y and it is a subdistribution of V1 . And so on.

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

112

∂ The distribution Vk local generated by is integrable and of dimension ∂y (k)i n. It is a subdistribution of the distribution Vk−1 . So we have the following sequence: V1 ⊃ V2 ⊃ ... ⊃ Vk Using again (1.2.1) we deduce: Theorem 1.2.1. The following operators in the algebra of functions F(T k M ) : ∂ , ∂y (k)i 2 ∂ (2)i ∂ , Γ= y (1)i (k−1)i + 2y ∂y ∂y (k)i .............................................. k ∂ (2)i ∂ (k)i ∂ + ... + ky Γ= y (1)i (1)i + 2y ∂y ∂y (2)i ∂y (k)i 1

Γ= y (1)i

(1.2)

k M and are the vector fields on T k M . They are independent on the manifold T] 1

2

k

Γ⊂ Vk , Γ⊂ Vk−1 ,...,Γ⊂ V1 . 1

2

k

The vector fields Γ, Γ, ..., Γ are called the Liouville vector fields. Also we have: k M ), the following entries are 1-form Theorem 1.2.2. For any function L ∈ F (T] kM : fields on the manifold T] d0 L =

∂L dxi , (k)i ∂y

∂L ∂L i dx + dy (1)i , (k−1)i (k)i ∂y ∂y .............................................. ∂L ∂L ∂L dk L = i dxi + (1)i dy (1)i + ... + (k)i dy (k)i . ∂x ∂y ∂y

d1 L =

(1.3)

Evidently, dk L = dL. In applications we shall use also the following nonlinear operator Γ = y (1)i

∂ ∂ (2)i ∂ (k)i + 2y + ... + ky . ∂xi ∂y (1)i ∂y (k−1)i

kM . Γ is not a vector field on T]

(1.4)

1.3. VARIATIONAL PROBLEM

113

Definition 1.2.1. A k-tangent structure J on T k M is the F(T k M )-linear mapping J : X (T k M ) → X (T k M ), defined by: ∂ ∂ ∂ ∂ = J , J = , ..., ∂xi ∂y (1)i ∂y (1)i ∂y (2)i (1.5) ∂ ∂ ∂ J = , J = 0. ∂y (k−1)i ∂y (k)i ∂y (k)i It is not diﬀicult to see that J has the properties: Proposition 1.2.1. We have: 1◦ . J is globally defined on T k M , 2◦ . J is an integrable structure, 3◦ . J is locally expressed by J=

∂ ∂ ∂ i (1)i ⊗ dx + ⊗ dy + ... + ⊗ dy (k−1)i (1)i (2)i (k)i ∂y ∂y ∂y

4◦ . ImJ = KerJ, 5◦ . rankJ = kn, k

k−1

(1.6)

KerJ = Vk , 2

1

1

6◦ . J Γ= Γ , ..., J Γ=Γ, J Γ= 0, 7◦ . J ◦ J ◦ ... ◦ J = 0, (k + 1 factors). In the next section we shall use the functions I 1 (L) = L 1 L, ..., I k (L) = L k L, Γ

Γ

∀L ∈ F (T k M ),

(1.7)

where L α is the operator of Lie derivation with respect to the Liouville vector field α

Γ

Γ. The functions I 1 (L), ..., I k (L) are called the main invariants of the function L. They play an important role in the variational calculus.

1.3 Variational Problem Definition 1.3.1. A differentiable Lagrangian of order k is a mapping L : (x, y (1) , ..., y (k) ) ∈ k M and continuous on the null secT k M → L(x, y (1) , y (k) ) ∈ R, differentiable on T] k M of the projection π k : T k M → M . tion 0 : M → T]

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

114

If c : t ∈ [0, 1] → (xi (t)) ∈ U ⊂ M is a curve, with extremities c(0) = (xi (0)) k M from (1.1.6) is its extension. Then the and c(1) = (xi (1)) and e c : [0, 1] → T] integral of action of L ◦ e c is defined by Z 1 dx 1 dk x I(c) = L x(t), (t), ..., (t) dt. (1.1) dt k! dtk 0 Remark 1.3.0.1. One proves that if I(c) does not depend on the parametrization of curve c then the following Zermelo conditions holds: I 1 (L) = ... = I k−1 (L) = 0,

I k (L) = L.

(1.2)

Generally, these conditions are not verified. The variational problem involving the functional I(c) from (1.3.1) will be studied as a natural extension of the theory expounded in §2.2, Ch. 2. On the open set U we consider the curves cε : t ∈ [0, 1] → (xi (t) + εV i (t)) ∈ M,

(1.3)

where ε is a real number, suﬀiciently small in absolute value such that Imcε ⊂ U , V i (t) = V i (x(t)) being a regular vector field on U , restricted to c. We assume all curves cε have the same end points c(0) and c(1) and their osculator spaces of order 1, 2, ..., k − 1 coincident at the points c(0), c(1). This means: i

i

V (0) = V (1) = 0;

dα V i dα V i (0) = (1) = 0, dtα dtα

(1.3.3’)

(α = 1, ..., k − 1). The integral of action I(cε ) of the Lagrangian L is: Z 1 dV 1 dk x dk V dx + ε , ..., +ε k dt. I(cε ) = L x + εV, dt dt k! dtk dt 0

(1.4)

A necessary condition for I(c) to be an extremal value for I(cε ) is dI(cε ) |ε=0 = 0. dε Thus, we have dI(cε ) = dε

Z

1 0

d dx dV 1 dk x dk V L x + εV, + ε , ..., +ε k dt. dε dt dt k! dtk dt

(1.5)

1.3. VARIATIONAL PROBLEM

115

The Taylor expansion of L for ε = 0, gives: Z 1 dI(cε ) ∂L i 1 ∂L dk V i ∂L dV i |ε=0 = V + (1)i +...+ dt. dε ∂xi dt k! ∂y (k)i dtk ∂y 0

(1.6)

Now, with notations k ∂L d ∂L ∂L k 1 d + ... + (−1) E i (L) := i − ∂x dt ∂y (1)i k! dtk ∂y (k)i ◦

and IV1 L

∂L , =V ∂y (k)i i

IV2 (L)

dV i ∂L ∂L + , ..., =V dt ∂y (k)i ∂y (k−1)i

(1.7)

i

dV i ∂L 1 dk−1 V v ∂L k i ∂L IV = V + + ... + dt ∂y (2)i (k − 1)! dtk−1 ∂y (k)i ∂y (1)i we obtain an important identity: ∂L i ∂L dV i 1 ∂L dk V i ◦ V + (1)i + ... + =E i (L)+ ∂xi dt k! ∂y (k)i dtk ∂y k−1 1 d d 1 d IVk (L)− IV1 (L) + IVk−1 (L) +...+ (−1)k−1 k−1 dt 2! dt k! dt

(1.8)

(1.9)

Now, applying (1.3.7) and taking into account (1.3.3’) with IVα (L)(c(0)) = IVα (L)c(1) = 0, we obtain dI(cε ) |ε=0 = dε

Z

1 0

(α = 1, 2, ..., k)

i E i (L)V dt.

(1.10)

0

But V i (t) is an arbitrary vector field. Therefore the equalities (1.3.5) and (1.3.10) lead to the following result: Theorem 1.3.1. In order that the integral of action I(c) be an extremal value for the functionals I(cε ), (1.3.4) is necessary that the following Euler - Lagrange equations hold: k 0 ∂L d ∂L ∂L k 1 d E i (L) := ∂xi − dt (1)i +...+ (−1) k! dtk (k) = 0, ∂y ∂y (1.11) i k i dx 1d x y (1)i = , · · · , y (k)i = . dt k! dtk

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

116

0

One proves [161] that E i (L) is a covector field. Consequence the equation 0

E i (L) = 0 has a geometrical meaning. Consider the scalar field E k (L) = I k (L) −

1 d k−1 1 dk−1 1 I (L) +...+(−1)k−1 I (L)−L. 2! dt k! dtk−1

(1.12)

It is called the energy of order k of the Lagrangian L. The next result is known, [161]: Theorem 1.3.2. For any Lagrangian L(x, y (1) , · · · , y (k) ) the energy of order k, E k (L) is conserved along every solution curve of the Euler - Lagrange equations 0 dxi 1 d k xi (1)i (k)i ,··· ,y = . E i (L) = 0, y = dt k! dtk Remark 1.3.2.1. Introducing the notion of energy of order 1, 2, · · · , k − 1 we can prove a Nöther theorem for the Lagrangians of order k. Now we remark that for any C ∞ −function ϕ(t) and any differentiable Lagrangian L(x, y (1) , · · · , y (k) ) the following equality holds: dk ϕ k dϕ 1 Ei (L) + · · · + k Ei (L), E i (ϕL) = ϕ Ei (L) + dt dt 0

0

1

(1.13)

k

where Ei (L), · · · , Ei (L) are d-covector fields - called Graig - Synge covectors, k−1

[63]. We consider the covector Ei (L): k−1 Ei

1 (L) = (−1)k−1 (k − 1)!

∂L d ∂L − ∂y (k−1)i dt ∂y (k)i

,

(1.14)

It is important in the theory of k−semisprays from the Lagrange spaces of order k. The Hamilton - Jacobi equations, of a space Ln = (M, L(x, y)) introduced in the section 4 of Chapter 2 can be extended in the higher order Lagrange spaces by using the Jacobi - Ostrogradski momenta. Indeed, the energy of order k, E k (L) dxi d k xi from (1.3.13) is a polynomial function in , · · · , k given by dt dt d 2 xi d k xi dxi + p(2)i 2 + · · · + p(k)i k − L, E (L) = p(1)i dt dt dt k

(1.15)

1.4. SEMISPRAYS. NONLINEAR CONNECTIONS

where p(1)i

p(1)i , ..., p(k)i

117

k−1 ∂L 1 d ∂L ∂L k−1 1 d = (1)i − +...+(−1) , k−1 (2)i 2! dt ∂y k! dt ∂y (k)i ∂y

k−2 1 ∂L 1 d ∂L ∂L k−2 1 d p(2)i = − +...+(−1) , 2! ∂y (2)i 3! dt ∂y (3)i k! dtk−2 ∂y (k)i ....................................................................... 1 ∂L p(k)i = . k! ∂y (k)i are called the Jacobi-Ostrogradski momenta.

(1.16)

The following important result has been established by M. de Léon and others, [161]: 0

Theorem 1.3.3. Along the integral curves of Euler Lagrange equations Ei (L) = 0 the following Hamilton - Jacobi - Ostrogradski equations holds: ∂E k (L) dα xi = α, ∂p(α)i dt

(α = 1, ..., k),

dp(1)i ∂E k (L) = − , ∂xi dt dp(α+1)i 1 ∂E k (L) , = − α! ∂y (α)i dt

(α = 1, ..., k − 1).

Remark 1.3.3.1. The Jacobi - Ostrogradski momenta allow the introduction of the 1-forms: p(1) = p(1)i dxi + p(2)i dy (1)i + · · · + p(k)i dy (k−1)i , p(2) = p(2)i dxi + p(3)i dy (1)i + · · · + p(k)i dy (k−2)i , .............................................................. p(k) = p(k)i dxi .

1.4 Semisprays. Nonlinear connections A vector field S ∈ χ(T k M ) with the property k

JS =Γ

(1.1)

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

118

is called a k−semispray on T k M . S can be uniquely written in the form: S = y (1)i

∂ ∂ ∂ (k)i i (1) (k) + ... + ky − (k + 1)G (x, y , ..., y ) , ∂xi ∂y (k−1)i ∂y (k)i

(1.2)

or shortly S = Γ − (k + 1)Gi

∂ . ∂y (k)i

(1.4.2’)

The set of functions Gi is the set of coeﬀicients of S. With respect to (1.1.5) Ch. 4 Gi are transformed as following: i

x ei = (k + 1)Gj ∂e (k + 1)G − Γe y (k)i . j ∂x

(1.3)

A curve c : I → M is called a k-path on M with respect to S if its extension e c is an integral curve of S. A k-path is characterized by the (k + 1)− differential equations: k dk+1 xi 1 d x dx + (k + 1)Gi x, , ..., = 0. (1.4) k+1 dt dt k! dtk We shall show that a k−semispray determine the main geometrical object fields on T k M as: the nonlinear connections N , the N −linear connections D and them structure equations. Evidently, N and D are basic for the geometry of manifold T kM . Definition 1.4.1. A subbundle HT k M of the tangent bundle (T T k M, dπ k , T k M ) which is supplementary to the vertical subbundle V1 T k M : T T k M = HT k M ⊕ V1 T k M

(1.5)

is called a nonlinear connection. The fibres of HT k M determine a horizontal distribution N : u ∈ T k M → Nu = Hu T k M ⊂ Tu T k M,

∀u ∈ T k M

supplementary to the vertical distribution V1 , i.e. Tu T k M = Nu ⊕ V1,u ,

∀u ∈ T k M.

(1.4.5’)

If the base manifold M is paracompact on T k M there exist the nonlinear connections. The local dimension of N is n = dim M .

1.4. SEMISPRAYS. NONLINEAR CONNECTIONS

119

Consider a nonlinear connection N and denote by h and v the horizontal and vertical projectors with respect to N and V1 : h + v = I,

hv = vh = 0,

h2 = h,

v 2 = v.

As usual we denote X H = hX,

X V = vX,

∀X ∈ χ(T k M ).

An horizontal lift, with respect to N is a F(M )-linear mapping lh : X (M ) → X (T k M ) which has the properties v ◦ lh = 0, dπ k ◦ lh = Id . There exists an unique local basis adapted to the horizontal distribution N . It is given by δ ∂ = lh , (i = 1, ..., n). (1.6) δxi ∂xi The linearly independent vector fields of this basis can be uniquely written in the form: δ ∂ ∂ j ∂ j = − N − ... − N . (1.7) i i (1)j (k)j δxi ∂xi ∂y ∂y (1) (k) The systems of differential functions on T k M : (Nij , ..., Nij ), gives the coeﬀi(1)

(k)

cients of the nonlinear connection N . By means of (1.4.6) it follows: Proposition 1.4.1. With respect of a change of local coordinates on the manifold T k M we have ∂e xj δ δ = , (1.4.7’) δxi ∂xi δe xj and xi ∂e y (1)i xm m ∂e i ∂e e Nm j = Nj − , m j ∂x ∂x (1) ∂x (1) .............................................................. xi y (k−1)i ∂e y (k)i xm m ∂e m ∂e i ∂e em N = N + ... + N − , j j j m m j ∂x ∂x ∂x ∂x (k) (k) (1)

(1.8)

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

120

Remark 1.4.0.1. The equalities (1.4.8) characterize a nonlinear connection N with the coeﬀicients Nij , · · · , Nij . (1)

(k)

These considerations lead to an important result, given by: Theorem 1.4.1 (I. Bucătaru [47]). If S if a k−semispray on T k M , with the coeﬀicients Gi , then the following system of functions: ∂Gi ∂Gi ∂Gi i i Nj = (k)i , Nj = (k−1)i , · · · , Nj = (1)j ∂y ∂y ∂y (2) (k) (1) i

(4.9)

gives the coeﬀicients of a nonlinear connection N . The k−tangent structure J, defined in (1.2.5), Ch. 4. applies the horizontal distribution N0 = N into a vertical distribution N1 ⊂ V1 of dimension n, supplementary to the distribution V2 . Then it applies the distribution N1 in distribution N2 ⊂ V2 , supplementary to the distribution V3 and so on. Of course we have dim N0 = dim N1 = · · · = dim Nk−1 = n. Therefore we can write: N1 = J(N0 ),

N2 = J(N1 ), · · · , Nk−1 = J(Nk−2 )

(1.9)

and we obtain the direct decomposition: Tu T k M = N0,u ⊕ N1,u ⊕ · · · ⊕ Nk−1,u ⊕ Vk,u ,

∀u ∈ T k M.

An adapted basis to N0 , N1 , ..., Nk−1 , Vk is given by: δ δ δ ∂ , , · · · , (k−1)i , (k)i , (i = 1, ..., n), δxi δy (1)i δy ∂y where

δ is in (1.4.7) and δxi δ ∂ ∂ j ∂ j = − N − · · · − N , i i (1)i (1)i (2)i (k)i δy ∂y ∂y ∂y (1) (k−1) .................................................................. δ ∂ ∂ j = − N . i δy (k−1)i (k)j ∂y (k−1)i (1) ∂y

(1.10)

(1.11)

(1.12)

1.5. THE DUAL COEFFICIENTS OF A NONLINEAR CONNECTION

121

With respect to (1.1.5), Ch. 4 we have: δ ∂xj δ δ ∂ (0)i i = , α = 0, 1, ..., k; y = x , (k)i = (k)i . ∂e xi δe δy (α)i y (α)j δy ∂y

(1.13)

Let h, v1 , ..., vk be the projectors determined by (1.4.10): h+

k X

vα = I, h2 = h, vα vα = vα , hvα = 0;

1

vα h = 0, vα vβ = vβ vα = 0, (α 6= β). If we denote X H = hX, X Vα = vα X, ∀X ∈ X (T k M )

(1.14)

X = X H + X V1 + ... + X Vk .

(1.15)

we have, uniquely, In the adapted basis (1.4.12) we have: X H = X (0)i

δ , XV i δx

α

= X (α)i

δ , (α = 1, ..., k). δy (α)i

A first result on the nonlinear connection N is as follows: Theorem 1.4.2. The nonlinear connection N is integrable if, and only if: [X H , Y H ]Vα = 0,

∀X, Y ∈ χ(T k M ),

(α = 1, · · · k).

1.5 The dual coeﬀicients of a nonlinear connection Consider a nonlinear connection N , with the coeﬀicients (Nji , ..., Nji ). The dual (1)

(k)

basis, of the adapted basis (1.4.12) is of the form (δxi , δy (1)i , · · · , δy (k)i ) where

(1.1)

δxi = dxi , δy (1)i = dy (1)i + Mji dxj , (1)

............................................. δy (k)i = dy (k)i + Mji dy (k−1)j + ... + Mji dy (1)j + Mji dxj , (1)

(k−1)

(k)

(1.2)

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

122

and where

i i Mj = Nji , Mji = Nji + Njm Mm , ...., (1) (1) (1) (2) (2) (1) i i m i m i Mj = Nj + Nj Mm + .... + Nj Mm . (k)

(k)

(k−1) (1)

(1.3)

(1) (k−1)

The system of functions (Mji , ..., Mji ) is called the system of dual coeﬀicients (1)

(k)

of the nonlinear connection N. If the dual coeﬀicients of N are given, then we uniquely obtain from (1.5.3) the primal coeﬀicients (Nji , ..., Nji , ) of N . (1)

(k)

With respect to (4.1.4), Ch. 4 the dual coeﬀicients of N are transformed by the rule Mjm (1)

xm ∂e y (1)i ∂e xi i ∂e fm = M + , j j ∂xm ∂x ∂x (1)

............................................. ∂e xi xm y (1)m y (k−1)m ∂e y (k)i i ∂e i ∂e i ∂e fm fm fm Mjm m = M + M + · · · + M + . j j j j ∂x ∂x ∂x ∂x ∂x (k) (k−1) (1) (k)

(1.4)

These transformations of the dual coeﬀicients characterize the nonlinear connection N . This property allows to prove an important result: Theorem 1.5.1 (R. Miron). For any k−semispray S with the coeﬀicients Gi the following system of functions Mji (1)

∂Gi 1 i = (k)j , Mji = (SMji + Mm Mjm ), · · · , 2 (1) ∂y (1) (1) (2)

1 i Mji = (S Mji + Mm Mjm ) k (k−1) (1) (k−1) (k)

(1.5)

gives the system of dual coeﬀicients of a nonlinear connection, which depend on the k−semispray S, only. Remark 1.5.1.1. Gh. Atanasiu [168] has a real contribution in demonstration of this theorem for k = 2. As an application we can prove:

1.5. THE DUAL COEFFICIENTS OF A NONLINEAR CONNECTION

Theorem 1.5.2. 1

123

1) In the adapted basis (4.4.12) Ch. 4, the Liouville vector k

fields Γ, ..., Γ can be expressed in the form 2 δ δ δ , = z (1)i (k−1)i + 2z (2)i (k)i , Γ (k)i δy δy δy ................................................................. k δ (2)i δ (k)i δ + · · · + kz , Γ= z (1)i (1)i + 2z δy δy (2)i δy (k)i 1

Γ= z (1)i

where

z (1)i = y (1)i ,

(1.6)

i (1)m 2z (2)i = 2y (2)i + Mm y , ..., (1)

(k)i i (k−1)m i (1)m = ky (k)i + (k − 1)Mm y + · · · + Mm y kz (1)

(1.7)

(k−1)

2) With respect to (1.1.4) we have: ze

(α)i

∂e xi (α)j = jz , ∂x

(α = 1, ..., k).

(1.5.7’)

This is reason in which we call z (1)i , ..., z (k)i the distinguished Liouville vector fields (shortly, d-vector fields). These vectors are important in the geometry of the manifold T k M . A field of 1-forms ω ∈ X ∗ (T k M ) can be uniquely written as ω = ω H + ω V1 + · · · + ω Vk where ω H = ω ◦ h, ω Vα = ω ◦ vα ,

(α = 1, ..., k).

For any function f ∈ F (T k M ), the 1-form df is df = (df )H + (df )V1 + · · · + (df )Vk . In the adapted cobasis we have: (df )H =

δf i δf Vα δy (α)i , dx , (df ) = i (α)i δx δy

(α = 1, ..., k).

Let γ : I → T k M be a parametrized curve, locally expressed by xi = xi (t),

y (α)i = y (α)i (t),

(t ∈ I),

(α = 1, ..., k).

(1.8)

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

124

dγ is given by: dt H V 1 Vk dγ dγ dγ dγ + = + ··· + = dt dt dt dt

The tangent vector field

δy (k)i δ dxi δ δy (1)i δ + ··· + . = + dt δxi dt δy (1)i dt δy (k)i H dγ dγ = . It is characterized by the The curve γ is called horizontal if dt dt system of differential equations δy (1)i δy (k)i x = x (t), = 0, ..., = 0. dt dt i

i

(1.9)

A horizontal curve γ is called autoparallel curve of the nonlinear connection if γ = e c, where e e c is the extension of a curve c : I → M . The autoparallel curves of the nonlinear connection N are characterized by the system of differential equations δy (k)i δy (1)i = 0, · · · , = 0, dt dt dxi 1 d k xi (k)i (1)i y = . ,··· ,y = dt k! dtk

(1.5.9’)

1.6 Prolongation to the manifold T k M of the Riemannian structures given on the base manifold M Applying the previous theory of the notion of nonlinear connection on the total space of acceleration bundle T k M we can solve the old problem of the prolongation of the Riemann (or pseudo Riemann) structure g given on the base manifold M . This problem was formulated by L. Bianchi and was studied by several remarkable mathematicians as: E. Bompiani, Ch. Ehresmann, A. Morimoto, S. Kobayashi. But the solution of this problem, as well as the solution of the prolongation to T k M of the Finsler or Lagrange structures were been recently given by R. Miron [172]. We will expound it here with very few demonstrations. Let Rn = (M, g) be a Riemann space, g being a Riemannian metric defined on the base manifold M , having the local coordinate gij (x), x ∈ U ⊂ M . We extend

1.6. PROLONGATION TO THE MANIFOLD T K M OF THE RIEMANNIAN STRUCTURES125

gij to π −1 (U ) ⊂ T k M , setting (gij ◦ π k )(u) = gij (x),

∀u ∈ π −1 (U ), π k (u) = x.

gij ◦ π k will be denoted by gij . The problems of prolongation of the Riemannian structure g to T k M can be formulated as follows: The Riemannian structure g on the manifold M being a priori given, determine a Riemannian structure G on T k M so that G be provided only by structure g. As usually, we denoted by γ i jk (x) the Christoffel symbols of g and prove, according theorem 1.5.1: kM Theorem 1.6.1. There exists nonlinear connections N on the manifold T] determined only by the given Riemannian structure g(x). One of them has the following dual coeﬀicients i Mji = γjm (x)y (1)m , (1)

Mji = (2)

1 2

(

) i ΓMji + Mm Mjm (1)

(1)

,

(1.1)

(1)

......................................... ( ) 1 i Γ Mji + Mm Mjm , Mji = k (1) (k−1) (k−1) (k) where Γ is the operator (1.2.4), Ch. 4. Remark 1.6.1.1. Γ can be substituted with any k−semispray S, since Mji , ..., Mji (1)

(k−1)

k

do not depend on the variables y . One proves, also: N is integrable if and only if the Riemann space Rn = (M, g) is locally flat. Let us consider the adapted cobasis (δxi , δy (1)i , ..., δy (k)i ) (1.5.2) to the nonlinear connection N and to the vertical distributions N , ..., N , Vk . It depend on the (1)

(k−1)

dual coeﬀicients (1.6.1). So it depend on the structure g(x) only. Now, on T k M consider the following lift of g(x): G = gij (x)dxi ⊗ dxj + gij (x)δy (1)i ⊗ δy (1)j + (1.2) +gij (x)δy (k)i ⊗ δy (k)j

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

126

Finally, we obtain: k M , G) is a Riemann space of dimension Theorem 1.6.2. The pair Prolk Rn = (T] (k + 1)n, whose metric G, (1.6.2) depends on the a priori given Riemann structure g(x), only.

The announced problem is solved. Some remarks: 1◦ The d−Liouville vector fields z (1)i , · · · , z (k)i , from (1.5.7) are constructed only by means of the Riemannian structure g; 2◦ The following function L(x, y (1) , ..., y (k) ) = gij (x)z (k)i z (k)j

(1.3)

is a regular Lagrangian which depend only on the Riemann structure gij (x). 3◦ The previous theory, for pseudo-Riemann structure gij (x), holds.

1.7 N −linear connections on T k M The notion of N −linear connection on the manifold T k M can be studied as a natural extension of that of N −linear connection on T M , given in the section 1.4. Let N be a nonlinear connection on T k M having the primal coeﬀicients Nji , · · · , Nji (1)

and the dual coeﬀicients

Mji , · · · (1)

(k)

, Mji . (k)

Definition 1.7.1. A linear connection D on the manifold T k M is called distinguished if D preserves by parallelism the horizontal distribution N . It is an N −connection if has the following property, too: DJ = 0.

(1.1)

We have: Theorem 1.7.1. A linear connection D on T k M is an N -linear connection if and only if H Vα D Y = 0, (α = 1, · · · , k), (DX Y Vα )H = 0; X (1.2) (DX Y Vα )Vβ = 0 (α 6= β) D (JY H ) = JD Y H ; D (JV α ) = JD V α . X X X X

1.7. N −LINEAR CONNECTIONS ON T K M

127

Of course, for any N −linear connection D we have Dh = 0, Dv α = 0, (α = 1, · · · , k). Since DX Y = DX H Y + DX V1 Y + · · · + DX Vk Y, setting Vα H DX = DX H , DX = DX Vα , (α = 1, · · · , k),

we can write: V1 Vk H DX Y = DX Y + DX Y + · · · + DX Y.

(1.3)

The operators DH , DVα are not the covariant derivations but they have similar properties with the covariant derivations. The notion of d−tensor fields (d−means “distinguished”) can be introduced and studied exactly as in the section 1.3. In the adapted basis (1.4.12) and in adapted cobasis (1.5.1) we represent a d−tensor field of type (r, s) in the form ...ir T = Tji11...j s

δ δ ⊗ · · · ⊗ ⊗ dxj1 ⊗ · · · ⊗ δy (k)js . i (k)i 1 r δx δy

(1.4)

A transformation of coordinates (1.1.5), Ch. 4, has as effect the following rule of transformation: ∂e x i1 ∂e xir ∂xk1 ∂xks h1... hr i1... ir e Tj1 ...js = h · · · hr j · · · j Tk1 ...ks . (1.7.3’) ∂x 1 ∂x ∂x 1 ∂x s δ δ So, 1, i , · · · , (k)i generate the tensor algebra of d−tensor fields. δx δy The theory of N −linear connection described in the chapter 1 for case k = 1 can be extended step by step for the N −linear connection on the manifold T k M . In the adapted basis (1.4.11) an N −linear connection D has the following form: D

D

δ δxj

δ δ δ m δ m δ = L , D = L , (α = 1, · · · , k), ij ij δxj δy (α)i δxi δxm δy (α)m

δ δy (β)j

δ δ δ m δ m δ = C , D = C , ij ij m (α)i (α)m δy (β)j δy δxi δx δy (β) (β) (α, β = 1, · · · , k).

(1.5)

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

128

The system of functions DΓ(N ) = (Lmij , C mij , ..., C mij ) (1)

(1.6)

(k)

represents the coeﬀicients of D. With respect to (1.1.5), Lmij are transformed by the same rule as the coeﬀicients of a linear connection defined on the base manifold M . Others coeﬀicients C hij , (α)

(α = 1, .., k) are transformed like d-tensors of type (1, 2). If T is a d-tensor field of type (r, s), given by (1.7.4) and X = X H = X i H then, by means of (1.7.5), DX T is: ...ir H DX T = X m Tji11...j s |m

δ δ ⊗ · · · ⊗ ⊗ dxj1 ⊗ · · · ⊗ δy (k)js , i (k)i 1 r δx δy

δ , δxi

(1.7)

where

...ir δTji11...j ...ir s 1 2 ...ir = + Lihm Tjhi1 ...j + · · · − Lhjs m Tji11...h . s m δx The operator “| ” will be called the h-covariant derivative. ...ir Tji11...j s |m

Consider the vα -covariant derivatives

Vα DX ,

(α) i

for X =X

(1.8)

δ , (α = 1, ..., k). δy (α)i

Then, (1.7.3) and (1.7.5) lead to: (α)

(α)

Vα ...ir DX T =X m Tji11...j | s

m

δ δ ⊗ dxj1 ⊗ · · · ⊗ dxjs , ⊗ · · · ⊗ i (k)i 1 r δx δy

(1.9)

where ...ir Tji11...j s

...ir δTji11...j i1 ...ir 2 ...ir | m = (α)ms + Chm Tjhi1 ...j + · · · − Cjhs m Tji11...j , (α = 1, .., k). s s−1 h δy (α) (α)

(α)

(α)

The operators “ | ”, in number of k are called vα -covariant derivatives. (α)

Each of operators “| ” and “ | ” has the usual properties with respect to sum of d-tensor or them tensor product. Now, in the adapted basis (1.4.11) we can determine the torsion T and curvature R of an N −linear connection D, follows the same method as in the case k = 1.

1.7. N −LINEAR CONNECTIONS ON T K M

129

We remark the following of d−tensors of torsion: T ijk = Lijk − Likj , S ijk = C ijk − C ikj = 0, (α = 1, .., k) (α)

(α)

(1.10)

(α)

and d−tensors of curvature Rhi jm ,

P

i

i

, S

(α)h jm

(βα)h jm

, (α, β = 1, · · · , k)

(1.11)

The 1-forms connection of the N −linear connection D are: ω i j = Lijh dxh + C ijh δy (1)h + · · · + C ijh δy (k)h . (1)

(1.12)

(k)

The following important theorem holds: Theorem 1.7.2. The structure equations of an N −linear connection D on the manifold T k M are given by: d(dx ) − dx ∧ ω i

d(δy

m

(α)i

) − δy

i

(α)m

m

(0) i

=−Ω

∧ω

i

m

(α) i

=−Ω,

(1.13)

dω i j − ω mj ∧ ω i m = −Ωi j , (0) (α) i i

where Ω , Ω are the 2-forms of torsion and where Ωi j are the 2-forms of curvature: 1 Ωi j = Rj i pq dxp ∧ dxq + 2 +

k X

Pj ipq dxp α=1 (α)

∧ δy

(α)q

+

k X

Sj ipq δy (α)p ∧ δy (β)q .

α,β=1 (αβ)

Now, the Bianchi identities of D can be derived from (1.7.13). The nonlinear connection N and the N −linear connection D allow to study the geometrical properties of the manifold T k M equipped with these two geometrical object fields.

130

CHAPTER 1. THE GEOMETRY OF THE MANIFOLD T K M

Chapter 2 Lagrange Spaces of Higher–order The concept of higher - order Lagrange space was introduced and studied by the author of the present monograph, [155], [161]. A Lagrange space of order k is defined as a pair L(k)n = (M, L) where L : T k M → R is a differentiable regular Lagrangian having the fundamental tensor of constant signature. Applying the variational problem to the integral of action of L we determine: a canonical k−semispray, a canonical nonlinear connection and a canonical metrical connection. All these are basic for the geometry of space L(k)n .

2.1 The spaces L(k)n = (M, L) Definition 2.1.1. A Lagrange space of order k ≥ 1 is a pair L(k)m = (M, L) formed by a real n−dimensional manifold M and a differentiable Lagrangian L : (x, y (1) , · · · , y (k) ) ∈ T k M → L(x, y (1) , · · · , y (k) ) ∈ R for which the Hessian with the elements: 1 ∂ 2L gij = (2.1) 2 ∂y (k)i ∂y (k)j has the property kM rank(gij ) = n on T] (2.2) and the d−tensor gij has a constant signature. Of course, we can prove that gij from (2.1.1) is a d−tensor field, of type (0,2), symmetric. It is called the fundamental (or metric) tensor of the space L(k)n , while L is called its fundamental function. The geometry of the manifold T k M equipped with L(x, y (1) , · · · , y (k) ) is called the geometry of the space L(k)n . We shall study this geometry using the theory from the last chapter. Consequently, starting from the integral of action 131

132

CHAPTER 2. LAGRANGE SPACES OF HIGHER–ORDER

Z

1

I(c) = 0

dx 1 dk x L x, , · · · , dt we determine the Euler - Lagrange equations dt k! dtk

0

1

k

E i (L) = 0 and the Craig - Synge covectors E i (L), · · · , E i (L). According to (3.1.14) one remarks that we have: k−1

Theorem 2.1.1. The equations g ij E i (L) = 0 determine a k−semispray S = y (1)i

∂ ∂ (2)i ∂ (k)i i ∂ + 2y + · · · + ky − (k + 1)G ∂xi ∂y (1)i ∂y (k−1)i ∂y (k)i

where the coeﬀicients Gi are given by 1 ∂ ∂ (k + 1)Gi = g ij Γ − (k−1)i 2 ∂y (k)i ∂y

(2.3)

(2.4)

Γ being the operator (1.2.4). The semispray S depend on the fundamental function L, only. S is called kM . canonical. It is globally defined on the manifold T] Taking into account Theorem 1.5.1, we have: Theorem 2.1.2. The systems of functions

!

i

Mji = (1)

Mji = (k)

∂G 1 i , M = j 2 ∂y (k)j (2) 1 k

i SMji + Mm Mjm , ..., (1)

!

(1)

(1)

(2.5)

i S Mji + Mm Mjm (k−1)

(1) (k−1)

are the dual coeﬀicients of a nonlinear connection N determined only on the fundamental function L of the space L(k)n . N is the canonical nonlinear connection ofL(k)n . δ δ δ , · · · , has its dual {δxi , δy (1)i , ..., δy (k)i }. The adapted basis , i (1)i (k)i δx δy δy They are constructed by the canonical nonlinear connection. So, the horizontal curves are characterized by the system of differential equations Part 2, Ch. 2, and the autoparallel curves of N are given by Part II, Ch. 1. Vα δ δ The condition that N be integrable is expressed by , = 0, (α = δxi δxj 1, · · · , k).

2.2. EXAMPLES OF SPACES L(K)N

133

2.2 Examples of spaces L(k)n 1◦ Let us consider the Lagrangian: L(x, y (1) , ..., y (k) ) = gij (x)z (k)i z (k)j

(2.1)

where gij (x) is a Riemannian (or pseudo Riemannian) metric on the base manifold M and the z (k)i is the Liuoville d−vector field: i (k−1)m i (1)m kz (k)i = ky (k)i + (k − 1)Mm y + · · · + k Mm y (1)

(2.2)

(k−1)

constructed by means of the dual coeﬀicients (Part II, Ch. 1) of the canonical nonlinear connection N from the problem of prolongation to T k M of gij (x). So that the Lagrangian (2.2.1) depend on gij (x) only. The pair L(k)n = (M, L), (2.2.1) is a Lagrange space of order k. Its fundamental tensor is gij (x), since the d−vector z (k)i is linearly in the variables y (k) . ◦

2◦ Let L (x, y (1) ) be the Lagrangian from electrodynamics ◦

(1) (1)i (1)j + L (x, y ) = mcγij (x)y y

2e bi (x)y (1)i m

(2.3)

Let N be the nonlinear connection given by the theorem Part II, Ch. 1, from the problem of prolongations to T k M of the Riemannian (or pseudo Riemannian) structure γij (x) and the Liouville tensor z (k)i constructed by means of N . Then the pair L(k)n = (M, L), with L(x, y (1) , ..., y (k) ) = mcγij (x)z (k)i z (k)j +

2e bi (x)z (k)i m

(2.4)

is a Lagrange space of order k. It is the prolongation to the manifold T k M ◦ of the Lagrangian L (2.2.3) of electrodynamics. These examples prove the existence of the Lagrange spaces of order k.

2.3 Canonical metrical N −connection Consider the canonical nonlinear connection N of a Lagrange space of order k, L(k)n = (M, L).

134

CHAPTER 2. LAGRANGE SPACES OF HIGHER–ORDER

An N −linear connection D with the coeﬀicients DΓ(N ) = (Lijk , C ijh , · · · , C ijh ) (1)

(k)

is called metrical with respect to metric tensor gij if (α)

gij|h = 0,

gij | h = 0,

(α = 1, · · · , k).

(2.1)

Now we can prove the following theorem: Theorem 2.3.1. The following properties hold: k M verifying the axioms: 1) There exists a unique N -linear connection D on T] 1◦ N − is the canonical nonlinear connection of space L(k)n . 2◦ gij|h = 0, (D is h−metrical) ◦

(α)

3 gij | h = 0, (α = 1, · · · , k), (D is vα −metrical) 4◦ T ijh = 0, (D is h−torsion free) 5◦ S i jh = 0, (α = 1, · · · , k), (D is vα −torsion free). (α)

2) The coeﬀicients CΓ(N ) = (Lhij , C hij , ..., C hij ) of D are given by the general(1)

(k)

ized Christoffel symbols: Lhij

1 = g hs 2

1 C hij = g hs 2 (α)

δgis δgsj δgij + i − s δxj δx δx

,

δgsj δgij δgis + (α)i − (α)s (α)j δy δy δy

(2.2)

, (α = 1, ..., k).

3) D depends only on the fundamental function L of the space L(k)n . The connection D from the previous theorem is called canonical metrical N connection and its coeﬀicients (2.3.2) are denoted by CΓ(N ). Now, the geometry of the Lagrange spaces L(k)n can be developed by means of these two canonical connection N and D.

2.4. THE RIEMANNIAN (K − 1)N −CONTACT MODEL OF THE SPACE L(K)N

135

2.4 The Riemannian (k − 1)n−contact model of the space L(k)n The almost Kählerian model of the Lagrange spaces Ln expound in the section 7, Ch. 2, can be extended in a corresponding model of the higher order Lagrange spaces. But now, it is a Riemannian almost (k − 1)n−contact structure on the kM . manifold T] The canonical nonlinear connection N of the space L(k)n = (M, L) determines ] ] k M )−linear mapping F : X (T k M ) → X (T k M ) defined on the the following F(T] adapted basis to N and to Nα , by δ ∂ = − F δxi ∂y (k)i F F

δ δy (α)i ∂ ∂y (k)i

= 0, =

δ , δxi

(α = 1, · · · , k − 1)

(2.1)

(i = 1, · · · , n)

We can prove: Theorem 2.4.1. We have: kM . 1◦ F is globally defined on T] kM 2◦ F is a tensor field of type (1, 1) on T]

3◦ KerF = N1 ⊕ N2 ⊕ · · · ⊕ Nk−1 , ImF = N0 ⊕ Vk 4◦ rankkFk = 2n 5◦ F3 + F = 0. k M determined by N . Thus F is an almost (k − 1)n−contact structure on T] !

Let

, (a = 1, ..., n) be a local basis adapted to the direct (k−1)a 1a 2a decomposition N1 ⊕ · · · ⊕ Nk−1 and η , η , ..., η , its dual. ξ , ξ , ..., ξa

1a 2a

(k−1)a

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CHAPTER 2. LAGRANGE SPACES OF HIGHER–ORDER

Thus the set

! F, ξ , ..., 1a

1a

ξ , η , ...,

(k−1)a

η

(a = 1, · · · , n − 1)

,

(2.2)

(k−1)a

is a (k − 1)n almost contact structure. Indeed, (2.4.1) imply: a αa δ b for F( ξ ) = 0, η ( ξ ) = 0 for αa βb

α=β α 6= β, (α, β = 1, · · · , (k − 1)).

n X k−1 X αa η (X) ξ , F (X) = −X + 2

αa

a=1 α=1

αa

∀X ∈ X (T k M ), η ◦F = 0.

Let NF be the Nijenhuis tensor of the structure F. NF (X, Y ) = [FX, FY ] + F2 [X, Y ] − F[FX, Y ] − F[X, FY ]. The structure (2.4.2) is said to be normal if: NF (X, Y ) +

n X k−1 X

αa

d η (X, Y ) = 0,

∀X, Y ∈ X (T k M ).

a=1 α=1

So we obtain a characterization of the normal structure F given by the following theorem: Theorem 2.4.2. The almost (k − 1)n−contact structure (2.4.2) is normal if and k M ) we have: only if for any X, Y ∈ X (T] NF (X, Y ) +

k−1 n X X

d(δy (α)a ) (X, Y ) = 0.

a=1 α=1

The lift of fundamental tensor gij of the space L(k)n with respect to N is defined by: G = gij dxi ⊗ dxj + gij δy (1)i ⊗ δy (1)j + · · · + gij δy (k)i ⊗ δy (k)j . (2.3) k M , deterEvidently, G is a pseudo-Riemannian structure on the manifold T] mined only by space L(k)n . Now, it is not diﬀicult to prove:

2.5. THE GENERALIZED LAGRANGE SPACES OF ORDER K

137

Theorem 2.4.3. The pair (G, F) is a Riemannian (k − 1)n-almost contact struckM . ture on T] In this case, the next condition holds: G(FX, Y ) = −G(FY, X),

k M ). ∀X, Y ∈ X (T]

k M , G, F) is an metrical (k − 1)n-almost contact space Therefore the triple (T] named the geometrical model of the Lagrange space of order k, L(k)n . Using this space we can study the electromagnetic and gravitational fields in the spaces L(k)n , [161].

2.5 The generalized Lagrange spaces of order k The notion of generalized Lagrange space of higher order is a natural extension of that studied in chapter 2. Definition 2.5.1. A generalized Lagrange space of order k is a pair GL(k)n = (M, gij ) formed by a real differentiable n−dimensional manifold M and a C ∞ −covariant k M , having the properties: of type (0,2), symmetric d−tensor field gij on T] kM ; a. gij has a constant signature on T] kM . b. rank(gij ) = n on T]

gij is called the fundamental tensor of GL(k)n . Evidently, any Lagrange space of order k, L(k)n = (M, L) determines a space GL(k)n with fundamental tensor 1 ∂ 2L gij = . 2 ∂y (k)i ∂y (k)j

(2.1)

But not and conversely. If gij (x, y (1) , ..., y (k) ) is a priori given, it is possible that the system of differential partial equations (2.5.1) does not admit any solution in L(x, y (1) , ..., y (k) ). A necessary condition that the system (2.5.1) admits solutions in the function L is that d-tensor field C (k) ijh

=

1 ∂gij 2 ∂y (k)h

(2.2)

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CHAPTER 2. LAGRANGE SPACES OF HIGHER–ORDER

be completely symmetric. If the system (2.5.1) has solutions, with respect to L we say that the space GL(k)n is reducible to a Lagrange space of order k. If this property is not true, then GLn is said to be nonreducible to a Lagrange space L(k)n . Examples 1◦ Let R = (M, γij (x)) be a Riemannian space and σ ∈ F (T k M ). Consider the d-tensor field: gij = e2σ (γij ◦ π k ). (2.3) ∂σ k M , then the pair is a nonvanishes d−covector on the manifold T] ∂y (k)h GL(k)n = (M, gij ) is a generalized Lagrange space of order k and it is not reducible to a Lagrange space L(k)n .

If

2◦ Let Rn = (M, γij (x)) be a Riemann space and Prolk Rn be its prolongation kM . of order k to T] Consider the Liouville d−vector field z (k)i of Prolk Rn . It is expressed in the (k) formula (2.2.2). We can introduce the d−covector field z i = γij z (k)j . kM . We assume that exists a function n(x, y (1) , · · · , y (k) ) ≥ 1 on T]

Thus

1 gij = γij + 1 − 2 n

(k) (k)

zi zj

(2.4)

is the fundamental tensor of a space GL(k)n . Evidently this space is not reducible to a space L(k)n , if the function n 6= 1. These two examples prove the existence of the generalized Lagrange space of order k. In the last example, k = 1 leads to the metric Part I, Ch. 2 of the Relativistic Optics, (n being the refractive index). In a generalized Lagrange space GL(k)n is diﬀicult to find a nonlinear connection N derived only by the fundamental tensor gij . Therefore, assuming that N is a priori given , we shall study the pair (N, GL(k)n ). Thus of theorem of the existence and uniqueness metrical N −linear connection holds: Theorem 2.5.1. We have:

2.5. THE GENERALIZED LAGRANGE SPACES OF ORDER K

139

1◦ There exists an unique N -linear connection D for which (α)

gij|h = 0, gij | h = 0, (α = 1, ..., k), T ijk = 0, S i jk = 0, (α = 1, ..., k). (α)

2◦ The coeﬀicients of D are given by the generalized Christoffel symbols Part II, Ch. 2. 3◦ D depends on gij and N only. Using this theorem it is not diﬀicult to study the geometry of Generalized Lagrange spaces.

140

CHAPTER 2. LAGRANGE SPACES OF HIGHER–ORDER

Chapter 3 Higher-Order Finsler spaces The notion of Finsler spaces of order k, introduced by the author of this monograph and presented in the book The Geometry of Higher-Order Finsler Spaces, Hadronic k M of the theory of Finsler Press, 1998, is a natural extension to the manifold T] spaces given in the Part I, Ch. 3. A substantial contribution in the studying of these spaces have H. Shimada and S. Sabău [223]. The impact of this geometry in Differential Geometry, Variational Calculus, Analytical Mechanics and Theoretical Physics is decisive. Finsler spaces play a role in applications to Biology, Engineering, Physics or Optimal Control. Also, the introduction of the notion of Finsler space of order k is demanded by the solution of problem of prolongation to T k M of the Riemannian or Finslerian structures defined on the base manifold M .

3.1 Notion of Finsler space of order k In order to introduce the Finsler space of order k are necessary some considerations on the concept of homogeneity of functions on the manifold T k M , [161]. k M and continuous on the null A function f : T k M → R of C ∞ −class on T] section of π k : T k M → M is called homogeneous of degree r ∈ Z on the fibres of T k M (briefly r−homogeneous) if for any a ∈ R+ we have f (x, ay (1) , a2 y (2) , · · · , ak y (k) ) = ar f (x, y (1) , · · · , y (k) ). An Euler Theorem hold: k M and continuous on the null A function f ∈ F (T k M ), differentiable on T] section of π k is r−homogeneous if and only if L k f = rf, Γ

141

(3.1)

142

CHAPTER 3. HIGHER-ORDER FINSLER SPACES k

L k being the Lie derivative with respect to the Lioville vector field Γ. Γ A vector field X ∈ X (T k M ) is r−homogeneous if L k X = (r − 1)X

(3.1.1’)

Γ

Definition 3.1.1. A Finsler space of order k, k ≥ 1, is a pair F (k)n = (M, F ) determined by a real differentiable manifold M of dimension n and a function F : T k M → R having the following properties: k M and continuous on the null section on π k . 1◦ F is differentiable on T]

2◦ F is positive. 3◦ F is k-homogeneous on the fibres of the bundle T k M . 4◦ The Hessian of F 2 with the elements: 1 ∂ 2F 2 gij = 2 ∂y (k)i ∂y (k)j

(3.2)

kM . is positively defined on T]

From this definition it follows that the fundamental tensor gij is nonsingular and 0-homogeneous on the fibres of T k M . Also, we remark: Any Finlser space F (k)n can be considered as a Lagrange space L(k)n = (M, L), whose fundamental function L is F 2 . By means of the solution of the problem of prolongation of a Finsler structure F (x, y (1) ) to T k M we can construct some important examples of spaces F (k)n . A Finsler space with the property gij depend only on the points x ∈ M is called a Riemann space of order k and denoted by R(k)n . Consequently, we have the following sequence of inclusions, similar with that from Ch. 3: n o n o n o n o (k)n (k)n (k)n (k)n R ⊂ F ⊂ L ⊂ GL . (3.3) So, the Lagrange geometry of order k is the geometrical theory of the sequence (1.1.3). Of course the geometry of F (k)n can be studied as the geometry of Lagrange space of order k, L(k)n = (M, F 2 ). Thus the canonical nonlinear connection N is the Cartan nonlinear connection of F (k)n and the metrical N −linear connection D is the Cartan N −metrical connection of the space F (k)n , [161].

Chapter 4 The Geometry of k−cotangent bundle 4.1 Notion of k−cotangent bundle, T ∗k M The k−cotangent bundle (T ∗k M, π ∗k , M ) is a natural extension of that of cotangent bundle (T ∗ M, π ∗ , M ). It is basic for the Hamilton spaces of order k. The manifold T ∗k M must have some important properties: 1◦ T ∗1 M = T ∗ M ; 2◦ dimT ∗k M =dimT k M = (k + 1)n; 3◦ T ∗k M carries a natural Poisson structure; 4◦ T ∗k M is local diffeomorphic to T k M . These properties are satisfied by considering the differentiable bundle (T ∗k M, π ∗k , M ) as the fibered bundle (T k−1 M xM T ∗ M, π k−1 xM π ∗ , M ). So we have T ∗k M = T k−1 M xM T ∗ M, π ∗k = π k−1 xM π ∗ .

(4.1)

A point u ∈ T ∗k M is of the form u = (x, y (1) , ..., y (k−1) , p). It is determined by dxi 1 dk−1 xi i (1)i (k−1)i the point x = (x ) ∈ M , the acceleration y = , ..., y = dt (k − 1)! dtk−1 and the momenta p = (pi ). The geometries of the manifolds T k M and T ∗k M are dual via Legendre transformation. Then, (xi , y (1)i , ..., y (k−1)i , pi ) are the local coordinates of a point u ∈ T ∗k M . 143

144

CHAPTER 4. THE GEOMETRY OF K−COTANGENT BUNDLE

The change of local coordinates on T ∗k M is:

∂e xi i i 1 n x e =x e (x , ..., x ), det ∂xj ye

(1)i

6= 0,

∂e xi (1)j = jy , ∂x (4.2)

........... (k − 1)e y

(k−1)i

∂e y (k−1)i (1)j ∂e y (k−2)i (k−1)j = y + ... + (k − 1) (k−2)j y ∂xj ∂y

∂xj pei = pj ∂e xi where the following equalities hold ∂e y (α)i ∂e y (α+1)i ∂e y (k−1)i = = ... = ; (α = 0, ..., k − 2; y (0) = x). j (1)j (k−1−α)j ∂x ∂y ∂y

(4.3)

The Jacobian matrix Jk of (4.1.2) have the property

∂e xj det Jk (u) = det (u) ∂xi

k−1 .

(4.4)

∂ The vector fields ∂˙ i = (∂˙ 1 , .., ∂˙ n ) = generate a vertical distribution Wk , while ∂pi ∂ ∂ ∂ determine a vertical distribution Vk−1 , ..., , ..., (k−1)i deter∂y (k−1)i ∂y (1)i ∂y mine a vertical distribution V1 . We have the sequence of inclusions: Vk−1 ⊂ Vk−2 ⊂ ... ⊂ V1 ⊂ V, Vu = V1,u ⊕ Wk,u , ∀u ∈ T ∗k M. We obtain without diﬀiculties: Theorem 4.1.1.

1◦ The following operators in the algebra of functions on T ∗k M

4.1. NOTION OF K−COTANGENT BUNDLE, T ∗K M

145

are the independent vector field on T ∗k M : ∂

1

Γ = y (1)i

∂y (k−1)i ∂

2

Γ = y (1)i

∂y (k−2)i

+ 2y (2)i

∂ ∂y (k−1)i (4.5)

........... k−1

Γ = y (1)i

∂ ∂ (k−1)i + ... + (k − 1)y ∂y (1)i ∂y (k−1)i

C∗ = pi ∂˙ i . 2◦ The function φ = pi y (1)i

(4.6)

is a scalar function on T ∗k M . 1

k−1

Γ, ..., Γ are called the Liouville vector fields. ∗k M → R, T ∗ kM = ^ Theorem 4.1.2. 1◦ For any differentiable function H : T^ T ∗k M \ {0}, d0 H, ..., dk−2 H defined by

d0 H =

∂H dxi (k−1)i ∂y

d1 H =

∂H ∂H i dx + dy (1)i (k−2)i (k−1)i ∂y ∂y

(4.7)

........... dk−2 H =

∂H ∂H ∂H i (1)i dx + dy + ... + dy (k−2)i (1)i (2)i (k−1)i ∂y ∂y ∂y

∗k M . are fields of 1−forms on T^

2◦ While dk−1 H = is not a field of 1−form.

∂H i ∂H dx + ... + (k−1)i dy (k−1)i i ∂x ∂y

(4.8)

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CHAPTER 4. THE GEOMETRY OF K−COTANGENT BUNDLE

3◦ We have dH = dk−1 H + ∂˙ i Hpi .

(4.9)

If H = φ = pi y (1)i , then d0 H = ... = dk−1 H = 0 and ω = dk−2 φ = pi dxi

(4.10)

ω is called the Liouville 1−form. Its exterior differential is expressed by θ = dω = dpi ∧ dxi .

(4.11)

θ is a 2−form of rank 2n < (k + 1)n = dim T ∗k M , for k > 1. Consequence θ is a presymplectic structure on T ∗k M . Let us consider the tensor field of type (1, 1) on T ∗k M : J=

∂ ∂ ∂ ⊗ dxi + (2)i ⊗ dy (1)i + ... + (k−1)i ⊗ dy (k−2)i . (1)i ∂y ∂y ∂y

(4.12)

Theorem 4.1.3. We have: 1◦ J is globally defined. 2◦ J is integrable. 3◦ J ◦ J ◦ ... ◦ J = J k = 0. 4◦ ker J = Vk−1 ⊕ Wk . 5◦ rankJ = (k − 1)n. 1

2

1

k−1

k−2

6◦ J(Γ) = 0, J(Γ) = Γ, ..., J( Γ ) = Γ , J(C∗ ) = 0. Theorem 4.1.4.

1

k−1

1◦ For any vector field X ∈ X (T ∗k M ), X, ..., X given by 1

2

k−1

X = J(X), X = J 2 X, ..., X = J k−1 X

(4.13)

are vector fields. (0)i

(1)i ∂ (k−1)i ∂ ∂ 2 If X = X i + X (1)i + ... + X + Xi ∂˙ i . Then ∂x ∂y ∂y (k−1)i ◦

(k−2)i k−1 0i ∂ ∂ ∂ , ..., X = X . X = J(X) = X (1)i + ... + X ∂y ∂y (k−1)i ∂y (k−1)i 1

(0)i

(4.14)

4.1. NOTION OF K−COTANGENT BUNDLE, T ∗K M

147

On the manifold T ∗k M there exists a Poisson structure given by {f, g}0 = {f, g}α =

∂f ∂g ∂f ∂g − ∂xi ∂pi ∂pi ∂xi ∂f ∂g ∂f ∂g − (α = 1, ..., k − 1). ∂y (α)i ∂pi ∂pi ∂y (α)i

It is not diﬀicult to study the properties of this structure. For details see the book [163].

(4.15)

148

CHAPTER 4. THE GEOMETRY OF K−COTANGENT BUNDLE

Part III Analytical Mechanics of Lagrangian and Hamiltonian Mechanical Systems

149

151

This part is devoted to applications of the Lagrangian and Hamiltonian geometries of order k = 1 and k > 1 to Analytical Mechanics. Firstly, to classical Mechanics of Riemannian mechanical systems ΣR = (M, T, F e) for which the external forces F e depend on the material point x ∈ M and on the dxi velocities . So, in general, ΣR are the nonconservative systems. Then we dt are obliged to take F e as a vertical field on the phase space T M and apply the Lagrange geometry for study the geometrical theory of ΣR . More general, we introduce the notion on Finslerian mechanical system ΣF = (M, F, F e), where M is the configuration space, T M is the velocity space, F is the fundamental funcdx tion of a given Finsler space F n = (M, F (x, y)) and F e x, are the external dt forces defined as vertical vector field on the velocity space T M . The fundamental equations of ΣF are the Lagrange equations d ∂F 2 ∂F 2 dxi i − = Fi (x, y), y = , dt ∂y i ∂xi dt F 2 being the energy of Finsler space F n . More general, consider a triple ΣL = (M, L, F e), where M is space of configurations, Ln = (M, L) is a Lagrange space, and F e are the external forces. The fundamental equations ate the Lagrange equations, too. The dual theory leads to the Cartan and Hamiltonian mechanical systems and is based on the Hamilton equations. Finally, we remark the extension of such kind of analytical mechanics, to the higher order. Some applications will be done. These considerations are based on the paper [175] and on the papers of J. Klein [123], M. Crampin [64], Manuel de Leon [138]. Also, we use the papers of R. Miron, M. Anastasiei, I. Bucataru [166], and of R. Miron, H. Shimada, S. Sabau and M. Roman [174], [180].

152

Chapter 1 Riemannian mechanical systems 1.1 Riemannian mechanical systems Let gij (x) be Riemannian tensor field on the configuration space M . So its kinetic energy is 1 dxi i j i T = gij (x)y y , y = = x˙ i . (1.1) 2 dt Following J. Klein [123], we can give: Definition 1.1.1. A Riemannian Mechanical system (shortly RMS) is a triple ΣR = (M, T, F e), where 1◦ M is an n−dimensional, real, differentiable manifold (called configuration space). 1 2◦ T = gij (x)x˙ i x˙ j is the kinetic energy of an a priori given Riemannian space 2 Rn = (M, gij (x)). ∂ is a vertical vector field on the velocity space T M (F e ∂y i are called external forces).

3◦ F e(x, y) = F i (x, y)

Of course, ΣR is a scleronomic mechanical system. The covariant components of F e are: Fi (x, y) = gij (x)F i (x, y). (1.2) Examples: 1. RMS - for which F e(x, y) = a(x, y)C, a 6= 0. Thus F i = a(x, y)y i and ΣR is called a Liouville RMS. 153

154

CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

2. The RMS ΣR , where F e(x, y) = F i (x) ∂y∂ i , and Fi (x) = gradi f (x), called conservative systems. 3. The RMS ΣR , where F e(x, y) = F i (x) ∂y∂ i , but Fi (x) 6= gradi f (x), called non-conservative systems. Remark 1.1.0.1. 1. A conservative system ΣR is called by J. Klein [123] a Lagrangian system. 2. One should pay attention to not make confusion of this kind of mechanical systems with the “Lagrangian mechanical systems” ΣL = (M, L(x, y), F e(x, y)) introduced by R. Miron [175], where L : T M → R is a regular Lagrangian. Starting from Definition 1.1.1, in a very similar manner as in the geometrical theory of mechanical systems, one introduces Postulate. The evolution equations of a RSM ΣR are the Lagrange equations: d ∂L ∂L − = Fi (x, y), dt ∂y i ∂xi

dxi y = , L = 2T. dt i

(1.3)

This postulate will be geometrically justified by the existence of a semispray S on T M whose integral curves are given by the equations (1.1.3). Therefore, the integral curves of Lagrange equations will be called the evolution curves of the RSM ΣR . The Lagrangian L = 2T has the fundamental tensor gij (x). Remark 1.1.0.2. In classical Analytical Mechanics, the coordinates (xi ) of a dxi dq i material point x ∈ M are denoted by (q i ), and the velocities y i = by q˙i = . dt dt However, we prefer to use the notations (xi ) and (y i ) which are often used in the geometry of the tangent manifold T M . The external forces F e(x, y) give rise to the one-form σ = Fi (x, y)dxi .

(1.4)

Since F e is a vertical vector field it follows that σ is semibasic one form. Conversely, if σ from (1.1.4) is semibasic one form, then F e = F i (x, y) ∂y∂ i , with F i = g ij Fj , is a vertical vector field on the manifold T M . J Klein introduced the the external forces by means of a one-form σ, while R. Miron [175] defined F e as a vertical vector field on T M .

1.1. RIEMANNIAN MECHANICAL SYSTEMS

155

The RMS ΣR is a regular mechanical system because the Hessian matrix with ∂ 2T elements i j = gij (x) is nonsingular. ∂y ∂y We have the following important result. Proposition 1.1.1. The system of evolution equations (1.1.3) are equivalent to the following second order differential equations: d 2 xi dxj dxk 1 i dx i + γ (x) = F (x, ), jk dt2 dt dt 2 dt

(1.5)

where γijk (x) are the Christoffel symbols of the metric tensor gij (x). In general, for a RMS ΣR , the system of differential equations (1.1.5) is not autoadjoint, consequently, it can not be written as the Euler-Lagrange equations for a certain Lagrangian. ∂U (x) 1 , here U (x) a potential In the case of conservative RMS, with Fi (x) = − 2 ∂xi function, the equations (1.1.2) can be written as Euler-Lagrange equations for the Lagrangian T + U . They have T + U =constant as a prime integral. This is the reason that the nonconservative RMS ΣR , with F e depending on dxi i y = cannot be studied by the methods of classical mechanics. A good geodt metrical theory of the RMS ΣR should be based on the geometry of the velocity space T M . From (1.1.4) we can see that in the canonical parametrization t = s (s being the arc length in the Riemannian space Rn ), we obtain the following result: Proposition 1.1.2. If the external forces are identically zero, then the evolution curves of the system ΣR are the geodesics of the Riemannian space Rn . In the following we will study how the evolution equations change when the ¯ n = (M, g¯) such space Rn = (M, g) is replaced by another Riemannian space R that: ¯ n have the same parallelism of directions; 1◦ Rn and R ¯ n have same geodesics; 2◦ Rn and R ¯ n is conformal to Rn . 3◦ R In each of these cases, the Levi-Civita connections of these two Riemmanian spaces are transformed by the rule: 1◦ γ¯ i jk (x) = γ i jk (x) + δji αk (x);

156

CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

2◦ γ¯ i jk (x) = γ i jk (x) + δji αk (x) + δki αj (x); 3◦ γ¯ i jk (x) = γ i jk (x) + δji αk (x) + δki αj (x) − gjk (x)αi (x), where αk (x) is an arbitrary covector field on M , and αi (x) = g ij (x)αj (x). It follows that the evolution equations (1.1.3) change to the evolution equations of the system ΣR¯ as follows: 1◦ In the first case we obtain dxj dxk dxi 1 i dx dxk d 2 xi i + γ jk (x) = −α + F (x, ); α = αk (x) dt2 dt dt dt 2 dt dt

(1.6)

Therefore, even though ΣR is a conservative system, the mechanical system ΣR¯ is nonconservative system having the external forces 1 ∂ dxk i i ¯ F e = −α(x, y)y + F (x, y) , α = αk (x) . 2 ∂y i dt 2◦ In the second case we have

j k i d 2 xi dx dx dx 1 dx + γ i jk (x) = −2α + F i x, ; 2 dt dt dt dt 2 dt

and

α = αk (x)

dxk dt

(1.7)

dxk 1 ∂ , α = α (x) . F¯ e = −2α(x, y)y i + F i (x, y) k 2 ∂y i dt 3◦ In the third case F¯ e is 1 dxk ∂ i i i ¯ F e = 2(−αy + T α ) + F , α = αk (x) , 2 ∂y i dt

αi (x) = g ij (x)αj (x). The previous properties lead to examples with very interesting properties.

1.2 Examples of Riemannian mechanical systems Recall that in the case of classical conservative mechanical systems we have 2F e = grad U , where U (x) is a potential function. Therefore, the Lagrange equations are given by ∂ d ∂ (T + U ) − (T + U ) = 0. dt ∂y i ∂xi

1.2. EXAMPLES OF RIEMANNIAN MECHANICAL SYSTEMS

157

We obtain from here a prime integral T + U = h (constant) which give us the energy conservation law. In the nonconservative case we have numerous examples suggested by 1◦ , 2◦ , 3◦ from the previous section, where we take F i (x, y) = 0. Other examples of RMS can be obtained as follows 1. 2F e = −β(x, y)y i

∂ , ∂y i

(1.1)

where β = βi (x)y i is determined by the electromagnetic potentials βi (x), (i = 1, .., n). 2. F e = (T − β)y i

∂ , ∂y i

(1.2)

where β = βi (x)y i and T is the kinetic energy. 3. In the three-body problem, M. Bărbosu [36] applied the following conformal transformation: d¯ s2 = (T + U )ds2 to the classic Lagrange equations and had obtained a nonconservative mechanical system with external force field F e = (T + U )y i

∂ . ∂y i

∂ lead to classical nonconservative Rieman∂y i nian mechanical systems. For instance, for Fi = −gradi U + Ri (x) where Ri (x) are the resistance forces, and the configuration space M is R3 .

4. The external forces F e = F i (x)

1 ∂ 5. If M = R3 , T = mδij y i y j and F e = 2F i (x) i , then the evolution equations 2 ∂y are d 2 xi m 2 = F i (x), dt which is the Newton’s law.

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CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

6. The harmonic oscillator. M = Rn , gij = δij , 2Fi = −ωi2 xi (the summation convention is not applied) and ωi are positive numbers, (i = 1, .., n). The functions i 2

hi = (x ) +

ωi2 xi ,

and H =

n X

hi

i=1

are prime integrals. 7. Suggested by the example 6◦ , we consider a system ΣR with F e = −2ω(x)C, where ω(x) is a positive function and C is the Liouville vector field. The evolution equations, in the case M = Rn , are given by d 2 xi dxi + ω(x) = 0. dt2 dt dxi Putting y = , we can write dt i

dy i + ω(x)y i = 0, dt So,Z we obtain y i R C i e− ω(x(t))dt dt.

=

C i e−

R

(i = 1, .., n).

ω(x(t))dt

and therefore xi

=

C0i +

8. We can consider the systems ΣR having F e = 2aijk (x)y j y k

∂ , ∂y i

where aijk (x) is a symmetric tensor field on M . The external force field F e has homogeneous components of degree 2 with respect to y i . 9. Relativistic nonconservative mechanical systems can be obtained for a Minkowski metric in the space-time R4 . 10. A particular case of example 1◦ above [223] is the case when the external force field coeﬀicients F i (x, y) are linear in y i , i.e. F e = 2F i (x, y)

∂ i k ∂ = 2Y (x)y , k ∂y i ∂y i

1.3. THE EVOLUTION SEMISPRAY OF THE MECHANICAL SYSTEM ΣR

159

where Y : T M → T M is a fiber diffeomorphism called Lorentz force, namely for any x ∈ M , we have ∂ ∂ = Yij (x) j . Yx : Tx M → Tx M, Yx i ∂x ∂x Let us remark that in this case, formally, we can write the Lagrange equations of this RMS in the form ∇γ˙ γ˙ = Y (γ), ˙ where ∇ is the Levi-Civita connection of the Riemannian space (M, g) and γ˙ is the tangent vector along the evolution curves γ : [a, b] → M . This type of RMS is important because of the global behavior of its evolution curves. Let us denote by S the evolutionary semispray, i.e. S is a vector field on T M which is tangent to the canonical lift γˆ = (γ, γ) ˙ of the evolution curves (see the following section for a detailed discussion on the evolution semispray). We will denote by T a the energy levels of the Riemannian metric g, i.e. a2 a T = (x, y) ∈ T M : T (x, y) = , 2 where T is the kinetic energy of g, and a is a positive constant. One can easily see that T a is the hypersurface in T M of constant Riemannian length vectors, namely for any X = (x, y) ∈ T a , we must have |X|g = a, where |X|g is the Riemannian length of the vector field X on M . If we restrict ourselves for a moment to the two dimensional case, then it is known that for suﬀiciently small values of c the restriction of the flow of the semispray S to T c contains no less than two closed curves when M is the 2-dimensional sphere, and at least three otherwise. These curves projected to the base manifold M will give closed evolution curves for the given Riemannian mechanical system.

1.3 The evolution semispray of the mechanical system ΣR Let us assume that F e is global defined on M , and consider the mechanical system ΣR = (M, T, F e). We have

160

CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

Theorem 1.3.1 ([175]). The following properties hold good: 1◦ The quantity S defined by ◦ 1 i ∂ i ∂ i S = y ∂xi − (2 G − 2 F ) ∂y i , ◦ 2 Gi = γ i jk y j y k ,

(1.1)

is a vector field on the velocity space T M . 2◦ S is a semispray, which depends on ΣR only. 3◦ The integral curves of the semispray S are the evolution curves of the system ΣR . Proof. 1◦ Writing S in the form ◦ 1 S =S + F e, 2

(1.2) ◦

◦

where S is the canonical semispray with the coeﬀicients Gi , we can see immediately that S is a vector field on T M . ◦ ◦ 2 Since S is a semispray and F e a vertical vector field, it follows S is a semispray. From (1.3.1) we can see that S depends on ΣR , only. 3◦ The integral curves of S are given by dxi = yi; dt

◦ dy i 1 + 2 Gi (x, y) = F i (x, y). dt 2

(1.3)

Replacing y i in the second equation we obtain (1.1.2) S will be called the evolution or canonical semispray of the nonconservative Riemannian mechanical system ΣR . In the terminology of J. Klein [123], S is the dynamical system of ΣR . Based on S we can develop the geometry of the mechanical system ΣR on T M . Let us remark that S can also be written as follows: S = yi

∂ ∂ i − 2G (x, y) , ∂xi ∂y i

with the coeﬀicients

◦

1 2G = 2 Gi − F i . 2 i

(1.3.1’)

(1.4)

1.3. THE EVOLUTION SEMISPRAY OF THE MECHANICAL SYSTEM ΣR

161

We point out that S is homogeneous of degree 2 if and only if F i (x, y) is 2∂F i i homogeneous with respect to y . This property is not satisfied in the case ≡ 0, ∂y j and it is satisfied for examples 1 and 7 from section 1.2. We have Theorem 1.3.2. The variation of the kinetic energy T of a mechanical system ΣR , along the evolution curves (1.1.2), is given by: 1 dxi dT = Fi . dt 2 dt

(1.5)

Proof. A straightforward computation gives i dT ∂T dxi ∂T dy i d ∂T 1 dx ∂T dy i = i + i = − Fi + i = dt ∂x dt ∂y dt dt ∂y i 2 dt ∂y dt ∂T d 1 dxi dT 1 dxi i = y i − Fi =2 − Fi , dt ∂y 2 dt dt 2 dt and the relation (1.3.5) holds good. Corollary 1.3.2.1. T =constant along the evolution curves if and only if the Liouville vector C and the external force F e are orthogonal vectors along the evolution curves of Σ. Corollary 1.3.2.2. If Fi = gradi U then ΣR is conservative and T + U = h (constant) on the evolution curves of ΣR . If the external forces F e are dissipative, i.e. hC, F ei < 0, then from the previous theorem, it follows a result of Bucataru-Miron (see [49]): Corollary 1.3.2.3. The kinetic energy T decreases along the evolution curves if and only if the external forces F e are dissipative. Since the energy of ΣR is T (the kinetic energy), the Theorem 1.3.2 holds good in this case. The variation of T is given by (1.3.5) and hence we obtain: T is conserved along the evolution curves of ΣR if and only if the vector field F e and the Liouville vector field C are orthogonal.

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CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

1.4 The nonlinear connection of ΣR Let us consider the evolution semispray S of ΣR given by ◦i ∂ 1 i ∂ S=y − (2 − F ) i G ∂xi 2 ∂y i

with the coeﬀicients

(1.1)

◦

1 2G = 2 Gi − F i . (1.2) 2 Consequently, the evolution nonlinear connection N of the mechanical system ΣR has the coeﬀicients: i

N

i

◦

j

1 ∂F i 1 ∂F i i k = γjk y − . =N j − 4 ∂y j 4 ∂y j i

(1.3)

◦ dxi If the external forces F e does not depend by velocities y = , then N =N . dt Let us consider the helicoidal vector field (see Bucataru-Miron, [49], [50]) 1 ∂Fi ∂Fj Pij = − i (1.4) 2 ∂y j ∂y i

∂Fi : ∂y j 1 ∂Fi ∂Fj Qij = + i . 2 ∂y j ∂y

and the symmetric part of tensor

(1.5)

On T M, P gives rise to the 2-form: (1.4.40 )

P = Pij dxi ∧ dxj

and Q is the symmetric vertical tensor: (1.4.50 )

Q = Qij dxi ⊗ dxj .

Denoting by ∇ the dynamical derivative with respect to the pair (S, N ), one proves the theorem of Bucataru-Miron: Theorem 1.4.1. For a Riemannian mechanical system ΣR = (M, T, F e) the evolution nonlinear connection is the unique nonlinear connection that satisfies the following conditions: 1 (1)∇g = − Q 2

1.4. THE NONLINEAR CONNECTION OF ΣR

163

1 (2)θL (hX, hY ) = P (X, Y ), ∀X, Y ∈ χ(T M ), 2 where

◦

θL = 4gij δ y j ∧ dxi

(1.6)

is the symplectic structure determined by the metric tensor gij and the nonlinear ◦

connection N .

The adapted basis of the distributions N and V is given by

δ ∂ , , where δxi ∂y i

◦

∂ 1 ∂F j ∂ δ δ j ∂ = i −N i j = i + δxi ∂x ∂y δx 4 ∂y i ∂y j

(1.7)

and its dual basis (dxi , δy i ) has the 1-forms δy i expressed by 1 ∂F i j δy = dy + N j dx =δy − dx . 4 ∂y j i

i

i

j

◦

i

It follows that the curvature tensor Ri jk of N (from (1.4.3) is ◦ k k δN δN δ ∂ δ ∂ 1 k i j k Rk ij = − = − F ) ( G − δxj δxi δxj ∂y i δxi ∂y j 4 and the torsion tensor of N is: ◦ ∂ ∂ 1 k ∂N ki ∂N kj ∂ ∂ k k t ij = − = − ( G − F ) = 0. ∂y j ∂y i ∂y j ∂y i ∂y i ∂y j 4

(1.8)

(1.9)

(1.10)

These formulas have the following consequences. 1. The evolution nonlinear connection N of ΣR is integrable if and only if the curvature tensor Ri jk vanishes. 2. The nonlinear connection is torsion free, i.e. tkij = 0. The autoparallel curves of the evolution nonlinear connection N are given by the system of differential equations d 2 xi dx dxj i + N j x, = 0, dt2 dt dt which is equivalent to

◦ dx dxj 1 ∂F i dxj d 2 xi i + N j x, = . dt2 dt dt 4 ∂y j dt

(1.11)

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CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

In the initial conditions

dx x0 , dt

, locally one uniquely determines the 0

autoparallel curves of N . If F i is 2-homogeneous with respect to y i , then the previous system coincides with the Lagrange equations (1.1.4). Therefore, we have: Theorem 1.4.2. If the external forces F e are 2-homogeneous with respect velocities dxi i y = , then the evolution curves of ΣR coincide to the autoparallel curves of dt the evolution nonlinear connection N of ΣR . In order to proceed further, we need the exterior differential of 1-forms δy i . One obtains 1 d(δy i ) = dN ij ∧ dxj = Rikj dxj ∧ dxk + B ikj δy j ∧ dxk , (1.12) 2 where ∂ 2 Gi (1.13) B ikj = B ijk = k j ∂y ∂y are the coeﬀicients of the Berwald connection determined by the nonlinear connection N .

1.5 The canonical metrical connection CΓ(N ) The coeﬀicients of the canonical metrical connection CΓ(N ) = F ijk , C ijk are given by the generalized Christoffel symbols [166]: 1 δg δg δg sk js jk F ijk = g is + k − s , j 2 δx δx δx (1.1) 1 ∂gsk ∂gjs ∂gjk + − C ijk = g is 2 ∂y j ∂y k ∂y s where gij (x) is the metric tensor of ΣR . δgjk ∂gjk ∂gjk On the other hand, we have = and = 0, and therefore we obtain: δxi ∂xi ∂y i Theorem 1.5.1. The canonical metrical connection CΓ(N ) of the mechanical system ΣR has the coeﬀicients F ijk (x, y) = γ ijk (x),

C ijk (x, y) = 0.

(1.2)

1.5. THE CANONICAL METRICAL CONNECTION CΓ(N )

165

Let ω ij be the connection forms of CΓ(N ): ω ij = F ijk dxk + C ijk δy k = γ ijk (x)dxk .

(1.3)

Then, we have ([166]): Theorem 1.5.2. The structure equation of CΓ(N ) can be expressed by 1 i k i d(dx ) − dx ∧ ω = − Ω i, k 2

d(δy i ) − δy k ∧ ω ik = − Ω i , dω i − ω k ∧ ω i = − i , Ωj j j k 1

(1.4)

2

where the 2-forms of torsion Ω i , Ω i are as follows 1

Ω i = C ijk dxj ∧ δy k = 0, (1.5) 2

Ω i = Rijk dxj ∧ dxk + P ijk dxj ∧ δy k . Here Rijk is the curvature tensor of N and P ijk = γ ijk − γ ikj = 0. The curvature 2-form Ωij is given by 1 1 Ωij = Rj ikh dxk ∧ dxh + Pj ikh dxk ∧ δy h + Sj ikh δy k ∧ δy h , 2 2 where Rhi jk

δF ihj δF ihk − + F shj F isk − F shk F isj + C ihs Rsjk = = k j δx δx =

∂γ ihj ∂xk

−

∂γ ihk ∂xj

(1.6)

(1.7)

+ γ shj γ isk − γ shk γ isj = rhi jk

i is the Riemannian tensor of curvature of the Levi-Civita connection γjk (x) and the curvature tensors Pj i kh , Sj i kh vanish. Therefore, the tensors of torsion of CΓ(N ) are

Ri jk , T i jk = 0, S i jk = 0, P i jk = 0, C i jk = 0

(1.8)

and the curvature tensors of CΓ(N ) are Rj i kh (x, y) = rj i kh (x), Pj i kh (x, y) = 0, Sj i kh (x, y) = 0.

(1.9)

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CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

The Bianchi identities can be obtained directly from (1.5.4), taking into account the conditions (1.5.8) and (1.5.9). The h- and v-covariant derivatives of d-tensor fields with respect to CΓ(N ) = (γ ijk , 0) are expressed, for instance, by ∇k tij =

δtij − γ sik tsj − γ sjk tis , k δx

˙ k tij = ∂tij − C sik tsj − C sjk tis = ∂tij . ∇ ∂y k ∂y k Therefore, CΓ(N ) being a metric connection with respect to gij (x), we have ◦

∇k gij = ∇k gij = 0,

(1.10)

◦

(∇ is the covariant derivative with respect to Levi-Civita connection of gij ) and ˙ k gij = 0. ∇

(1.5.10’)

The deflection tensors of CΓ(N ) are Dij

δy i = ∇j y = j + y s γ isj = −N ij + y s γ isj δx i

and ˙ j y i = δ ij . dij = ∇ The evolution nonlinear connection of a Riemannian mechanical system ΣR given by (1.4.3) implies Dij ◦

where we have used Dj y

i

=−

◦i Nj

◦i =D j =

1 ∂F i 1 ∂F i s i + + y γsj = , 4 ∂y j 4 ∂y j

0. It follows

Proposition 1.5.1. For a Riemannian mechanical system the deflection tensors Dij and dij of the connection CΓ(N ) are expressed by Dij

1 ∂F i , = 4 ∂y j

dij = δji .

(1.11)

1.6. THE ELECTROMAGNETISM IN THE THEORY OF THE RIEMANNIAN MECHANICAL SYSTEMS

1.6 The electromagnetism in the theory of the Riemannian mechanical systems ΣR In a Riemannian mechanical system ΣR = (M, T, F e) whose external forces F e dedxi pend on the point x and on the velocity y i = , the electromagnetic phenomena dt appears because the deflection tensors Dij and dij from (1.5.11) nonvanish. Hence the d-tensors Dij = gih Dhj , dij = gih δjh = gij determine the h-electromagnetic tensor Fij and v-electromagnetic tensor fij by the formulas, [166]: 1 fij = (dij − dji ). 2

1 Fij = (Dij − Dji ), 2 By means of (1.5.11), we have

(1.1)

Proposition 1.6.1. The h- and v-tensor fields Fij and fij are given by 1 Fij = Pij , fij = 0. 4 where Pij is the helicoidal tensor (1.4.4) of ΣR . Indeed, we have 1 1 Fij = (Dij − Dji ) = 4 8

∂F i ∂F j − ∂y j ∂y i

(1.2)

1 = Pij . 4

If we denote Rijk := gih Rhjk , then we can prove: Theorem 1.6.1. The electromagnetic tensor Fij of the mechanical system ΣR = (M, T, F e) satisfies the following generalized Maxwell equations: ∇k Fji + ∇i Fkj + ∇j Fik = −(Rkji + Rikj + Rjik ). ˙ k Fji = 0. ∇

(1.3) (1.6.3’)

Proof. Applying the Ricci identities to the Liouville vector field, we obtain k ˙ j Dki = 0 ∇i Dkj − ∇j Dki = y h rhk ij − Rij , ∇i dkj − ∇

and this leads to ∇i Dkj − ∇j Dki = y h rhkij − Rkij , ˙ j Dki = 0. ∇

(1.4) (1.6.4’)

By taking cyclic permutations of the indices i,k,j and adding in (1.6.4), by taking into account the identity rhijk +rhjki +rhkij = 0 we deduce (1.6.3), and analogously (1.6.3’).

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CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

From equations (1.6.2) and (1.6.4’) we obtain as consequences: Corollary 1.6.1.1. The electromagnetic tensor Fij of the mechanical system ΣR = dxi i (M, T, F e) does not depend on the velocities y = . dt ˙ j Fik = ∂Fik = 0. Indeed, by means of (1.6.3’) we have ∇ ∂y j In other words, the helicoidal tensor Pij of ΣR does not depend on the velocities dxi i . y = dt We end the present section with a remark: this theory has applications to the mechanical systems given by example 1◦ in Section 1.2.

1.7 The almost Hermitian model of the RMS ΣR Let us consider a RMS ΣR = (M, T (x, y), F e(x, y)) endowed with the evolution nonlinear connection with coeﬀicients Nji from (1.4.3) and with the canonical N i metrical connection CΓ(N ) = (γjk (x), 0). Thus, on the velocity space Tg M = T M \ {0} we can determine an almost Hermitian structure H 2n = (Tg M , G, F) which depends onthe RMS only. δ ∂ Let , be the adapted basis to the distributions N and V and its δxi ∂y i adapted cobasis (dxi , δy i ), where ◦

δ δ 1 ∂F s ∂ = + i s δxi δxi 4 ∂y ∂y

(1.1)

◦

1 ∂F i s s dx 4 ∂y The lift of the fundamental tensor gij (x) of the Riemannian space Rn = (M, gij (x)) is defined by δy i =δ y i −

G = gij dxi ⊗ dxj + gij δy i ⊗ δy j ,

(1.2)

and the almost complex structure F, determined by the nonlinear connection N , is expressed by ∂ δ F = − i ⊗ dxi + i ⊗ δy i . (1.3) ∂y δx Thus, the following theorems hold good.

1.7. THE ALMOST HERMITIAN MODEL OF THE RMS ΣR

169

Theorem 1.7.1. We have: 1. The pair (Tg M , G) is a pseudo-Riemannian space. 2. The tensor G depends on ΣR only. 3. The distributions N and V are orthogonal with respect to G. Theorem 1.7.2.

1. The pair (Tg M , F) is an almost complex space.

2. The almost complex structure F depends on ΣR only. i 3. F is integrable on the manifold Tg M if and only if the d-tensor field Rjk (x, y) vanishes.

Also, it is not diﬀicult to prove Theorem 1.7.3. We have 1. The triple H 2n = (Tg M , G, F) is an almost Hermitian space. 2. The space H 2n depends on ΣR only. 3. The almost symplectic structure of H 2n is θ = gij δy i ∧ dxj .

(1.4)

If the almost symplectic structure θ is a symplectic one (i.e. dθ = 0), then the space H 2n is almost Kählerian. On the other hand, using the formulas (1.7.4), (1.4.12) one obtains dθ =

1 (Rijk + Rjki + Rkij )dxi ∧ dxj ∧ dxk 3!

1 s s + (gis Bjk − gjs Bik )δy k ∧ dxj ∧ dxi . 2 Therefore, we deduce Theorem 1.7.4. The almost Hermitian space H 2n is almost Kählerian if and only if the following relations hold good Rijk + Rjki + Rkij = 0,

s s gis Bjk − gjs Bik = 0.

(1.5)

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CHAPTER 1. RIEMANNIAN MECHANICAL SYSTEMS

The space H 2n = (Tg M , G, F) is called the almost Hermitian model of the RMS ΣR . One can use the almost Hermitian model H 2n to study the geometrical theory of the mechanical system ΣR . For instance the Einstein equations of the RMS ΣR are the Einstein equations of the pseudo-Riemannian space (Tg M , G) (cf. Chapter 6, part 1). Remark 1.7.4.1. The previous theory can be applied without diﬀiculties to the examples 1-8 in Section 1.2.

Chapter 2 Finslerian Mechanical systems The present chapter is devoted to the Analytical Mechanics of the Finslerian Mechanical systems. These systems are defined by a triple ΣF = (M, F 2 , F e) where M is the configuration space, F (x, y) is the fundamental function of a semidefinite Finsler space F n = (M, F (x, y)) and F e(x, y) are the external forces. Of course, F 2 is the kinetic energy of the space. The fundamental equations are the Lagrange equations: d ∂F 2 ∂F 2 2 Ei (F ) ≡ − = Fi (x, x). ˙ dt ∂ x˙ i ∂xi We study here the canonical semispray S of ΣF and the geometry of the pair (T M, S), where T M is velocity space. One obtain a generalization of the theory of Riemannian Mechanical systems, which has numerous applications and justifies the introduction of such new kind of analytical mechanics.

2.1 Semidefinite Finsler spaces Definition 2.1.1. A Finsler space with semidefinite Finsler metric is a pair F n = (M, F (x, y)) where the function F : T M → R satisfies the following axioms: 1◦ F is differentiable on Tg M and continuous on the null section of π : T M → M ; 2◦ F ≥ 0 on T M ; 3◦ F is positive 1-homogeneous with respect to velocities x˙ i = y i . 4◦ The fundamental tensor gij (x, y) 1 ∂ 2F 2 gij = 2 ∂y i ∂y j 171

(2.1)

172

CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

has a constant signature on Tg M; 5◦ The Hessian of fundamental function F 2 with elements gij (x, y) is nonsingular: det(gij (x, y)) 6= 0 on Tg M. (2.2) Example. If gij (x) is a semidefinite Riemannian metric on M , then q F = |gij (x)y i y j |

(2.3)

is a function with the property F n = (M, F ) is a semidefinite Finsler space. Any Finsler space F n = (M, F (x, y)), in the sense of definition 3.1.1, part I, is a definite Finsler space. In this case the property 5◦ is automatical verified. But, these two kind of Finsler spaces have a lot of common properties. Therefore, we will speak in general on Finsler spaces. The following properties hold: 1◦ The fundamental tensor gij (x, y) is 0-homogeneous; 2◦ F 2 = gij (x, y)y i y j ; 1 ∂F 2 is d−covariant vector field; 3 pi = 2 ∂y i ◦

4◦ The Cartan tensor Cijk

1 ∂gij 1 ∂ 3F 2 = = 4 ∂y i ∂y j ∂y k 2 ∂y k

(2.4)

y i Cijk = C0jk = 0.

(2.5)

is totally symmetric and 5◦ ω = pi dxi is 1-form on Tg M (the Cartan 1-form); 6◦ θ = dω = dpi ∧ dxi is 2-form (the Cartan 2-form); 7◦ The Euler-Lagrange equations of F n are d ∂F 2 ∂F 2 dxi i Ei (F ) = − = 0, y = dt ∂y i ∂xi dt 2

(2.6)

8◦ The energy EF of F n is EF = y

i ∂F

2

∂y i

− F2 = F2

(2.7)

2.1. SEMIDEFINITE FINSLER SPACES

173

9◦

The energy EF is conserved along to every integral curve of Euler-Lagrange equations (2.1.6);

10◦

In the canonical parametrization, the equations (2.1.6) give the geodesics of F n ;

11◦

The Euler-Lagrange equations (2.1.6) can be written in the equivalent form

i where γjk

dx x, dt

dx dxj dxk d 2 xi i + γjk (x, ) = 0, dt2 dt dt dt

(2.8)

are the Christoffel symbols of the fundamental tensor

gij (x, y). 12◦

The canonical semispray S is S = yi

∂ ∂ i − 2G (x, y) ∂xi ∂y i

(2.9)

with the coeﬀicients: i i 2Gi (x, y) = γjk (x, y)y j y k = γ00 (x, y),

(2.1.9’)

(the index “0” means the contraction with y i ). 13◦

The canonical semispray S is 2-homogeneous with respect to y i . So, S is a spray.

14◦

The nonlinear connection N determined by S is also canonical and it is exactly the famous Cartan nonlinear connection of the space F n . Its coeﬀicients are ∂Gi (x, y) 1 ∂ i i = γ (x, y). N j (x, y) = (2.10) ∂y j 2 ∂y j 00 An equivalent form for the coeﬀicients Nji is as follows i i k N i j = γj0 (x, y) − Cjk (x, y)γ00 (x, y).

(2.1.10’)

Consequently, we have i N i 0 = γ00 = 2Gi .

(2.11)

Therefore, we can say: The semispray S 0 determined by the Cartan nonlinear connection N is the canonical spray S of space F n .

174

15◦

CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

The Cartan nonlinear connection N determines a splitting of vector space Tu T M, ∀u ∈ T M of the form: Tu T M = Nu ⊕ Vu , ∀u ∈ T M (2.12) δ ∂ Thus, the adapted basis , , (i = 1, .., n), to the previous splitting δxi ∂y i δ has the local vector fields i given by: δx δ ∂ ∂ = i − N j i (x, y) j , i δx ∂x ∂y

i = 1, .., n,

(2.13)

with the coeﬀicients N i j (x, y) from (2.1.6). Its dual basis is dxi , δy i , where δy i = dy i + N i j (x, y)dxj .

(2.14)

The autoparallel curves of the nonlinear connection N are given by, [166], dx dxj d 2 xi i + N j (x, ) = 0. dt2 dt dt

(2.15)

Using the dynamic derivative ∇ defined by N , the equations (2.1.11) can be written as follows dxi ∇( ) = 0. (2.1.11’) dt 16◦

The variational equations of the autoparallel curves (2.1.11) give the Jacobi equations: i j j k ∂N dx j 2 i i k i dx dx ∇ξ + − N k ∇ξ + Rjk = 0. (2.16) ∂y k dt dt dt The vector field ξ i (t) along a solution c(t) of the equations (2.1.11) and which verifies the previous equations is called a Jacobi field. In the Riemannian ∂gij = 0, the Jacobi equations (2.1.12) are exactly the classical Jacobi case, ∂y k equations: dxl dxj k 2 i i ξ =0 (2.17) ∇ ξ + Rjlk (x) dt dt

2.1. SEMIDEFINITE FINSLER SPACES

17◦

175

A distinguished metric connections D with the coeﬀicients i i CΓ(N ) = Fjk , Cjk is defined as a N -linear connection on T M , metric with respect to the fundamental tensor gij (x, y) of Finsler space F n , i.e. we have gij|k =

δgij s − Fiks gsj − Fjk gis = 0, k δx

(2.18)

∂gij s s − Cik gsj − Cjk gis = 0. gij |k = k ∂y 18◦

The following theorem holds: Theorem 2.1.1. 1◦ There is an unique N -linear connection D, with coefficients CΓ(N ) which satisfies the following system of axioms: A1 . N is the Cartan nonlinear connection of Finsler space F n . A2 . D is metrical, (i.e. D satisfies (2.1.14)). i i i i i i A3 . Tjk = Fjk − Fkj = 0, Sjk = Cjk − Ckj = 0. i i 2◦ The metric N -linear connection D has the coeﬀicients CΓ(N ) = Fjk , Cjk given by the generalized Christoffel symbols i Fjk

1 = g is 2

1 i Cjk = g is 2 20◦

δgsj δgsk δgjk + j − s δxk δx δx

∂gsj ∂gsk ∂gjk + − ∂y k ∂y j ∂y s

, (2.19)

.

By means of this theorem, it is not diﬀicult to see that we have i Cjk = g is Csjk

(2.20)

y i |k = 0.

(2.21)

and

21◦

The Cartan nonlinear connection N determines on Tg M an almost complex structure F, as follows: F(

∂ δ ) = − , δxi ∂y i

F(

∂ δ ) = , ∂y i δxi

i = 1, .., n.

(2.22)

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

But one can see that F is the tensor field on Tg M: ∂ δ F = − i ⊗ dxi + i ⊗ δy i , ∂y δx

(2.1.22’)

δ given by (2.1.10), (2.1.9), (2.1.6). δxi It is not diﬀicult to prove that: The almost complex structure F is integrable if and only if the distribution N is integrable on T M .

with the 1-forms δy i and the vector field

22◦

The Sasaki-Matsumoto lift of the fundamental tensor gij of Finsler space F is G(x, y) = gij (x, y)dxi ⊗ dxj + gij (x, y)δy i ⊗ δy j . (2.23) n

The tensor field G determines a pseudo-Riemannian structure on T M . 23◦

The following theorem is known: Theorem 2.1.2. 1◦ The pair (G, F) is an almost Hermitian structure on Tg M n determined only by the Finsler space F . 2◦ The symplectic structure associate to the structure (G, F) is the Cartan 2-form: θ = 2gij δy i ∧ dxj . (2.24) 3◦ The space (Tg M , G, F) is almost Kählerian. The space H 2n = (Tg M , G, F) is called the almost Kählerian model of the n Finsler space F .

G.S. Asanov in the paper [27] proved that the metric G from (2.1.23) does not satisfies the principle of the Post-Newtonian Calculus. This is due to the fact that the horizontal and vertical terms of G do not have the same physical dimensions. This is the reason for R. Miron to introduce a new lift of the fundamental tensor gij , [166], in the form: a2 i j e G(x, y) = gij (x, y)dx ⊗ dx + gij (x, y)δy i ⊗ δy j 2 ||y|| where a > 0 is a constant imposed by applications in Theoretical Physics and where kyk2 = gij (x, y)y i y j = F 2 has the property F 2 > 0. The lift G is 2homogeneous with respect to y i . The Sasaki-Matsumoto lift G has not the property of homogeneity.

2.2. THE NOTION OF FINSLERIAN MECHANICAL SYSTEM

177

Two examples: 1. Randers spaces. They have been defined by R. S. Ingarden as a triple RF n = (M, α + β, N ), where α + β is a Randers metric and N is the Cartan nonlinear connection of the Finsler space F n = (M, α + β), [175]. 2. Ingarden spaces. These spaces have been defined by R. Miron, [166], as a triple IF n = (M, α + β, NL ), where α + β is a Randers metric and NL is the Lorentz nonlinear connection of F n = (M, α + β) having the coeﬀicients ◦ ◦ ◦ ∂bj ∂bs 1 is i i k i i − . (2.25) Nj (x, y) =γ jk (x)y − F j (x), F j = a (x) 2 ∂xj ∂xs The Christoffel symbols are constructed with the Riemannian metric tensor ◦ aij (x) of the Riemann space (M, α2 ) and F ij (x) is the electromagnetic tensor determined by the electromagnetic form (α + β).

2.2 The notion of Finslerian mechanical system As we know from the previous chapter, the Riemannian mechanical systems ΣR = (M, T, F e) is defined as a triple in which M is the configuration space, T is the kinetic energy and F e are the external forces, which depend on the material point dxi i . x ∈ M and depend on velocities y = dt Extending the previous ideas, we introduce the notion of Finslerian Mechanical System, studied by author in the paper [175]. The shortly theory of this analytical mechanics can be find in the joint book Finsler-Lagrange Geometry. Applications to Dynamical Systems, by Ioan Bucataru and Radu Miron, Romanian Academy Press, Bucharest, 2007. In a different manner, M. de Leon and colab. [138], M. Crampin et colab. [66], have studied such kind of new Mechanics. A Finslerian mechanical system ΣF is defined as a triple ΣF = (M, EF 2 , F e)

(2.1)

where M is a real differentiable manifold of dimension n, called the configuration space, EF 2 is the energy of an a priori given Finsler space F n = (M, F (x, y)), which can be positive defined or semidefined, and F e(x, y) are the external forces given as a vertical vector field on the tangent manifold T M . We continue to say that T M is the velocity space of M .

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

Evidently, the Finslerian mechanical system ΣF is a straightforward generalization of the known notion of Riemannian mechanical system ΣR obtained for EF 2 as kinetic energy of a Riemann space Rn = (M, g). Therefore, we can introduce the evolution (or fundamental) equations of ΣF by means of the following Postulate: Postulate. The evolution equations of the Finslerian mechanical system ΣF are the Lagrange equations: d ∂EF 2 ∂EF 2 − = Fi (x, y), dt ∂y i ∂xi

yi =

dxi dt

(2.2)

where the energy is EF 2 = y

i ∂F

2

∂y i

− F 2 = F 2,

(2.3)

and Fi (x, y), (i = 1, .., n), are the covariant components of the external forces F e: ∂ F e(x, y) = F i (x, y) i ∂y (2.4) Fi (x, y) = gij (x, y)F i (x, y), and

1 ∂ 2F 2 gij (x, y) = , det(gij (x, y)) 6= 0, (2.5) 2 ∂y i ∂y j is the fundamental (or metric) tensor of Finsler space F n = (M, F (x, y)). Finally, the Lagrange equations of the Finslerian mechanical system are: d ∂F 2 ∂F 2 − = Fi (x, y), dt ∂y i ∂xi

dxi y = . dt i

(2.6)

A more convenient form of the previous equations is given by: Theorem 2.2.1. The Lagrange equations (2.2.6) are equivalent to the second order differential equations: dx dxj dxk 1 i dx d 2 xi i + γ (x, ) = F (x, ), jk dt2 dt dt dt 2 dt

(2.7)

i (x, y) are the Christoffel symbols of the metric tensor gij (x, y) of the where γjk Finsler space F n .

2.2. THE NOTION OF FINSLERIAN MECHANICAL SYSTEM

179

Proof. Writing the kinetic energy F 2 (x, y) in the form: F 2 (x, y) = gij (x, y)y i y j ,

(2.8)

the equivalence of the systems of equations (2.2.6) and (2.2.7) is not diﬀicult to establish. But, the form (2.2.7) is very convenient in applications. So, we obtain a first result expressed in the following theorems: Theorem 2.2.2. The trajectories of the Finslerian mechanical system ΣF , without external forces (F e ≡ 0), are the geodesics of the Finsler space F n . Indeed, F i (x, y) ≡ 0 and the SODE (2.2.7) imply the equations (2.2.4) of geodesics of space F n . A second important result is a consequence of the Lagrange equations, too. Theorem 2.2.3. The variation of kinetic energy EF 2 = F 2 of the mechanical system ΣF along the evolution curves (2.2.6) is given by dEF 2 dxi = Fi . dt dt

(2.9)

Consequently: Theorem 2.2.4. The kinetic energy EF 2 of the system ΣF is conserved along the evolution curves (2.2.6) if the external forces F e are orthogonal to the evolution curves. The external forces F e are called dissipative if the scalar product hC, F ei is negative, [175]. The formula (2.2.9) leads to the following property expressed by: Theorem 2.2.5. The kinetic energy EF 2 decreases along the evolution curves of the Finslerian mechanical system ΣF if and only if the external forces F e are dissipative.

Some examples of Finslerian mechanical systems 1◦ The systems ΣF = (M, EF 2 , F e) given by F n = (M, α + β) as a Randers space ∂ and F e = βC = βy i i . Evidently F e is 2-homogeneous with respect to y i . ∂y

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

2◦ ΣF determined by F n = (M, α + β) and F e = αC. 3◦ ΣF with F n = (M, α + β) and F e = (α + β)C. 4◦ ΣF defined by a Finsler space F n = (M, F ) and F e = aijk (x)y j y k

∂ , aijk (x) i ∂y

being a symmetric tensor on the configuration space M of type (1, 2).

2.3 The evolution semispray of the system ΣF The Lagrange equations (2.2.6) give us the integral curves of a remarkable semispray on the velocity space T M , which governed the geometry of Finslerian mechanical system ΣF . So, if the external forces F e are global defined on the manifold T M , we obtain: Theorem 2.3.1 (Miron, [175]). For the Finslerian mechanical systems ΣF , the following properties hold good: 1◦ The operator S defined by ◦ ◦ 1 ∂ ∂ i i S = yi i − 2 G i − F i ; 2 = γjk (x, y)y j y k (2.1) G i ∂x 2 ∂y is a vector field, global defined on the phase space T M . 2◦ S is a semispray which depends only on ΣF and it is a spray if F e is 2-homogeneous with respect to y i . 3◦ The integral curves of the vector field S are the evolution curves given by the Lagrange equations (2.2.7) of ΣF . ◦

Proof. 1◦ Let us consider the canonical semispray S of the Finsler space F n . Thus from (2.3.1) we have ◦ 1 S =S + F e. (2.2) 2 It follows that S is a vector field on T M . 2◦ Since F e is a vertical vector field, then S is a semispray. Evidently, S depends on ΣF , only. 3◦ The integral curves of S are given by: ◦ dy i 1 dxi i =y; + 2 Gi (x, y) = F i (x, y). (2.3) dt dt 2 The previous system of differential equations is equivalent to system (2.2.7). In the book of I. Bucataru and R. Miron [49], one proves the following important result, which extend a known J. Klein theorem:

2.4. THE CANONICAL NONLINEAR CONNECTION OF THE FINSLERIAN MECHANICAL SYSTEMS Σ

Theorem 2.3.2. The semispray S, given by the formula (2.3.1), is the unique vector field on Tg M , solution of the equation: ◦

iS ω= −dT + σ,

(2.4)

1 ◦ where ω is the symplectic structure of the Finsler space F n = (M, F ), T = F 2 = 2 1 dxi dxj gij and σ is the 1-form of external forces: 2 dt dt σ = Fi (x, y)dxi .

(2.5)

In the terminology of J. Klein, [123], S is the dynamical system of ΣF , defined on the tangent manifold T M . We will say that S is the evolution semispray of ΣF . By means of semispray S (or spray S) we can develop the geometry of the Finslerian mechanical system ΣF . So, all geometrical notion derived from S, as nonlinear connections, N-linear connections etc. will be considered as belong to the system ΣF .

2.4 The canonical nonlinear connection of the Finslerian mechanical systems ΣF The evolution semispray S (2.3.1) has the coeﬀicients Gi expressed by ◦

1 1 i (x, y)y j y k − F i (x, y). (2.1) 2G (x, y) = 2 Gi (x, y) − F i (x, y) = γjk 2 2 Thus, the evolution nonlinear connection N (or canonical nonlinear connection) of the Finslerian mechanical system ΣF has the coeﬀicients: i

N ◦

i

j

◦ ∂Gi 1 ∂F i i = =N j − ∂y j 4 ∂y j

(2.2)

◦

where N with coeﬀicients N i j is the Cartan nonlinear connection of Finsler space F n = (M, F (x, y)). N depends on the mechanical system ΣF , only. It is called canonical for F n . The nonlinear connection N determines the horizontal distribution, denoted by N too, with the property T u T M = Nu ⊕ V u ,

∀u ∈ T M,

(2.3)

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

Vu being the natural vertical distribution on the tangent manifold T M . A localadapted basis to the horizontal and vertical vector spaces Nu and Vu is δ ∂ given by , , (i = 1, .., n), where δxi ∂y i ◦ ∂ ∂ 1 ∂F j ∂ δ j ∂ j = − N = − − , i = 1, .., n, (2.4) N i i δxi ∂xi ∂y j ∂xi 4 ∂y i ∂y j and the adapted cobasis (dxi , δy i ) with i ◦ 1 ∂F dxj , i = 1, .., n. δy i = dy i + N i j dxj = dy i + N i j − j 4 ∂y From (2.4.4) and (2.4.4’) it follows ◦ j δ ∂F ∂ δ 1 = + , 4 δxi δxi ∂y i ∂y j ◦ 1 ∂F i j i i dx . δy =δy − 4 ∂y j

(2.4.4’)

(2.5)

Now we easy determine the curvature Ri jk and torsion ti jk of the canonical nonlinear connection N . One obtains ◦ i i δN δ δN δ ∂ ∂ k j − = − j k (Gi − 21 F i ), Ri jk = k j k j δx δx δx ∂y δx ∂y (2.6) i i ∂N j ∂N k ti jk = − =0 ∂y k ∂y j such that: 1◦ The torsion of the canonical nonlinear connection N vanishes. 2◦ The condition Ri jk = 0 is necessary and suﬀicient for N to be integrable. Another important geometric object field determined by the canonical nonlinear i ∂N j connection N is the Berwald connection BΓ(N ) = (B i jk , 0) = , 0 . Its ∂y k coeﬀicients are: ◦ 1 ∂ 2F i i i B jk =B jk − , (2.7) 4 ∂y j ∂y k ◦

where Bi jk are coeﬀicients of Berwald connection of Finsler space F n . The coeﬀicients (2.4.7) are symmetric. We can prove:

2.5. THE DYNAMICAL DERIVATIVE DETERMINED BY THE EVOLUTION NONLINEAR CONNECTIO

Theorem 2.4.1. The autoparallel curves of the evolution nonlinear connection N are given by the following SODE: dxi i y = , dt

◦

δy i δ y i 1 ∂F i dxj = − = 0. dt dt 4 ∂y j dt

(2.8)

Corollary 2.4.1.1. If the external forces F e vanish, then the evolution nonlinear connection N is the Cartan nonlinear connection of Finsler space F n . Corollary 2.4.1.2. If the external forces F e are 2-homogeneous with respect to velocities y i , then the equations (2.4.8) coincide with the evolution equations of the Finslerian mechanical system ΣF . It is not diﬀicult to determine the evolution nonlinear connection N from examples in section 2.3. Lemma 2.4.1. The exterior differential of the 1-forms δy i are given by formula: 1 d(δy i ) = dN i j ∧ dxj = Ri kj dxj ∧ dxk + B i kj δy j ∧ dxk . 2

(2.9)

2.5 The dynamical derivative determined by the evolution nonlinear connection N The dynamical covariant derivation induced by the evolution nonlinear connection N is expressed by (?) using the coeﬀicients N i j from (2.4.2). It is given by ∂ ∂ ∇ X i i = SX i + X j N i j . (2.1) ∂y ∂y i Applied to a d-vector field X i (x, y), we have the formula: ∇X i = SX i | = SX i + X j N i j

(2.2)

and for a d-tensor gij , we have ∇gij = gij | = Sgij − gsj N s i − gis N s j , where

(2.5.2’)

◦ 1 S =S + F e 2 ◦

N i j =N i j −

i

1 ∂F . 4 ∂y j

(2.3)

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

∂F i Remarking that is a d-tensor field of type (1,1), one can introduce two d∂y j tensors, important in the geometrical theory of the Finslerian mechanical systems ΣF : 1 ∂Fi ∂Fj 1 ∂Fi ∂Fj Pij = (2.4) − i , Qij = + i . 2 ∂y j ∂y 2 ∂y j ∂y The first one Pij is called the helicoidal tensor of the Finslerian mechanical system ΣF , [166]. Also, on the phase space T M , the elicoidal d-tensor Pij give rise to the 2-form P = Pij dxi ∧ dxj

(2.5)

and Qij allows to consider the symmetric tensor Q = Qij dxi ⊗ dxj .

(2.6)

The following Bucataru-Miron theorem holds: Theorem 2.5.1. For a Finslerian mechanical system ΣF = (M, EF 2 , F e) the evolution nonlinear connection N is the unique nonlinear connection that satisfies the conditions: ∇g = −

1 Q, 2

1 ◦ ω (hX, hY ) = P (X, Y ), ∀X, Y ∈ χ(T M ), 2

(2.7)

◦

where ω is the symplectic structure of the Finsler space F n : ◦

◦

ω= 2gij δ y j ∧ dxi .

(2.8)

dxi If F e does not depend on velocities y = we have P ≡ 0, Q ≡ 0. One dt obtains the the case of Riemannian mechanical systems ΣR , studied in the previous chapter. i

2.6 Metric N-linear connection of ΣF The metric, or canonic, N -linear connection D, with coeﬀicients CΓ(N ) = (F i jk , C i jk ) of the Finslerian mechanical system ΣF is uniquely determined by the following axioms, (Miron [161]): A1 . N is the canonical nonlinear connection of ΣF .

2.6. METRIC N-LINEAR CONNECTION OF ΣF

185

A2 . D is h-metric, i.e gij |k = 0. A3 . D is h-symmetric, i.e. T i jk = F i jk − F i kj = 0. A4 . D is v-metric, i.e. gij |k = 0. A5 . D is v-symmetric, i.e. S i jk = C i jk − C i kj = 0. The following important result holds: Theorem 2.6.1. The local coeﬀicients DΓ(N ) = (F i jk , C i jk ) of the canonical N connection D of the Finslerian mechanical system ΣF are given by the generalized Christoffel symbols: δg δg δg 1 sk js jk F ijk = g is + k − s , 2 δxj δx δx (2.1) 1 ∂gsk ∂gjs ∂gjk + − . C ijk = g is 2 ∂y j ∂y k ∂y s Using the expression (?) of the operator

δ , one obtains: δxi

Theorem 2.6.2. The coeﬀicients F i jk , C i jk of canonical N -connection D have the expressions: ◦

F i jk =F i jk + Cˇ i jk , Cˇ i jk ◦

◦

◦

C i jk =C i jk

∂F h ◦ ∂F h ◦ ∂F h 1 is ◦ = g + C jsh − C jkh , C skh 4 ∂y j ∂y k ∂y s

(2.2)

◦

where CΓ(N ) = (F i jk , C i jk ) are the coeﬀicients of canonic Cartan metric connection of Finsler space F n . A consequence of the previous formulas is the following relation: 1 ◦ i s ∂F k i ˇ y C jh = C hk y . 4 ∂y s j

(2.3)

Let ω i j be the connection forms of CΓ(N ): ω i j = F i jk dxk + C i jk δy k . Then, we have

(2.4)

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

Theorem 2.6.3. The structure equations of the canonical connection CΓ(N ) are given by: 1 d(dxi ) − dxk ∧ ω ik = − Ω i , 2 (2.5) i k i i d(δy ) − δy ∧ ω = − , Ω k dω i − ω k ∧ ω i = − i , Ωj j j k 1

2

where the 2-forms of torsion Ω i , Ω i are as follows 1

Ω i = C ijk dxj ∧ δy k , 1 Ω i = Rijk dxj ∧ dxk + P ijk dxj ∧ δy k , 2

(2.6)

2

with Rijk

δN i j δN i k − , = δxk δxj

P ijk

∂N i j − F ikj = k ∂y

(2.7)

and the 2-form of curvature Ωi j is given by

where

1 1 Ωi j = Rj ikh dxk ∧ dxh + Pj ikh dxk ∧ δy h + Sj ikh δy k ∧ δy h , 2 2

(2.8)

δF ijk δF ijh i Rj kh = − + F sjk F ish − F sjh F isk + C ihs Rskh , h k δx δx ∂F ijk i i s Pj i kh = − Cjh |k + C js P kh , h ∂y ∂C ijk ∂C ijh i s i s i = S − + Cjk Csh − Cjh Cjk . j kh h k ∂y ∂y

(2.9)

Taking into account that the coeﬀicients F ijk and C ijk are expressed in the formulas (2.6.2), the calculus of curvature tensors is not diﬀicult. Exterior differentiating (2.6.5) and using them again one obtains the Bianchi identities of CΓ(N ). The h- and v-covariant derivatives, denoted by “| ” and “|”, with respect to canonical connection D have the properties given by the axioms A1 − A4 .

2.7. THE ELECTROMAGNETISM IN THE THEORY OF THE FINSLERIAN MECHANICAL SYSTEMS Σ

So, we obtain

g ij

|k

=

δgij − gsj F sik − gis F sjk = 0, δxk

∂g gij |k = ijk − gsj C sik − gis C sjk = 0. ∂y

(2.10)

The Ricci identities applied to the fundamental tensor gij give us: Rijkh + Rjikh = 0,

Pijkh + Pjikh = 0,

Sijkh + Sjikh = 0,

where Rijkh = gjs Ri s kh , etc. Also, Pijk = gis P s jk is totally symmetric. The deflection tensors of D are Dij

=y

i

δy i = j + y s F isj ; δx

|j

dij = y i |j .

(2.11)

dij = δji .

(2.12)

Taking into account (2.6.11), we obtain: Dij

1 ∂F i i ˇ , = y C sj + 4 ∂y j s

2.7 The electromagnetism in the theory of the Finslerian mechanical systems ΣF For a Finslerian mechanical system ΣF = (M, EF 2 , F e) whose external forces F e dxi i depend on the material point x ∈ M and on velocity y = , the electromagnetic dt phenomena appears because the deflection tensors Dij and dij nonvanish. Setting Dij = gih Dhj , dij = gih δjh , the h-electromagnetic tensor Fij and the v-electromagnetic tensor fij are defined by 1 Fij = (Dij − Dji ), 2

1 fij = (dij − dji ). 2

(2.1)

By using the equalities (2.6.11), we have 1 Fij = Pij , 4

fij = 0.

If we denote Rijk := gih Rhjk , then one proves:

(2.2)

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

Theorem 2.7.1. The electromagnetic tensor Fij of the Finslerian mechanical system ΣF = (M, T, F e) satisfies the following generalized Maxwell equations: Fij

|k

+ Fjk

|i

+ Fki

|j

1 = {y s (Rsijk + Rsjki + Rskij )− 2 −(Rijk + Rjki + Rkij )}, (2.3)

1 Fij |k + Fjk |i + Fki |j = {y s [(Psijk − Psikj )+ 2 +(Psjki − Psjik ) + (Pskij − Pskji )]} where Fij is expressed in (2.7.2). Corollary 2.7.1.1. If the external forces F e does not depend on the velocity y i then the electromagnetic fields Fij and fij vanish. Remark 2.7.1.1. 1◦ The application of the previous theory to the examples from section 2.2 is immediate. 2◦ The theory of the gravitational field gij and the Einstein equations can be realized by same method as in the papers, [166].

2.8 The almost Hermitian model on the tangent manifold T M of the Finslerian mechanical systems ΣF Consider a Finslerian mechanical system ΣF = (M, F (x, y), F e(x, y)) endowed with the evolution nonlinear connection N and also endowed with the canonical N −metrical connection having the coeﬀicients CΓ(N ) = (F ijk , C ijk ). On the velocity manifold Tg M = T M \ {0} we can see that the previous geometrical object fields determine an almost Hermitian structure H 2n = (Tg M , G, F). Moreover, the theory of gravitational and electromagnetic fields can be geometrically studied much better on such model, since the symplectic structure θ, the almost complex structure F and the Riemannian structure G on Tg M are well determined by the Finslerian mechanical system ΣF .

2.8. THE ALMOST HERMITIAN MODEL ON THE TANGENT MANIFOLD T M OF THE FINSLERIAN M

One obtains:

where

G = gij dxi ⊗ dxj + gij δy i ⊗ δy j F=−

∂ δ i ⊗ dx + ⊗ δy i i i ∂y δx

P=−

∂ δ i ⊗ δy + ⊗ dxi , i i ∂y δx

◦ 1 ∂F s ∂ δ δ = + δxi δxi 4 ∂y i ∂y s ◦ 1 ∂F i s i i δy = y − dx . δ 4 ∂y s

(2.1)

(2.2)

Thus, the following theorem holds: Theorem 2.8.1. We have: 1◦ The pair (Tg M , G) is a pseudo Riemannian space. 2◦ The tensor G depends on ΣF , only. 3◦ The distributions N and V are orthogonal with respect to G. Theorem 2.8.2. 1◦ The pair (Tg M , F) is an almost complex space. ◦ 2 F depends on ΣR , only. 3◦ F is integrable on Tg M iff the tensors Ri jk vanishes. Theorem 2.8.3. 1◦ The pair (Tg M , P) is an almost product space. ◦ 2 P depends on ΣF , only. 3◦ P is integrable iff the tensors Ri jk vanishes. The integrability of F and P are studied by means of Nijenhuis tensors NP and NF : NF (X, Y ) = F2 (X, Y ) + [FX, FY ] − F [FX, Y ] − −F [X, FY ] , ∀X, Y ∈ χ(T M ). δ ∂ , i NF = 0 if, and only if Ri jk = 0, ti jk = 0. But In the adapted basis i δx ∂y i the torsion tensor t jk vanishes. It is not diﬀicult to prove the following results:

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

g Theorem 2.8.4. 1 The triple T M , G, F is an almost Hermitian space. ◦ g 2 T M , G, F depends on Finslerian mechanical system ΣF , only. ◦ g 3 The almost symplectic structure θ of the space T M , G, F is ◦

θ = gij δy i ∧ dxj ,

(2.3)

with δy i from (2.8.2). If the almost symplectic structure θ is a symplectic one (i.e. dθ = 0), then the space H 2n = Tg M , G, F is almost Kählerian. But, using the formulas (2.8.3) and (2.4.9) one obtains: dθ =

1 (Rijk + Rjki + Rkij )dxi ∧ dxj ∧ dxk + 3! 1 + (gis B s jk − gjs B s ik )δy k ∧ dxj ∧ dxi . 2

Therefore we deduce Theorem 2.8.5. The space H if the following equations hold:

2n

(2.4)

g = T M , G, F is almost Kählerian if, and only

Rijk + Rjki + Rkij = 0; gis B s jk − gjs B s ik = 0. (2.5) 2n g The space H = T M , G, F is called the almost Hermitian model of the Finslerian Mechanical System ΣF . 2n g Remark. We can study the space H = T M , G, P by similar way. 2n g One can use the model H = T M , G, F to study the geometrical theory of Finslerian Mechanical system ΣF = (M, F (x, y), F e(x, y)). For instance, the Einstein equations of the pseudo Riemannian space (Tg M , G) can be considered as the Einstein equations of the Finslerian Mechanical system ΣF . Remark. G.S. Asanov showed, [27], that the metric G given by the lift (2.8.1) does not satisfy the principle of the Post-Newtonian calculus. This is due to the fact that the horizontal and vertical terms of the metric G do not have the same physical dimensions. This is the reason for the author to introduce a new lift, [174], of the fundamental tensor gij (x, y) of ΣF that can be used in a gauge theory. This lift is similar with that introduced in section 2.1, adapted to ΣR .

2.8. THE ALMOST HERMITIAN MODEL ON THE TANGENT MANIFOLD T M OF THE FINSLERIAN M

It is expressed by a2 i j e G(x, y) = gij (x, y)dx ⊗ dx + 2 gij (x, y)δy i ⊗ δy j , F

(2.6)

1 ∂F i j where δy =δ y + dx and a > 0 is a constant imposed by applications. This 4 ∂y j e is to preserve the physical dimensions to the both terms of G. Let us consider also the tensor field on Tg M i

◦

i

and the 2-form

F ∂ a δ i Fe = − ⊗ dx + ⊗ δy i i i a ∂y F δx

(2.7)

a θe = θ. F

(2.8)

Then, we have: Theorem 2.8.6. ◦ e e g 1 The triple T M , G, F is an almost Hermitian space. 2◦ The 2-form θe given by (2.8.8) is the almost symplectic structure determined e F e . by G, 3◦ θe is conformal to θ. 2n e e g The space H = T M , G, F can be used to study the geometrical theory of the Finslerian mechanical system ΣF , too.

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CHAPTER 2. FINSLERIAN MECHANICAL SYSTEMS

Chapter 3 Lagrangian Mechanical systems A natural extension of the notion of the Finslerian mechanical system is that of the Lagrangian mechanical system. It is defined as a triple ΣL = (M, L(x, y), F e(x, y)) where: M is a real n−dimensional C ∞ manifold called the configuration space; L(x, y) is a regular Lagrangian with the property that the pair Ln = (M, L(x, y)) is a Lagrange space; F e(x, y) is an a priori given vertical vector field on the velocity space T M called the external forces. The number n is the number of freedom degree of ΣL . The equations of evolution, or fundamental equations of Lagrangian mechanical system ΣL are the Lagrange equations: (I)

d ∂L ∂L dxi i − = Fi (x, y), y = (i = 1, 2, ..., n) dt ∂y i ∂xi dt

where Fi are the covariant components of the external forces F e. These equations determine the integral curves of a canonical semispray S. Thus, the geometrical theory of the semispray S is the geometrical theory of the Lagrangian mechanical system ΣL . The Lagrangian L(x, y) and the external forces F e(x, y) do not explicitly depend on the time t. Therefore ΣL is a scleronomic Lagrangian mechanical system.

3.1 Lagrange Spaces. Preliminaries Let Ln = (M, L(x, y)) be a Lagrange space (see ch. 2, part I). L : T M → R being a regular Lagrangian for which 1 gij (x, y) = ∂˙i ∂˙j L(x, y) 2 193

(3.1)

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

is the fundamental tensor. Thus gij (x, y) is a covariant of order 2 symmetric d−tensor field of constant signature and non singular: det(gij (x, y)) 6= 0 on Tg M = T M \ {0}.

(3.2)

The Euler-Lagrange equations of the space Ln are: d ∂L ∂L dxi i − = 0, y = . dt ∂y i ∂xi dt

(3.3)

As we know, the system of differential equations (3.1.3) can be written in the equivalent form: ◦ d 2 xi dx i + 2G x, =0 (3.4) dt2 dt with ◦i 1 (3.1.4’) 2G = g is {(∂˙s ∂h L)y h − ∂s L} 2 ∂ ∂ ∂˙i = i , ∂i = i . ∂y ∂x ◦i

◦

G are the coeﬀicients of a semispray S: ◦

◦i

S = y ∂i − 2G (x, y)∂˙i i

(3.5)

◦

and S depend on the space Ln , only. ◦

S is called the canonical semispray of Lagrange space Ln . The energy of the Lagrangian L(x, y) is: EL = y i ∂˙i L − L.

(3.6)

In the chapter 2, part I, we prove the Law of conservation energy: Theorem 3.1.1. Along the solution curves of Euler-Lagrange equations (3.1.3) the energy EL is conserved. The following properties hold: ◦

1◦ The canonical nonlinear connection N of Lagrange space Ln has the coeﬀicients ◦i

◦i

N j = ∂j G

(3.7)

3.1. LAGRANGE SPACES. PRELIMINARIES

195

◦

2◦ N determine a differentiable distribution on the velocity space T M supplementary to the vertical distribution V : ◦

Tu T M = N u ⊕ Vu , ∀u ∈ T M.

(3.8)

◦

3◦ An adapted basis to (3.1.8) is (δ i , ∂˙i )u , (i = 1, ..., n), where ◦

◦j

δ i = ∂i − N i (x, y)∂˙j

(3.9)

◦

and an adapted cobasis (dxi , δy i )u , (i = 1, ..., n) with ◦

δy i = dy i + Nji (x, y)dxj ◦

◦

◦

(3.1.9’) ◦i

◦i

4 The canonical metrical N −connection CΓ(N ) = (Ljk , C jk ) has the coeﬀicients expressed by the generalized Christoffel symbols: ◦ ◦ ◦ 1 Ljh = g is (δ j gsh + δ h gjs − δ s gjh ) 2 ◦i

(3.10)

◦i

1 C jh = g is (∂˙j gsh + ∂˙h gjs − ∂˙s gjh ) 2 5◦ The Cartan 1-form of Ln is 1 ◦ ω = (∂˙i L)dxi 2

(3.11)

6◦ The Cartan-Poincaré 2-form of Ln is ◦

◦

◦

θ = dω = gij (x, y)δy i ∧ dxj ◦

(3.12)

◦

7 The 2-form θ determine a symplectic structure on the velocity manifold T M . Example 3.1.1. The function 2e Ai (x)y i + U (x) (3.13) m with m, c, e the known physical constants, γij (x) are the gravitational potentials, γij (x) being a Riemannian metric, Ai (x) are the electromagnetic potentials and U (x) is a potential function. Thus, Ln = (M, L(x, y)) is a Lagrange space. It is the Lagrange space of electrodynamics (cd. ch. 2, part I). L(x, y) = mcγij (x)y i y j +

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

3.2 Lagrangian Mechanical systems, ΣL Definition 3.2.1. A Lagrangian mechanical system is a triple: ΣL = (M, L(x, y), F e(x, y)),

(3.1)

where Ln = (M, L(x, y)) is a Lagrange space, F e(x, y) is an a priori given vertical vector field: F e(x, y) = F i (x, y)∂˙i , (3.2) the number n = dim M is the number of freedom degree of ΣL ; the manifold M is real C ∞ differentiable manifold called the configuration space; F e(x, y) is the external forces and F i (x, y) are the contravariant components of F e. Of course, F i (x, y) is a vertical vector field and Fi (x, y) = gij (x, y)F j (x, y) are the covariant components of F e. Fi (x, y) is a d−covector field. The 1-form: σ = Fi (x, y)dxi

(3.3)

(3.4)

is a vertical 1-form on the velocity space T M . The Riemannian mechanical systems ΣR and the Finslerian mechanical systems ΣF are the particular Lagrangian mechanical systems ΣL . Remark 3.2.0.1. The notion of Lagrangian mechanical system ΣL is different of that of “Lagrangian mechanical system” defined by Joseph Klein [123], which is a Riemannian conservative mechanical system. The fundamental equations of ΣL are an extension of the Lagrange equations of a Finslerian mechanical system ΣF , from the previous chapter. So, we introduce the following Postulate: Postulate. The evolution equations of the Lagrangian mechanical system ΣL = (M, L, F e) are the following Lagrange equations: d ∂L dxi ∂L i . (3.5) − i = Fi (x, y), y = dt ∂y i ∂x dt But the both members of the Lagrange equations (3.2.5) are d−covectors. Consequently, we have:

3.3. THE EVOLUTION SEMISPRAY OF ΣL

197

Theorem 3.2.1. The Lagrange equations (3.2.5) of a Lagrangian mechanical system ΣL = (M, L, F e) have a geometrical meaning. Theorem 3.2.2. The trajectories without external forces of the Lagrangian mechanical system ΣL = (M, L, F e) are the geodesics of the Lagrange space Ln = (M, L). Indeed, Fi (x, y) ≡ 0 implies the previous aﬀirmations. For a regular Lagrangian L(x, y) (in this case (3.1.2) holds), the Lagrange equations (3.2.5) are equivalent to the second order differential equations (SODE): ◦i d 2 xi dx 1 i dx + 2G x, = F x, , (3.6) dt2 dt 2 dt ◦i

where the functions G (x, y) are the local coeﬀicients (3.1.4’) of canonical semispray ◦

S of Lagrange space Ln = (M, L(x, y)). The equations (3.2.6) are called fundamental equations of ΣL , too.

3.3 The evolution semispray of ΣL The Lagrange equations (3.2.6) determine a semispray S which depend on the Lagrangian mechanical system ΣL , only. Indeed, the vector field on T M : S = y i ∂i − 2Gi (x, y)∂˙i with

◦i

1 2G (x, y) = 2G (x, y) − F i (x, y) 2 has the property JS = C. So it is a semispray depending only on ΣL . i

(3.1)

(3.2)

Theorem 3.3.1 (Miron). For a Lagrangian mechanical system ΣL the following properties hold: 1◦ The semispray S is given by ◦ 1 S = S + Fe 2

2◦ S is a dynamical system on the velocity space T M .

(3.3.1’)

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

3◦ The integral curves of S are the evolution curves (3.2.6) of ΣL . In fact, 1◦ derives from the formulas (3.3.1) and (3.3.2). 2◦ S being a vector field on T M , compatible with the geometric structure of T M , (i.e. JS = C), it is a dynamical system on the manifold T M . 3◦ The integral curves of S are determined by the system of differential equations dxi dy i i =y, + 2Gi (x, y) = 0. (3.3) dt dt By means of the expression (3.3.2) of the coeﬀicients Gi (x, y) the system (3.3.4) is coincident to (3.2.6). The vector field S is called the evolution (or canonical) semispray of the Lagrangian mechanical system ΣL . Exactly as in the case of Riemannian or Finslerian mechanical systems, one can prove: Theorem 3.3.2 ([49]). The evolution semispray S is the unique vector field, on the velocity space T M , solution of the equation ◦

iS θ = −dEL + σ

(3.4)

with L from (3.1.6) and σ from (3.2.4). But S being a solution of the previous equations, we get: S(ΣL ) = dEL (S) = σ(S) = Fi y i = gij F i y j . So, we have: Theorem 3.3.3. The variation of energy EL along the evolution curves of mechanical system ΣL is given by dx dxi dEL = Fi x, . (3.5) dt dt dt The external forces field F e is called dissipative if g(C, F e) = gij F i y j ≤ 0. Thus, the previous theorem implies: Theorem 3.3.4. The energy of Lagrange space Ln = (M, L) is decreasing along the evolution curves of the mechanical system ΣL if and only if the external forces field F e is dissipative. Evidently, the semispray S being a dynamical system on the velocity space T M it can be used for study the important problems, as the stability of evolution curves of ΣL , the equilibrium points etc.

3.4. THE EVOLUTION NONLINEAR CONNECTION OF ΣL

199

3.4 The evolution nonlinear connection of ΣL The geometrical theory of the Lagrangian mechanical system ΣL is based on the evolution semispray S, with the coeﬀicients Gi (x, y) expressed in formula (3.3.2). Therefore, all notions or properties derived from S will be considered belonging to the mechanical system ΣL . Such that, the evolution nonlinear connection N of ΣL is characterized by the coeﬀicients: Nji (x, y)

◦ ◦i 1˙ i 1 i ˙ ˙ = ∂j G (x, y) = ∂ G(x, y) − ∂F (x, y) = N j (x, y) − ∂˙j F i (x, y). (3.1) 4 4

Since ∂˙j Fi is a d−tensor field, consider its symmetric and skewsymmetric parts: 1 1 Pij = (∂˙j Fi + ∂˙i Fj ), Fij = (∂˙j Fi − ∂˙i Fj ) (3.2) 2 2 The d−tensor Fij is the elicoidal tensor field of ΣL , [166]. The evolution nonlinear connection N allows to determine the dynamic derivative of the fundamental tensor gij (x, y) of ΣL : gij| = S(gij ) − gi|s Njs − gsj Nis . It is not diﬀicult to prove the following formula: 1 gij| = Pij . 2 N is metric nonlinear connection if gij| = 0. So, we have

(3.3)

(3.4)

Proposition 3.4.1. The evolution nonlinear connection N is metric if, and only if the d−tensor field Pij vanishes. The adapted basis (δi , ∂˙i ) to the distributions N and V has the operators δi of the form: ◦ 1 δi = ∂i − Nij ∂˙j = δ i + ∂˙i F s ∂˙s . (3.5) 4 The dual adapted cobasis (dxi , δy i ) has 1-forms δy i : ◦ 1 δy i = dy i + Nji dxj = δy i − ∂˙j F i dxj 4 In the adapted basis the evolution semispray S has the expression: i ◦ ◦i 1 1˙ i j ∂ i δ j i S = y i − 2G − N j y − F − ∂j F y . δx 2 2 ∂y i

Consequences:

(3.4.5’)

(3.4.5”)

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

1◦ S cannot be a vertical vector field. ◦i

2◦ If the coeﬀicients G are 2-homogeneous with respect to the vertical variables y i and the contravariant components F i (x, y) are 2-homogeneous in y i , then the canonical semispray S belongs to the horizontal distribution N . The tensor of integrability of the distribution N is i ◦ 1 i Rjh = δh Nji − δj Nhi = (δh ∂˙j − δj ∂˙h ) G − F i 4 i Thus: N is integrable iff Rjh = 0. The d−tensor of torsion of the nonlinear connection N vanishes. Indeed: tijh = ∂˙h Nji − ∂˙j Nhi = 0.

(3.6)

(3.7)

i The Berwald connection BΓ(N ) = (Bjk , 0) of the canonical nonlinear connection N has the coeﬀicients i Bjh

◦i

1 = B jh − ∂˙j ∂˙h F i , 4

(3.8)

◦i

where (B jh , 0) are the coeﬀicients of Berwald connection of Lagrange space Ln . i i It follows that BΓ(N ) is symmetric: Bjh − Bhj = tijh = 0. In the following section we need: Lemma 3.4.1. The exterior differential of 1-forms δy i are given by 1 i i dδy i = Rjh dxh ∧ dxj + Bjh δy j ∧ dxh . 2

(3.9)

Indeed, from (3.4.5’) we have: dδy i = dNji ∧ dxj = δh Nji dxh ∧ dxj + ∂˙h Nji δy h ∧ dxj , which are exactly (3.4.9). By means of the formula (3.4.5’) one gets: Theorem 3.4.1. The autoparallel curves of the canonical nonlinear connection N are given by the differential system of equations: ◦

δy i 1 ˙ i dxj dxi δy i i , = − ∂j F = 0. y = dt dt dt 4 dt

(3.10)

3.5. CANONICAL N −METRICAL CONNECTION OF ΣL . STRUCTURE EQUATIONS 201

Evidently, if F e = 0, then the canonical nonlinear connection N coincides with ◦

the canonical nonlinear connection N of the Lagrange space Ln . In particular, if ΣL is a Finslerian mechanical system ΣF , then the previous theory reduces to that studied in the previous chapter.

3.5 Canonical N −metrical connection of ΣL. Structure equations Taking into account the results from part I, the canonical N −metrical connection CΓ(N ) is characterized by: i Theorem 3.5.1. The local coeﬀicients DΓ(N ) = (Lijk , Cjk ) of the canonical N −metrical connection of Lagrangian mechanical system ΣL are given by the generalized Christoffel symbols:

1 Lijh = g is (δj gsh + δh gsj − δs gjk ) 2

(3.1)

1 i Cjh = g is (∂˙j gsh + ∂˙h gsj − ∂˙s gjh ). 2 Using the expression (3.4.5) of the operator δk and developing the terms δh gij from (3.5.1), we get: i Theorem 3.5.2. The coeﬀicients Lijh , Cjh of DΓ(N ) can be set in the form

Lijk i Cˇjk ◦

◦i

◦i

◦i

i i = Ljk + Cˇjk , Cjk = C jk ,

◦ ◦ 1 is ◦ ˙ h + C jsh ∂F ˙ h − C jkh ∂F ˙ h , = g C skh ∂F 4

(3.2)

◦i

where CΓ(N ) = (Ljk , C jk ) is the canonical N −metrical connection of the Lagrange space Ln = (M, L(x, y)). Let ωji be the connection 1-forms of the canonical N −metrical connection DΓ(N ): i ωji = Lijk dxk + Cjk δy k . (3.3) Then, we have:

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

Theorem 3.5.3. The structure equations of canonical N −metrical connection DΓ(N ) of Lagrangian mechanical system ΣL are given by: d(dx ) − dx ∧

ωki

d(δy ) − δy ∧

ωki

i

k

i

k

1i

= −Ω

2i

(3.4)

= −Ω

dωji − ωjk ∧ ωki = −Ωi , 1i

2i

where the 1-form of torsions Ω and Ω are: 1i

i Ω = Cjk dxj ∧ δy k

(3.5) 2i

1 i i Ω = Rjk dxj ∧ dxk + Pjk dxj ∧ δy k 2 and the 2-forms of curvature Ωij is as follows: 1 i 1 Ωij = Rjkh dxk ∧ dxh + Pj ikh dxk ∧ δy h + Sji 2 2

kh δy

k

∧ δy h

(3.6)

where the d−tensors of torsions are: i i i i Tjk = Lijk − Likj = 0; Sjk = Cjk − Ckj = 0, ◦i

(3.7)

i i i i Cjk = C jk , Rjk = δk Nji − δj Nki , Pjk = Bjk − Lijk

and the d−tensors of curvature are: i s Rj i kh = δk Lijk − δk Lijh + Lsjk Lish − Lsjh Lisk + Cjs Rkh i i + Cjs P skh Pj ikh = ∂˙h Lijk − Cjh|k

(3.8)

i i s i s i − ∂˙k Cjh + Cjk Csh − Cjh Csk . Sj ikh = ∂˙h Cjk i i Here Cjk|h is the h−covariant derivative of the tensor Cjk with respect to DΓ(N ).

3.5. CANONICAL N −METRICAL CONNECTION OF ΣL . STRUCTURE EQUATIONS 203

So, we have gij|h = δh gij − gsj Lsih − gis Lsjh = 0, (3.9) gij |h = ∂˙h gij −

s gsj Cih

−

s gis Cjh

= 0.

Applying the Ricci identities to the fundamental tensor gij , it is not diﬀicult to prove the identities Rijkh + Rjikh = 0, Pijkh + Pjikh = 0, Sijkh + Sjikh = 0,

(3.10)

where Rijkh = gjs Ri skh , etc. Also, Pijk = gis P skh is totally symmetric. Taking into account the form (3.5.2) of the coeﬀicients of DΓ(N ), the calculus of d−tensors of torsion (3.5.7) and of d−tensors of curvature (3.5.6) can be obtained. The exterior differentiating the structure equations (3.5.4) modulo the same system of equations (3.5.4) and using Lemma (3.4.1) one obtains the Bianchi identities of the N −metrical connection DΓ(N ). ∂ The h− and v−covariant derivative of Liouville vector field C = y i i lead to ∂y the deflection tensors of DΓ(N ): Dji = y|ji = δj y i + y s Lisj , di j = y i |j = ∂˙j y i + y s Cs ij .

(3.11)

Theorem 3.5.4. The expression of h−tensor of deflection Dji and v− tensor of deflection dij are given by i

Dj=y

s

Lisj

−

Nji ,

i

i

s

◦ i

d j = δ j + y Cs j.

(3.12)

Finally, we determine the horizontal paths, vertical paths of DΓ(N ) and its autoparallel curves. i The horizontal paths of metrical connection DΓ(N ) = (Lijk , Cjk ) are given by j k dx dxi d 2 xi dx dx i i + Ljk x, = 0, y = , (3.13) dt2 dt dt dt dt where the coeﬀicients Lijk are expressed by formula (3.5.2). i dx In the initial conditions xi = xi0 , y i = , local, this system of differential dt 0 equation has a unique solution xi = xi (t), y i = y i (t), t ∈ I.

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

The vertical paths, at a point (xi0 ) ∈ M are characterized by system of differential equations: ◦i dy i + C jk (x0 , y)y j y k = 0, xi = xi0 . (3.14) dt In the initial conditions xi = xi0 , y i = y0i the previous system of differential equations, locally, has a unique solution.

3.6 Electromagnetic field For Lagrangian mechanical system ΣL = (M, L, F e) the electromagnetic phenomena appear because the deflection tensor Dji nonvanishes. The covariant components Dij = gis Djs , dij = gis dsj determine h−electromagnetic field Fij and v−electromagnetic field fij as follows: 1 1 Fij = (Dij − Dji ), fij = (dij − dji ). 2 2

(3.1)

Taking into account the formula (3.5.11) one obtains: ◦

1 1 Fij = F ij + Fij + Fˇij 4 4

(3.2)

◦

where F ij is the electromagnetic tensor field of the Lagrange space Ln , Fij is the elicoidal tensor field of mechanical system ΣL and Fˇij is the skewsymmetric tensor Fˇij = {∂˙j F s Cirs − ∂˙i F r Cjrs }y s .

(3.3)

In the case of Finslerian mechanical system ΣF the electromagnetic tensor Fij 1 is equal to Fij . 4 Applying the method given in the chapter 2, part I, one gets the Maxwell equations of the Lagrangian mechanical system ΣL .

3.7 The almost Hermitian model of the Lagrangian mechanical system ΣL Let N be the canonical nonlinear connection of the mechanical system ΣL and (δi , ∂˙i ) the adapted basis to the distribution N and V . Its dual basis is (dxi , δy i ).

3.7. THE ALMOST HERMITIAN MODEL OF THE LAGRANGIAN MECHANICAL SYSTEM ΣL 205

The N −lift G of fundamental tensor gij on the velocity space Tg M = T M \ {0} is given by G = gij dxi ⊗ dxj + gij δy i ⊗ δy j . (3.1) The almost complex structure F determined by the nonlinear connection N is expressed by F = δi ⊗ dxj − ∂˙i ⊗ dxi . (3.2) Thus, one proves: Theorem 3.7.1. We have: 1◦ G is a pseudo-Riemannian structure and F is an almost complex structure on the manifold T M . They depend only on the Lagrangian mechanical system ΣL . 2◦ The pair (G, F) is an almost Hermitian structure. 3◦ The associated 2-form of (G, F) is given by θ = gij δy i ∧ dxj . 4◦ θ is an almost symplectic structure on the velocity space Tg M. 5◦ The following equality holds: ◦ 1 θ = θ − Fij dxi ∧ dxj , 4 ◦

◦

◦

θ being the symplectic structure of the Lagrange space Ln , i.e. θ = gij δy i ∧dxj . ◦

n

◦

Since θ is the symplectic structure of Lagrange space L , we have dθ = 0. So the exterior differential dθ is as follows: 1 dθ = − dFij ∧ dxi ∧ dxj . (3.3) 4 But dFij = δk Fij dxk + ∂˙k Fij δy k . Consequently one obtain from (2.2): dθ = −

1 1 (Fij|k + Fjk|i + Fki|j )dxk ∧ dxi ∧ dxj − ∂˙k Fij δy k ∧ dxi ∧ dxj . 12 4

Consequently: ◦

1◦ θ = θ, if and only if the helicoidal tensor Fij vanish.

(3.4)

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

2◦ The almost symplectic structure θ of the Lagrangian mechanical system ΣL is integrable, if and only if the helicoidal tensor Fij satisfies the following tensorial equations: Fij|k + Fjk|i + Fki|j = 0, ∂˙k Fij = 0, (3.5) where the operator “| ” is h−covariant derivation with respect to canonical N −linear connection DΓ(N ). Now we observe that some good applications of this theory can be done for the Lagrangian mechanical systems ΣL = (M, L(x, y), F e(x, y)) when the external forces F e are of Liouville type: F e = a(x, y)C.

3.8 Generalized Lagrangian mechanical systems As we know, a generalized Lagrange space GLn = (M, gij ) is given by the configuration space M and by a d−tensor field gij (x, y) on the velocity space Tg M= T M \ {0}, gij being symmetric, nonsingular and of constant signature. There are numerous examples of spaces GLn given by Miron R. [175], Anastasiei M. [14], R.G. Beil [40], [41], T. Kawaguchi [176] etc. 1◦ The GLn space with the fundamental tensor gij (x, y) = αij (x) + yi yj , y= αj y j

(3.1)

and αij (x) a semidefinite Riemann tensor. 1 ◦ 2 gij (x, y) = αij (x) + 1 − yi yj , yi = αij y j where n(x, y) ≥ 1 is a C ∞ n(x, y) function (the refractive index in Relativistic optics), [176]. 3◦ gij (x, y) = αij (x, y) + ayi yj , a ∈ R+ , yi = αij y j and where αij (x, y) is the fundamental tensor of a Finsler space. The space GLn = (M, gij (x, y)) is called reducible to a Lagrange space if there exists a regular Lagrangian L(x, y) such that 1 ∂ 2L = gij (x, y). 2 ∂y i ∂y j

(3.2)

A necessary condition that the space GLn = (M, gij (x, y)) be reducible to a Lagrange space is that the d−tensor field Cijk =

1 ∂gij 2 ∂y k

(3.3)

3.8. GENERALIZED LAGRANGIAN MECHANICAL SYSTEMS

207

be totally symmetric. The following property holds: Theorem 3.8.1. If we have a generalized Lagrange space GLn for which the fundamental tensor gij (x, y) is 0-homogeneous with respect to y i and it is reducible to a Lagrange space Ln = (M, L), then the function L(x, y) is given by: L(x, y) = gij (x, y)y i y j + 2Ai (x)y i + U (x).

(3.4)

The prove does not present some diﬀiculties, [166]. Also, we remark that in some conditions the fundamental tensor gij (x, y) of a space GLn determine a non linear connection, [241], [242]. For instance, in the cases 1◦ , 2◦ , 3◦ of the previous examples we can take: 1◦ , 2◦ , ◦i 1 i i N j = γjk (x)y j y k , γjk (x) being the Christoffel symbols of the Riemannian met2 1 i i ric γij (x). For example 3◦ we take Nji = γjk (x, y)y j y k where γjk (x, y) are the 2 Christoffel symbols of γij (x, y) (see ch. 2, part I). Let us consider the function E(x, y) on Tg M: E(x, y) = gij (x, y)y i y j

(3.5)

called the absolute energy [161] of the space GLn . The fundamental tensor gij (x, y) is say to be weakly regular if the absolute energy E(x, y) is a regular Lagrangian. 1 That means: the tensor gˇij (x, y) = ∂˙i ∂˙j E is nonsingular. Thus the pair (M, E(x, y)) 2 is a Lagrange space. Definition 3.8.1. A generalized Lagrangian mechanical system is a triple ΣGL = (M, gij , F e), where GL = (M, gij ) is a generalized Lagrange space and F e is the vector field of external forces. F e being a vertical vector field we can write: ˙ F e = F i (x, y)∂.

(3.6)

In the following we assume that the fundamental tensor gij (x, y) of ΣGL is weakly regular. Therefore we can give the following Postulate [49]: Postulate. The evolution equations (or Lagrange equations) of a generalized Lagrange mechanical system ΣGL are: ∂E dxi d ∂E i − = Fi (x, y), y = dt ∂y i ∂xi dt

(3.7)

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

with the covariant components of Fe: Fi (x, y) = gij (x, y)F j (x, y).

(3.8)

The Lagrange equations (3.8.7) are equivalent to the following system of second order differential equations: 2 i dx d x 1 dx ˇ i x, + 2G = F i x, 2 dt dt 2 dt (3.9) 2 ∂E ∂ E j ˇ i = 1 g is 2G y − . 2 ∂y s ∂xj ∂xs Therefore, one can apply the theory from this chapter for the Lagrangian mechanical system ΣL = (M, E(x, y), F e). So we have: Theorem 3.8.2.

1◦ The operator

1 ˇ i − F i ∂˙i Sˇ = y i ∂i − 2 G 2

(3.10)

is a semispray on the velocity space Tg M. 2◦ The integral curves of S are the evolution curves of ΣGL . 3◦ Sˇ is determined only by the generalized mechanical system ΣGL . ∂E − E are given by ∂y i dEE dxi dx = Fi x, . dt dt dt

Theorem 3.8.3. The variation of energy EE = y i

(3.11)

F e are called dissipative if gij F i y j ≤ 0. Theorem 3.8.4. The energy EE is decreasing on the evolution curves (3.8.9) of system ΣGL if and only if the external forces F e are dissipative. Other results: ˇ of ΣG,L has the local coeﬀicients 4◦ The canonical nonlinear connection N ◦i 1 i ˙ ˇ Nj = ∂j G − ∂˙j F i . 4

(3.12)

3.8. GENERALIZED LAGRANGIAN MECHANICAL SYSTEMS

209

ˇ −linear metric connection of mechanical system ΣGL , DΓ(N ˇ) = 5◦ The canonical N (Ljk , Cjk ) has the coeﬀicients: ˇ ˇ sk δg ˇ jk 1 δg δg js i is ˇ = g + j − s L jk 2 δxk δx δx (3.13) ∂gjs ∂gsk ∂gjk 1 i C˙ jk + − . = g is 2 ∂y k ∂y j ∂y s ˇ N ˇ , DΓ(N ˇ ) we can study the theory By using the geometrical object fields gij , S, of Generalized Lagrangian mechanical systems ΣG,L .

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CHAPTER 3. LAGRANGIAN MECHANICAL SYSTEMS

Chapter 4 Hamiltonian and Cartanian mechanical systems The theory of Hamiltonian mechanical systems can be constructed step by step following the theory of Lagrangian mechanical system, using the differential geometry of Hamiltonian spaces expound in part I. The legality of this theory is proved by means of Legendre duality between Lagrange and Hamilton spaces. The Cartanian mechanical system appears as a particular case of the Hamiltonian mechanical systems. We develop here these theories using the author’s papers [154] and the book of R. Miron, D. Hrimiuc, H. Shimada and S. Sabau [174].

4.1 Hamilton spaces. Preliminaries Let M be a C ∞ −real n−dimensional manifold, called configuration space and (T ∗ M, π ∗ , M ) be the cotangent bundle, T ∗ M is called momentum (or phase) space. A point u∗ = (x, p) ∈ T ∗ M , π ∗ (u∗ ) = x, has the local coordinate (xi , pi ). As we know from the previous chapter, a changing of local coordinate (x, p) → (e x, pe) of point u∗ is given by i ∂e x x ei = x ei (x1 , ..., xn ), det 6= 0 ∂xj (4.1) ∂xj pei = pj ∂e xi ∂ ∂ The tangent space Tu∗∗ T ∗ M has a natural basis = ∂i , = ∂˙ i with re∂xi ∂pi 211

212

CHAPTER 4. HAMILTONIAN AND CARTANIAN MECHANICAL SYSTEMS

spect to (4.1.1) this basis is transformed as follows ∂ ∂e xj ∂ ∂ pej ∂ = + ∂xi ∂xi ∂e xj ∂xi ∂ pej

(4.2)

i

∂x e˙ j ∂˙ i = j ∂ ∂e x Thus we have: 1◦ On T ∗ M there are globally defined the Liouville 1-form: pˇ = pi dxi

(4.3)

and the natural symplectic structure ◦

θ = dˇ p = dpi ∧ dxi .

(4.4)

2◦ The vertical distribution V has a local basis (∂˙ 1 , ..., ∂˙ n ). It is of dimension n and is integrable. 3◦ A supplementary distribution N to the distribution V is given by a splitting: Tu∗ T ∗ M = Nu∗ ⊕ Vu∗ , ∀u∗ ∈ T ∗ M.

(4.5)

N is called a horizontal distribution or a nonlinear connection on the momentum space T ∗ M . The dimension of N is n. 4◦ A local adapted basis to N and V are given by (δi , ∂˙ i ), (i = 1, ..., n), with δi = ∂i + Nji ∂˙ j .

(4.6)

The system of functions Nji (x, p) are the coeﬀicients of the nonlinear connection N. 5◦ tij = Nij − Nji (4.7) is a d−tensor on T ∗ M - called the torsion tensor of the nonlinear connection N . If tij = 0 we say that N is a symmetric nonlinear connection. 6◦ The dual basis (dxi , δpi ) of the adapted basis (δi , ∂˙ i ) has 1-form δpi expressed by δpi = dpi − Nij dxj . (4.8) 7◦

4.1. HAMILTON SPACES. PRELIMINARIES

213

Proposition 4.1.1. If N is a symmetric nonlinear connection then the symplectic ◦

structure θ can be written:

◦

θ = δpi ∧ dxi .

(4.9)

8◦ The integrability tensor of the horizontal distribution N is Rkij = δi Nkj − δj Nki

(4.10)

and the equations Rkij = 0 gives the necessary and suﬀicient condition for integrability of the distribution N . The notion of N −linear connection can be taken from the ch. 2, part I. Definition 4.1.1. A Hamilton space is a pair H n = (M, H(x, p)) where H(x, p) is a real scalar function on the momentum space T ∗ M having the following properties: ∗ M = T ∗ M \ {0} and continuous on the 1◦ H : T ∗ M → R is differentiable on T] null section of projection π∗ .

2◦ The Hessian of H, with the elements

is nonsingular, i.e.

1 g ij = ∂˙ i ∂˙ j H 2

(4.11)

∗M . det(g ij ) 6= 0 on T]

(4.12)

∗M . 3◦ The 2-form g ij (x, p)ηi ηj has a constant signature on T]

In the ch. 2, part I it is proved the following Miron’s result: Theorem 4.1.1. 1◦ In a Hamilton space H n = (M, H(x, p)) for which M is a paracompact manifold, there exist nonlinear connections determined only by the fundamental function H(x, p). ◦

2◦ One of the, N , has the coeﬀicients ◦ 1 1 ˙k N ij = − gjh gik ∂ {H, ∂˙ h H} + ∂˙ h ∂i H . 2 4 ◦

3◦ The nonlinear connection N is symmetric.

(4.13)

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CHAPTER 4. HAMILTONIAN AND CARTANIAN MECHANICAL SYSTEMS ◦

In the formula (4.1.13), {, } is the Poisson brackets. N is called the canonical nonlinear connection of H n . The variational problem applied to the integral of action of H n : Z 1 dxi − H(x(t), p(t)) dt, (4.14) I(c) = pi (t) dt 0 where

1 H(x, p) = H(x, p) 2

(4.15)

leads to the following results: Theorem 4.1.2. The necessary conditions as the functional I(c) be an extremal value of the functionals I(e c) imply that the curve c(t) = (xi (t), pi (t)) is a solution of the Hamilton–Jacobi equations: dpi ∂H dxi ∂H − = 0, + i = 0. dt ∂pi dt ∂x

(4.16)

◦

If N is the canonical nonlinear connection then the Hamilton–Jacobi equations can be written in the form dxi ∂H δpi δH − = 0, + i = 0. dt ∂pi dt δx

(4.17)

Evidently, the equations (4.1.16) have a geometrical meaning. The curves c(t) which verify the Hamilton–Jacobi equations are called the extremal curves (or geodesics) of H n . Other important results: Theorem 4.1.3. The fundamental function H(x, p) of a Hamilton space H n = (M, H) is constant along to every extremal curves. Theorem 4.1.4. The following properties hold:

◦

∗ M ) with the 1◦ For a Hamilton space H n there exists a vector field ξ ∈ X (T] property ◦

i◦ θ = −dH.

(4.18)

◦ 1 ξ = (∂˙ i H∂i − ∂i H ∂˙ i ). 2

(4.19)

ξ

◦

◦

2 ξ is given by ◦

3◦ The integral curve of ξ is given by the Hamilton–Jacobi equations (4.1.16).

4.2. THE HAMILTONIAN MECHANICAL SYSTEMS

215

◦

The vector field ξ from the formula (4.1.19) is called the Hamilton vector of the space H n . ◦

Proposition 4.1.2. In the adapted basis of the canonical nonlinear connection N ◦

the Hamilton vector ξ is expressed by ◦ 1 ξ = (∂˙ i Hδi − δi H ∂˙ i ). (4.20) 2 By means of the Theorem 4.1.4 we can say that the Hamilton vector field ξ is a dynamical system of the Hamilton space H n .

4.2 The Hamiltonian mechanical systems Following the ideas from the first part, ch. ..., we can introduce the next definition: Definition 4.2.1. A Hamiltonian mechanical system is a triple: ΣH = (M, H(x, p), F e(x, p)),

(4.1)

where H n = (M, H(x, p)) is a Hamilton space and F e(x, p) = Fi (x, p)∂˙ i

(4.2)

is a given vertical vector field on the momenta space T ∗ M . F e is called the external forces field. The evolution equations of ΣH can be defined by means of equations (4.1.16) from the variational problem. Postulate 4.2.1. The evolution equations of the Hamiltonian mechanical system ΣH are the following Hamilton equations: dxi dpi 1 1 − ∂˙ i calH = 0, + ∂i H = Fi (x, p), H = H. dt dt 2 2

(4.3)

Evidently, for F e = 0, the equations (4.2.3) give us the geodesics of the Hamilton space H n . ◦

Using the canonical nonlinear connection N we can write in an invariant form the Hamilton equations, which allow to prove the geometrical meaning of these equations.

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CHAPTER 4. HAMILTONIAN AND CARTANIAN MECHANICAL SYSTEMS

Examples. 1◦ Consider H n = (M, H(x, p)) the Hamilton spaces of electrodynamics, [19]: 1 ij 2e i e3 H(x, p) = γ (x)pi pj − A (x)pi + Ai (x)Ai (x) 2 3 mc mc mc and F e = pi ∂˙ i . Then ΣH is a Hamiltonian mechanical system determined only by H n. 2◦ H n = (M, K 2 (x, p)) is a Cartan space and F e = pi ∂˙ i . 3◦ H n = (M, E(x, p)) with E(x, p) = γ ij (x)pi pj and F e = a(x)pi ∂˙ i . Returning to the general theory, we can prove: Theorem 4.2.1. The following properties hold: 1◦ ξ given by 1 ξ = [∂˙ i H∂i − (∂i H − Fi )∂˙ i ] 2 ∗M . is a vector field on T] 2◦ ξ is determined only by the Hamiltonian mechanical system ΣH . 3◦ The integral curves of ξ are given by the Hamilton equation (4.2.3).

(4.4)

The previous Theorem is not diﬀicult to prove if we remark the following expression of ξ: ◦ 1 ξ = ξ + F e. (4.5) 2 Also we have: Proposition 4.2.1. The variation of the Hamiltonian H(x, p) along the evolution curves of ΣH is given by: dxi dH = Fi . (4.6) dt dt As we know, the external forces F e are dissipative if hF w, Ci ≥ 0. Looking at the formula (4.2.6) one can say: Proposition 4.2.2. The fundamental function H(x, p) of the Hamiltonian mechanical system ΣH is decreasing on the evolution curves of ΣH , if and only if, the external forces F e are dissipative. ∗ M is called the canonical dynamical system of the The vector field ξ on T] Hamilton mechanical system ΣH . Therefore we can say: The geometry of ΣH is the geometry of pair (H n , ξ).

4.3. CANONICAL NONLINEAR CONNECTION OF ΣH

217

4.3 Canonical nonlinear connection of ΣH The fundamental tensor g ij (x, p) of the Hamilton space H n is the fundamental or metric tensor of the mechanical system ΣH . But others fundamental geometric notions, as the canonical nonlinear connection of ΣH cannot be introduced in a straightforward manner. They will be defined by means of L−duality between the Lagrangian and the Hamiltonian mechanical systems ΣL and ΣH (see ch. ..., part I). Let ΣL = (M, L(x, y), F e(x, y)), F e = F i (x, y)∂˙i be a Lagrangian mechanical 1 1 1 system. The mapping 1 φ : (x, y) ∈ T M → (x, p) ∈ T ∗ M, pi = ∂˙i L 2 is a local diffeomorphism. It is called the Legendre transformation. Let ψ be the inverse of φ and H(x, p) = 2pi y i − L(x, y), y = ψ(x, p).

(4.1)

One prove that H n = (M, H(x, p)) is an Hamilton space. It is the L− dual of Lagrange space Ln = (M, L(x, y)). One proves that φ transform: ◦

◦

1◦ The canonical semispray S of Ln in the Hamilton vector ξ of H n . ◦

2◦ The canonical nonlinear connection N L of Ln into the canonical nonlinear ◦

connection N H of H n . 3◦ The external forces F e of ΣL into external forces F e of ΣH , with Fi (x, p) = 1

gij (x, p)F1j (x, ψ(x, p)). ◦

4 The canonical nonlinear connection NL of ΣL with coeﬀicients into the canonical nonlinear connection NH of ΣH with the coeﬀicients ◦ 1 Nij (x, p) =N ij (x, p) + gih ∂˙ h Fj . 4

(4.2)

◦

N ij are given by (...., ch. ...). Therefore we have: Proposition 4.3.1. The canonical nonlinear connection N of the Hamiltonian mechanical system ΣH has the coeﬀicients Nij , (4.3.2).

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CHAPTER 4. HAMILTONIAN AND CARTANIAN MECHANICAL SYSTEMS

Of course, directly we can prove that Nij are the coeﬀicients of a nonlinear connection. It is canonical for ΣH , since N depend only on the mechanical system ΣH . The torsion of N is 1 tij = (gih ∂˙ h Fj − gjh ∂˙ h Fi ). (4.3) 4 Evidently ∂˙ i Fj = 0, implies that N is symmetric nonlinear connection on the momenta space T ∗ M . Let (δi , ∂˙ i ) be the adapted basis to N and V and (dxi , δpi ) its adapted cobasis: 1 δi = ∂i + Nji ∂˙ j = δi + gjh ∂˙ h Fi ∂˙ j , 4 ◦

(4.4)

1 δpi = dpi − Nij dxj = δpi − gih ∂˙ h Fi dxj . 4 The tensor of integrability of the canonical nonlinear connection N is Rkij = δi Nkj − δj Nki .

(4.5)

Rkij = 0 characterize the integrability of the horizontal distribution N . i The canonical N −metrical connection CΓ(N ) = (Hjk , Cijk ) of the Hamiltonian mechanical system ΣH is given by the following Theorem: Theorem 4.3.1. The following properties hold: i 1) There exists only one N −linear connection CΓ = (Nij , Hjk , Cijk ) which depend on the Hamiltonian system ΣH and satisfies the axioms: 1◦ Nij from (4.3.2) is the canonical nonlinear connection. 2◦ CΓ is h− metric: ij g|k = 0. (4.6) 3◦ CΓ is v−metric: g ij |k = 0.

(4.3.6’)

4◦ CΓ is h−torsion free: i i i Tjk = Hjk − Hkj = 0.

(4.7)

Sijk = Cijk − Cikj = 0.

(4.3.7’)

5◦ CΓ is v−torsion free

4.4. THE CARTAN MECHANICAL SYSTEMS

219

2) The coeﬀicients of CΓ are given by the generalized Christoffel symbols: 1 i = g is (δj gsk + δk gjs − δs gjk ) Hjk 2

(4.8)

1 Cijk = − gis (∂˙ j g sk + ∂˙ k g js − ∂˙ s g jk ) 2 Now, we are in possession of all data to construct the geometry of Hamilton mechanical system ΣH . Such that we can investigate the electromagnetic and gravitational fields of ΣH .

4.4 The Cartan mechanical systems An important class of systems ΣH is obtained when the Hamiltonian H(x, p) is 2-homogeneous with respect to momenta pi . Definition 4.4.1. A Cartan mechanical system is a set ΣC = (M, K(x, p), F e(x, p))

(4.1)

where C n = (M, K(x, p)) is a Cartan space and F e = Fi (x, p)∂˙ i

(4.2)

are the external forces. The fact that C n is a Cartan spaces implies: 1◦ K(x, p) is a positive scalar function on T ∗ M . 2◦ K(x, p) is a positive 1-homogeneous with respect to momenta pi . 3◦ The pair H n = (M, K 2 (x, p)) is a Hamilton space. Consequently: a. The fundamental tensor g ij (x, p) of C n is 1 g ij = ∂˙ i ∂˙ j K 2 . 2

(4.3)

K 2 = g ij pi pj .

(4.4)

b. We have c. The Cartan tensor is 1 C ijk = ∂˙ i ∂˙ j ∂˙ k K 2 , pi C ijk = 0. 4

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CHAPTER 4. HAMILTONIAN AND CARTANIAN MECHANICAL SYSTEMS

d. C n = (M, K(x, p)) is L− dual of a Finsler space F n = (M, F (x, y)). ◦

e. The canonical nonlinear connection N , established by R. Miron [175], has the coeﬀicients ◦ 1 h N ij = γijh ph − (γsr ph pr )∂˙ s gij , (4.5) 2 i γjk (x, p) being the Christoffel symbols of gij (x, p). Clearly, the geometry of ΣC is obtained from the geometry of ΣH taking H = 2 K (x, p). So, we have: Postulate 4.4.1. The evolution equations of the Cartan mechanical system ΣC are the Hamilton equations: dxi 1 ˙ i 2 dpi 1 1 − ∂ K = 0, + ∂i K 2 = Fi (x, p). dt 2 dt 2 2

(4.6)

A first result is given by Proposition 4.4.1. 1◦ The energy of the Hamiltonian K 2 is given by EK 2 = pi ∂˙ i K 2 − K 2 = K 2 . 2◦ The variation of energy EK 2 = K 2 along to every evolution curve (4.4.6) is dxi dK 2 = Fi . dt dt

(4.7)

Example. ΣC , with K(x, p) = {γ ij (x)pi pj }1/2 , (M, γij (x)) being a Riemann spaces and F e = a(x, p)pi ∂˙ i . Theorem ... of part ... can be particularized in: Theorem 4.4.1. The following properties hold good: 1◦ ξ given by 1 ξ = [∂˙ i K 2 ∂i − (∂i K 2 − Fi )∂˙ i ] (4.8) 2 ∗M . is a vector field on T] 2◦ ξ is determined only by the Cartan mechanical system ΣC . 3◦ The integral curves of ξ are given by the evolution equations (4.4.6) of ΣC . The vector ξ is Hamiltonian vector field on T ∗ M or of the Cartan mechanical system ΣC . Therefore, the geometry of ΣC is the differential geometry of the pair (C n , ξ).

4.4. THE CARTAN MECHANICAL SYSTEMS

221

The fundamental object fields of this geometry are C n , ξ, F e, the canonical nonlinear connection N with the coeﬀicients ◦ 1 Nij =N ij + gih ∂˙ h Fj , 4

(4.9)

Taking into account that the vector fields δi = ∂i − Nji ∂˙ j determine an adapted basis to the nonlinear connection N , we can get the canonical N − metrical connection CΓ(N ) of ΣC . Theorem 4.4.2. The canonical N − metrical connection CΓ(N ) of the Cartan mechanical system ΣC has the coeﬀicients 1 i Hjk = g is (δj gsk + δk gjs − δs gjk ), Cijk = gis C sjk . 2

(4.10)

Using the canonical connections N and CΓ(N ) one can study the electromagnetic and gravitational fields on the momenta space T ∗ M of the Cartan mechanical systems ΣC , as well as the dynamical system ξ of ΣC .

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CHAPTER 4. HAMILTONIAN AND CARTANIAN MECHANICAL SYSTEMS

Chapter 5 Lagrangian, Finslerian and Hamiltonian mechanical systems of order k ≥ 1 The notion of Lagrange or Finsler spaces of higher order allows us to introduce the Lagrangian and Finslerian mechanical systems of order k ≥ 1. They will be defined as an natural extension of the Lagrangian and Finslerian mechanical systems. Now, the geometrical theory of these analytical mechanics is constructed by means of the differential geometry of the manifold T k M −space of accelerations of order k and of the notion of k−semispray, presented in the part II.

5.1 Lagrangian Mechanical systems of order k ≥ 1 We repeat shortly some fundamental notion about the differential geometry of the acceleration space T k M . Let T k M be the acceleration space of order k (see ch. 1, part II) and u ∈ T k M , u = (x, y (1) , ..., y (k) ) a point with the local coordinates (xi , y (1)i , ..., y (k)i ). A smooth curve c : I → M represented on a local chart U ⊂ M by xi = xi (t), t ∈ I can be extended to (π k )−1 (U ) ⊂ T k M by e c : I → T k M , given as follows: i

i

x = x (t), y

(1)i

1 dxi (t) 1 d k xi (k)i = , ..., y = (t), t ∈ I. 1! dt k! dtk

(5.1)

Consider the vertical distribution V1 , V2 , ..., Vk . They are integrable and have the properties V1 ⊃ V2 ⊃ ... ⊃ Vk . The dimension of Vk is n, dimVk−1 = 2n, ..., dimV1 = kn. 223

224CHAPTER 5. LAGRANGIAN, FINSLERIAN AND HAMILTONIAN MECHANICAL SYSTEMS OF OR

The Liouville vector fields on T k M are: 1 ∂ Γ = y (1)i (k)i ∂y ∂

2

Γ = y (1)i

∂y (k−1)i

+ 2y (2)i

∂ ∂y (k)i

(5.2)

............. k

Γ = y (1)i 1

2

∂ (2)i ∂ (k)i ∂ + 2y + ... + ky . ∂y (1)i ∂y (2)i ∂y (k)i k

Thus Γ ∈ Vk , Γ ∈ Vk−1 , ..., Γ ∈ V1 . The following operator: Γ = y (1)i

∂ ∂ (2)i ∂ (k)i + 2y + ... + ky ∂xi ∂y (1)i ∂y (k−1)i

(5.3)

is not a vector field on T k 2 it is not a differential system of equations of order 2, like in the classical Lagrange equations. We solve this problem by means of the prolongation of order k of the Riemannian space Rn = (M, γij (x)). 1 Let ΣR = (M, 2T, F e) a mechanical system of form (5.7.1) with T = γij (x)y (1)i y (1)j , 2 i dx as kinetic energy. Thus the Riemannian space Rn = (M, γij (x)) y (1)i = dt can be prolonged to the Riemannian space of order k ≥ 1 Prolk Rn = (T k M, G), where G = γij (x)δxi ⊗ δxj + γij (x)δy (1)i ⊗ δy (1)j + ... + γij (x)δy (k)i ⊗ δy (k)j .

(5.2)

The 1-forms δxi , δy (1)i , ..., δy (k)i being determined as follows: δxi = dxi , δy (1)i = dy (1)i + M i dxj , ..., δy (k)i = dy (k)i + (1) j

(5.3) i

+M dy (1) j

(k−1)i

i

j

+ ... + M dx . k j

In these formulas M i (α = 1, ..., k) are the dual coeﬀicients of a nonlinear connec(α) j

5.7. CLASSICAL RIEMANNIAN MECHANICAL SYSTEM WITH EXTERNAL FORCES DEPENDING ON ◦

tion N determined only on the Riemannian structure γij (x). That are i M i (x, y (1) ) = γjh (x)y (1)m , 1 j

!

1 ΓM i + M i M m 2 j 2 (1) j (1) m (1) j .................................. ! 1 M i (x, y (1) , ..., y (k) ) = Γ M i + Mi M m , k j 2 (k−1)j (1) m (k−1)j M i (x, y (1) , y (2) ) =

(5.4)

Γ being the nonlinear operator: Γ = y (1)i

∂ ∂ ∂ + 2y (2)i (1)i + ... + ky (k)i (k−1)i . i ∂x ∂y ∂y

(5.5)

Thus, (cf. ch. ...) the dual coeﬀicients M (5.7.4) and the primal coeﬀicients N (α)

(α)

(5.4.7) depend on the Riemannian structure γij (x), only. Now the Lagrangian mechanical system of order k: ΣP rolk Rn = (M, L(x, y (1) , ..., y (k) )), F e(x, y (1) , y (k) ), L = γij (x)z (k)i z (k)j ,

(5.6)

kz (k)i = ky (k)i + M i y (k−1)m + ... + M (1) m

i

y (1)m ,

(k−1)m

depend only on ΣRn . It is the prolongation of order k of the Riemannian mechanical system (4.7.1), ΣRn . We can postulate: The Lagrange equations of ΣRn are the Lagrange equations of the prolonged Lagrange equations of order k, ΣP rolk Rn : ∂L(x, y (1) , ..., y (k) ) d ∂L(x, y (1) , ..., y (k) ) − = Fi (x, y (1) , ..., y (k) ) (k)i (k−1)i dt ∂y ∂y (5.7) dxi 1 d k xi (1)i (k)i y = , ..., y = . dt k! dtk Applying the general theory from this chapter, we have

240CHAPTER 5. LAGRANGIAN, FINSLERIAN AND HAMILTONIAN MECHANICAL SYSTEMS OF OR

Theorem 5.7.1. The Lagrange equations of the classical Riemannian systems ΣR with external forces depending on higher order accelerations are given by ◦i dk+1 xi 1 (1) (k) + (k + 1)! G (x, y , ..., y ) = k!F i (x, y (1) , ..., y (k) ) k+1 dt 2 ◦i 1 ij ∂L ∂L (k + 1)G = γ Γ (k)j − (k−1) 2 ∂y ∂y

(5.8)

L = γij (x)z (k)i z (k)j . The canonical semispray of ΣR is expressed by S=y

(1)i

◦i ∂ ∂ ∂ 1 i ∂ (k)i + ... + ky − (k + 1) G + F . ∂xi ∂y (k−1)i ∂y (k)i 2 ∂y (k)i

(5.9)

The integral curves of S are given by the Lagrange equations (5.7.15). The canonical semispray S is the dynamical system of the classical Lagrangian mechanical system ΣR with the external forces depending on the higher order accelerations.

5.8 Finslerian mechanical systems of order k, ΣF k The theory of the Finslerian mechanical systems of order k is a natural particularization of that of Lagrangian mechanical systems of order k ≥ 1, described in the previous sections of the present chapter. Definition 5.8.1. A Finslerian mechanical system of order k ≥ 1 is a triple: ΣF k = (M, F (x, y (1) , ..., y (k) )), F e(x, y (1) , ..., y (k) ),

(5.1)

F (k)n = (M, F (x, y (1) , ..., y (k) ))

(5.2)

where is a Finsler space of order k ≥ 1, and F e(x, y (1) , ..., y (k) ) = F i (x, y (1) , ..., y (k) ) is an a priori given vertical vector field.

∂ ∂y (k)i

(5.3)

5.8. FINSLERIAN MECHANICAL SYSTEMS OF ORDER K, ΣF K

241

F (x, y (1) , ...y (k) ) is the fundamental function of ΣF k , 1 ∂˙ ∂ gij = F2 (k)i (k)j 2 ∂y ∂y

(5.4)

is the fundamental tensor of ΣF k and F e are the external forces. ◦

The Euler–Lagrange equations of F (k)n are given by (5.2.5) E i (F 2 ) = 0, y (1)i = dxi 1 d k xi (k)i , ..., y = and the Craig–Synge covector field is E k−1 (F 2 ) from (5.2.7). k dt k! dt Theorem 5.2.1 for L = F 2 can be applied. ◦

Thus, the canonical semispray S of F (k)n is: ◦

S = y (1)i

◦ ∂ ∂ ∂ (k)i (2)i ∂ + ... + ky − (k + 1) G , + 2y ∂xi ∂y (1)i ∂y (k−1)i ∂y (k)i

with the coeﬀicients:

1 ij ∂F 2 ∂F 2 (k + 1)G = g Γ (k)i − (k−1)i 2 ∂y ∂y ◦i

(5.5)

(5.6)

and with nonlinear operator Γ = y (1)i

∂ ∂ ∂ + 2y (2)i (1)i + ... + ky (k)i (k−1)i . i ∂x ∂y ∂y

(5.7)

◦

The canonical nonlinear connection N of F (k)n has dual coeﬀicients (5.2.10) particularized in the following form: ◦ i

M = 1 j

◦i

∂G ∂y (k)j

◦ i ◦ m ◦ i 1 ◦ ◦i M = (S M + M M ), 1 m 1 j 2 j 2 1j ................................. ◦ i ◦ i ◦ m 1 ◦ ◦ i M = (S M + M M ). k j k k−1j k−1m k−1j

(5.8)

The fundamental equations of the Finslerian mechanical systems of order n ≥ 1, ΣF n are given by

242CHAPTER 5. LAGRANGIAN, FINSLERIAN AND HAMILTONIAN MECHANICAL SYSTEMS OF OR

Postulate 4.8.1. The evolution equations of the Finslerian mechanical system ΣF k = (M, F, F e) are the following Lagrange equations d ∂F 2 ∂F 2 − = Fi , Fi = gij F j , (k)i (k−1)i dt ∂y ∂y

(5.9)

dxi 1 d k xi (k)i y = , ..., y = . dt k! dtk Remark 5.8.0.1. For k = 1, (5.8.8) are the Lagrange equations of a Finslerian mechanical system ΣF = (M, F, F e). (1)i

The Lagrange equations are equivalent to the following system of differential equations of order k + 1: dk+1 xi k! ij ∂F 2 ∂F 2 k! i + g Γ − = F . (5.10) dtk+1 2 2 ∂y (k)i ∂y (k−1) Integrating this system in initial conditions we obtain an unique solution xi = xi (t), which express the moving of Finslerian mechanical system ΣF k . But it is convenient to write the system (5.8.9) in the form (5.3.5): y

(1)i

1 d k xi dxi (2)i 1 d2 xi (k)i , y = , ..., y = = dt 2 dt2 k! dtk

(5.11)

◦i dy (k)i 1 + (k + 1)G (x, y (1) , ..., y (k) ) = F i (x, y (1) , ..., y (k) ). dt 2 These equations determine the trajectories of the vector field ◦ 1 S = S + F e. 2

(5.12)

Consequently, we have: Theorem 5.8.1. 1◦ For the Finslerian mechanical system of order k ≥ 1, ΣF k the operator S from (5.8.12) is a k−semispray which depend only on ΣF k . 2◦ The integral curves of S are the evolution curves of ΣF k . S is the canonical semispray of the mechanical system ΣF k . It is named the dynamical system of ΣF n , too. The geometrical theory of the Finslerian mechanical system ΣF k can be developed by means of the canonical semispray S, applying the theory from the sections 5.4, 5.5, 5.6 from this chapter.

5.9. HAMILTONIAN MECHANICAL SYSTEMS OF ORDER K ≥ 1

243

Remark 5.8.1.1. Let ΣF = (M, F, F e) be a Finslerian mechanical system, where F e depend on the higher order accelerations. What kind of Lagrange equations we have for ΣF ? The problem can be solved by means of the prolongation of Finsler space F n = (M, F ) to the manifold T k M , following the method used in the section 5.7 of the present chapter.

5.9 Hamiltonian mechanical systems of order k ≥ 1 The Hamiltonian mechanical systems of order k ≥ 1 can be studied as a natural extension of that of Hamiltonian mechanical systems of order k = 1, given by ΣH = (M, H(x, p), F e(x, p)), ch. 4, part II. So, a Hamiltonian mechanical system of order k ≥ 1 is a triple ΣH k = (M, H(x, y (1) , ..., y (k−1) ), p), F e(x, y (1) , ..., y (k−1) , p)

(5.1)

H (k)n = (M, H(x, y (1) , ..., y (k−1) ), p)

(5.2)

where is a Hamilton space of order k ≥ 1 (see definition 5.1.1) and F e(, y (1) , ..., y (k−1) , p) = Fi (x, y (1) , ..., y (k−1) , p)∂˙ i

(5.3)

are the external forces. ∂ Of course, ∂˙ i = . ∂pi Looking to the Hamilton–Jacobi equation of the space H (k)n we can formulate Postulate 5.9.1. The Hamilton equations of the Hamilton mechanical system ΣH k are as follows: H = 1H 2 i dx ∂H = ∂pi dt (5.4) k−1 ∂H ∂H 1 d d ∂H 1 dp d i =− + ... + (−1) = Fi + k−1 ∂y k−1 (1)i dt ∂x dt (k − 1)! dt dt 2 ∂y i (1)i dxi (2)i 1 d2 xi 1 dk−1 xi (k−1) , y = , ..., y = . y = dt 2 dt2 (k − 1)! dtk−1

244CHAPTER 5. LAGRANGIAN, FINSLERIAN AND HAMILTONIAN MECHANICAL SYSTEMS OF OR

It is not diﬀicult to see that these equations have a geometric meaning. For k = 1 the equations (5.9.4) are the Hamilton equations of a Hamiltonian mechanical system ΣH = (M, H(x, p), F e(x, p)) . These reasons allow us to say that the Hamilton equations (5.9.4) are the fundamental equations for the evolution of the Hamiltonian mechanical systems of order k ≥ 1. Example 5.9.1. The mechanical system ΣH k with H(x, y (1) , ..., y (k−1) , p) = αγ ij (x, y (1) , ..., y (k−1) )pi pj ,

(5.5)

γ ij being a Riemannian metric tensor on the manifold T (k−1)n and with F e = βi (x, y (1) , ..., y (k−1) )∂˙ i

(5.9.5’)

βi is a d−covector field on T k−1 M and α > 0 a constant. Let us consider the energy of order k − 1 of E k−1 (H) = I k−1 (H) −

1 d k−2 I (H) + ...+ 2! dt (5.6)

k−2

+(−1)k−2

1 d (H) − H, (k − 1)! dtk−2

where I (1) H = L 1 H, ..., I k−1 H = Lk−1 . Γ

Γ

Theorem 5.9.1. The variation of energy E k−1 (H) along the evolution curves (5.9.4) can be calculate without diﬀiculties. Thus, the canonical nonlinear connection N ∗ of ΣH k is given by the direct decomposition (5.1.11), and has the coeﬀicients N i ,..., N i , Nij from (5.1.12’) (1)j

(k−1)j

and its dual coeﬀicients M ,..., M from (5.1.14). We can calculate the Liouville (1) (k−1)

(k−1)

vector fields z (1) , ..., z from (5.1.17). ∗ The metrical N −linear connection has the coeﬀicients DΓ(N ∗ ) = (Hijs , C i , Cijs ) (k)jk

given by (5.5.2). Now we can say: The geometries of the Hamiltonian mechanical systems of order k ≥ 1, ΣH k can be constructed only by the fundamental equations (5.9.4), by the nonlinear connection of ΣH k and by the N ∗ −metrical connection CΓ(N ∗ ).

5.9. HAMILTONIAN MECHANICAL SYSTEMS OF ORDER K ≥ 1

245

Remark 5.9.1.1. The homogeneous case of Cartanian mechanical systems of order k, ΣC k = (M, K(x, y (1) , ..., y (k−1) ), p), F e(x, y (1) , ..., (k−1) (1) (k−1) y p) is a direct particularization of the previous theory for H(x, y , ..., y , p) = 2 (1) (k−1) (1) (k−1) K (x, y , ..., y , p), where K(x, y , ..., y , p) is k−homogeneous with re(1) (k−1) spect to variables (y , ..., y , p).

246CHAPTER 5. LAGRANGIAN, FINSLERIAN AND HAMILTONIAN MECHANICAL SYSTEMS OF OR

Bibliography [1] Abate, M., Patrizio, G., Finsler Metrics- A Global Approach with applications to geometric functions theory, Lecture Notes in Mathematics 1591, 1994, Springer-Verlag, Berlin. [2] Abraham, R., Marsden, J., Foundations of Mechanics, Benjamin, New-York, (1978). [3] Aikou, T., A partial connection on complex Finsler bundle and its applications, Illinois J. Math. 42(1998), 83–111. [4] Aikou, T., The Chern–Finsler connection and Finsler–Kähler manifolds, in Finsler Geometry, Advanced Study of Pure Mathematics 48, pp. 343–373, 2007. [5] Aikou, T., Differential Geometry of Finslerian Vector Bundles, PhD. Thesis, Iaşi, Romania, 1991. [6] Albu, I.D., Opriş, D., Densités Lagrangiennes invariantes par rapport aux automorphismes d’une variété fibrée, Proc. Int. Workshop on Diff. Geom. Sci. Bul. “Politehnica” Univ. of Bucharest, S.A. vol. 55, no. 3-4, 1993, 1-7. [7] Amici, O., Casciaro, B., An example of Lagrangian theory of relativity satisfying the conservation law, Proc. Nat. Sem. on Finsler and Lagrange Spaces, Braşov, Romania, 1986, 37–42. [8] Anastasiei, M., Symplectic structures and Lagrange geometry, in M. Anastasiei, P.L. Antonelli (eds.), Finsler and Lagrange Geometry, Kluwer Academic Publisher, pp. 1–6, 2003. [9] Anastasiei, M., Metrizable nonlinear connections, in Proceedings of the Annual Symposium on Mathematics applied in Biology and Biophysics, Iaşi, May, UASVM, pp. 1–6, 2004. 247

248

BIBLIOGRAPHY

[10] Anastasiei, M., The Geometry of Time-Dependent Lagrangians, Math. Comput. Modelling, vol. 20, no. 4/5, 1994, 67–81. [11] Anastasiei, M., A historical remark on the connections of Chern and Rund, Contemporary mathematics, Finsler Geometry 196, 1996, 171-176. [12] Anastasiei, M., Certain generalizations of Finsler metrics, Contemporary Mathematics, 196, 1996, 161-170. [13] Anastasiei, M., Antonelli, P.L., The differential geometry of Lagrangians which generate sprays, in vol. of Kluwer Acad. Publ. FTPH, no. 76, 1996, 15-34. [14] Anastasiei, M., Shimada, H., Deformation of Finsler metrics, Finslerian Geometries, Kluwer Academic Publishers, FTPH. 109, 2000, 53-66. [15] Anastasiei, M., Kawaguchi, H., A geometrical theory of time dependent Lagrangians, I: Non-linear connections, Tensor N.S., vol. 48, 273–282. [16] Anastasiei, M., Kawaguchi, H., A geometrical theory of time dependent Lagrangians, II: M-connections, Tensor N.S., vol. 48, 283–293. [17] Andrés, L.C., de Léon, M, Rodriques, P.R., Connections on tangent bundle of higher order associated to regular Lagrangians, Geometriae Dedicata 39, 17–28, 1991. [18] Antonelli, P.L., Ingarden, R.S., Matsumoto, M., The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Acad. Publ. FTPH no. 58, 1993. [19] Antonelli, P.L., Miron, R., (eds.), Lagrange and Finsler Geometry. Applications to Physics and Biology, Kluwer Acad. Publ. FTPH no. 76, 1996. [20] Antonelli, P.L. (ed.), Finslerian Geometries. A Meeting of Minds, Kluwer Acad. Publ. (FTPH), vol. 109(2000). [21] Antonelli, P.L. Handbook of Finsler Geometry, vol. I, II, Kluwer Acad. Publ. (2003). [22] Antonelli, P. L., Hrimiuc, D., A new class of spray-generating Lagrangians, Lagrange and Finsler Geometry. Applications to Physics and Biology, Kluwer Academic Publishers 76, 2000, 81-92.

BIBLIOGRAPHY

249

[23] Antonelli, P. L., Lackey, B. C., (eds.) The Theory of Finslerian Laplacians and Applications, Kluwer Academic Publishers, 459, 1997. [24] Antonelli, P. L., Zawstaniak, T.J., Lagrange Geometry, Finsler Spaces and Noise Applied in Biology and Physics, Math. Comp. Mod. 20, no. 4-5, 1994. [25] Aringazn, A.K., Asanov, G.S., Problems of Finslerian theory of Gauge fields and Gravitation, Rep. Math. Phys. 25, 1988, 35–93. [26] Arnold, V.I., Mathematical Methods of Classical Mechanics, Texts in Mathematics, Springer-Verlag, (1989). [27] Asanov, G.S., Finsler Geometry, Relativity and Gauge Theories, D. Reidel Publ. Co., Dordrecht, 1985. [28] Asanov, G.S., Stavrinos, P.C., Finslerian Deviation of Geodesics over Tangent Bundle, Rep. on Math. Physics, Vol. 30 No. 1, pp. 63–69, 1991. [29] Bácsó, S., Randers and Kropina spaces in geodesic correspondence, in vol. of Kluwer Acad. Publ. FTPH, no. 76, 1996, 61-64. [30] Bacso, S., Matsumoto, M., On Finsler Spaces of Douglas type, Publ. Math., Debrecen 51/3-4, 1997, 385-406. [31] Balan, V., On the generalized Einstein–Yang Mills equations, Publ. Math. Debrecen, 43, 1993, 273–282. [32] Balan, V., Stavrinos, P.C., Deviations of Geodesics in the Fibred Finslerian Approach, Kluwer Academic Publishers, 76, 1996, 65-75. [33] Barbu, V., Mathematical Methods in Optimization of Differential Systems, Kluwer Academic Publishers, 1994. [34] Barthel, W., Nichtlineare Zusammenhange und Deren Holonomie Gruppen, J. Reine Angew. Math. 212, 1963, 120–149. [35] Bao, D., Chern, S.S., Shen, Z., An introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics; 2000, Springer–Verlag, 2000. [36] Barbosu, M., Riemannian Curvatures in the Planar Three-Body ProblemStability, C.R. Acad. Sci. Paris, (1999), t.327, Serie II b, p. 959-962, with Elmabsout B..

250

BIBLIOGRAPHY

[37] Bao, D., Chern, S.S., Shen, Z. (eds.), Finsler Geometry, Contemporary Mathematics, AMS, no.196, 1995. [38] Bejancu, A., Finsler Geometry and Applications, Ellis Harwood Ltd. (1990). [39] Beil, R.G., On the physics of generalized Lagrange geometry, An. Şt. ale Univ. “Al. I. Cuza” Iaşi, Tom XLIX, 2003, 1, 2, 229–238. [40] Beil, R.G., Electrodynamics from a Metric, Int. J. of Theoretical Physics v.26, 189–197, 1987. [41] Beil, R.G., Finsler Gauge Transformations and General Relativity, Intern. J. Theor. Phys., 31, 6, 1992, 1025–1044. [42] Berwald, L., Ueber Finslersche und Cartansche Geometrie. I. Geometrische Erklärungen der Krümmung und des Hauptskalars eines zweidimensionalen Finslerschen Raumes, Mathematica, Timisoara 17, (1941), 34–58. [43] Blair, D.E., Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics 509, Springer-Verlag, Berlin, 1976. [44] Blănuţă, V., Almost r −p structures on the cotangent bundle, Tensor Society, Japan, New Series, Vol. 54, 1993, 17 – 21. [45] Blănuţă, V., Natural 2 − p structures in Lagrange space of higher order, Journal of the Egyptian Math. Soc. Cairo, Egipt, Vol. 9 (2), 2001, 151 – 165. [46] Boskoff, W. G., Hyperbolic geometry and Barbilian spaces, Hardronic Press, 1996, H.P. Inc. Fl. USA. [47] Bucătaru, I., Linear connections for systems of higher order differential equations, Houston Journal of Mathematics, 32(1), 315–332 (2005). [48] Bucataru, I., Connection map in the higher order geometry, An. St. Univ. “Al. I. Cuza” Iasi, Ser. I Mat., XLII, 1996, 253–258. [49] Bucataru, I., Miron, R., Finsler-Lagrange Geometry. Applications to Dynamical Systems, Romanian Academy Publ. House, (2007). [50] Bucataru, I., Miron, R., Nonlinear Connections for Nonconservative Mechanical Systems, Reports on Mathematical Physics, 59(2), p.225–241, (2007).

BIBLIOGRAPHY

251

[51] Buchner, K., Kräftige Lösungen der Einsteinschen Gleichungen, Tensor NS. 38, 65–68, 1982. [52] Busemann, H., The geometry of Finsler spaces, Bull. Amer. Math. Soc. 56 (1950), 5–16. [53] Cantrijn, F., Crampin, M., Sarlet, W., Higher order differential equations and Higher order Lagrangian Mechanics, Proc. Camb. Phil. Soc. 99, 1986, 565–587. [54] Capursi, M., Palombella, A., On the almost Hermitian structures associated to a generalized Lagrange space, Mem. Sect. St. Acad. Romane, VIII, 1985, 57–64. [55] Caratheodory, C., Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Teubner, 1935. [56] Cartan, E., Les Espaces de Finsler, Hermann, Paris, 1934. [57] Do Carmo, M.P., Riemannian Geometry, Birkhäuser, Boston, 1992. [58] Chen, Bang-yen, Differential geometry of real submanifolds in a Kähler manifold, Monat. für Math., Volume 91, Number 4, 257–274, 1981. [59] Chern, S.S., Local equivalence and Euclidean conditions in Finsler spaces, Science Reports Nat. Tsing Hua Univ. 5 (1948), 95–121. [60] Chern, S.S., On Finsler geometry, C.R. Acad. Sci. Paris 314 (1992), 757–761. [61] Comic, I., Generalized second order gauge connections, Proc. of 4th Internat. Conf. Of Geometry, Thessaloniki (1996), 114-128. [62] Comic, I., Curvature Theory of Generalized Mirons d-Connection, Diff. Geom. and Appl. Proc. Conf. Brno, 1989, 17–26. [63] Craig, H. V., On a generalized tangent vector, Amer. J. Math. 57 (1935), 457–462. [64] Crampin, M., Jet bundle techniques in Analytical Mechanics, Quaderni del Cons. naz. delle Ricerche, Gruppo Naz. di Fisica Matematica no. 47, (1995).

252

BIBLIOGRAPHY

[65] Crampin, M., On the inverse problem of the calculus of variations for systems of second-order ordinary differential equations, in the vol. 109, 2000, Kluwer Acad. Publ.. [66] Crampin, M., Pirani, F. A. E., Applicable Differential Geometry, Lecture Notes S. 59, Cambridge University Press 1986. [67] Crasmreanu, M., A Noetherian symmetry for 2D spinning particle, Internat. J. Non-Linear Mech., 35(2000), no. 5, 947–951. [68] Crasmareanu, M., Fibre bundle maps and complete sprays in Finslerian setting, J. Korean Math. Soc., 46(2009), no. 3, 551–560. [69] Cruceanu, V., On certain lifts in the tangent bundle, An. Şt. Univ. “Al. I. Cuza” Iaşi, 46(2000), 57–72. [70] Cruceanu, V., Sur la théorie des sous-fibrés vectoriels, C. R. Acad. Sci. Paris, Ser. I 302 (20), 1986. [71] Cruceanu, V., Sur certain morphismes des structures géométriques, Rendiconti di Mat., ser. VIII, V.G.N., 1986, 321–332. [72] Dekret, A., Natural connections of 1−forms on tangent bundles, Diff. Geom. and its Appl. Proc. Int. Conf. Opava, 1992. [73] Dillen, F.J.E., Verstraelen, L.C.A., Handbook of Differential Geometry, vol. I, II (2005, 2006). [74] Djukic, D., Vujanovic, B., On Some Geometrical Aspects of Classical Nonconservative Mechanics, Journal of Mathematical Physics, vol. 16, no. 10, p. 2099–2102, (1975). [75] Djukic, D., Vujanovic, B., Noether’s Theory in Classical Nonconservative Mechanics, Acta Mathematica, no.23, p.17-27, (1975). [76] Dodson, C.T.J., Radivoiovici, M.S., Tangent and frame bundles of order two, Analele Stiintifice ale Universitatii Al. I. Cuza 28 (1982), pp. 63–71. [77] Dombrowski, P., On the geometry of the tangent bundle, J. Reine Angew. Math., 210 (1962), 73–88. [78] Dragoş, L., Principiile Mecanicii Analitice, Ed. Tehnică, Bucuresti, 1969.

BIBLIOGRAPHY

253

[79] Duc, T.V., Sur la géométrie différentielle des fibrés vectoriels,Kodai Math. Sem. Rep. 26, 1975, 349–408. [80] Eck, D.J., Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc. 247 (1981), pp. 1–48. [81] Ehresmann, C., Les connections infinitésimales dans un espace fibré différentiable, Colloq. de Topol., CBRM, Bruxelles (1950) pp. 29–55. [82] Ehresmann, C., Les prolongements d’un espace fibré différentiable, Compte Rend. de l’Acad. Sci. Paris 240 (1955), 1755–1757. [83] Eliopoulos, H.A., A generalized metric space for electromagnetic theory, Bull. CI. Sci. Acad. R. Belg., No. 9, 986–995 (1965). [84] Endo, H., Submanifolds in contact metric manifolds and in almost cosymplectic manifolds, PhD Thesis, “Al.I.Cuza” Univ. of Iasi, Romania, 1993. [85] Frigioiu, C. [86] Fueki, S., Endo, H., A Sasakian structure on the indicatrix bundle of a Finsler space with the Chern-Rund connection, Proceedings of the 43-th Symposium on Finsler Geometry, Nov. 4–7, p. 15–19, (2008). [87] Garcia, X., Pons, J. M., Román-Roy, N., Higher order lagrangian systems: geometric structures, dynamics and constraints, J. Math. Phys. 32 (1991) 2744–2763. [88] Gancarzewicz, J., Mikulski, W., Pogoda, Z., Natural bundles and natural liftings. Prolongations of geometric structures, Proc. Conf. on Diff. Geometry and Its Applications, Opava, August 24-28, 1992, Silesian University, Opava (1993), 281-320. [89] Ghinea, I., Conexiuni Finsler compatibile cu anumite structuri geometrice, Teza de doctorat. Univ. “Babes—Bolyai”, Cluj-Napoca, 1978. [90] Greub, W., Halperin, S., Vanstone, R., Connections, Curvature and Cohomology, vol. I. Academic Press, New York (1972). [91] Grifone, J., Structure Presque Tangente et Connections I, II, Ann. Inst. Fourier, Grenoble, 22, 1(1972).

254

BIBLIOGRAPHY

[92] Grigore, D. R., A Generalized Lagrangian Formalism in Particle Mechanics and Classical field Theory, Fortschr. der Phys. 41 (1993). [93] Grigore, D.R., Generalized Lagrangian dynamics and Noetherian symmetries, Internat. J. Modern Phys. A 7 (1992), 7153–7168. [94] Grigore, D.R., The derivation of Einstein equations from invariance principles, 1992, Class. Quantum Grav. 9. [95] Ginzburg, V., On the existence and non-existence of closed trajectories for some Hamiltonian flows, Math. Z. 223, p.397–409, (1996). [96] Gîrţu, V., Torse forming Finsler vector fields and Beil metrics, Proc. of the 40-th Symposium on Finsler Geometry, Sept. 6–10, 2005, Sapporo (Hokkaido Tokai University), Japan, 45–48. [97] Gîrţu, M., A framed f (3, −1)– structure on a GL– tangent manifold, Balkan Journal of Geometry and its Applications, Vol. 13, no. 1, 47–54, 2008. [98] Grifone, J., Structure presque–tangente et connexions, I, II, Ann. Inst. Henri Poincaré 22(1) (1972), pp. 287–334; Ann. Inst. Henri Poincaré 22(3) (1972), pp. 291–338. [99] Haimovici, A., Sur les espaces à connexion non linéaire à parallelisme absolue, Rev. Roum. Math. Pures Appl., 10, 1965, 1121–1128. [100] Haimovici, M., Les formules fondamentales de la théorie des hypersurfaces d’un espace générale, Ann. Sci. Univ. Jassy, 20, 1935, 39–58. [101] Haimovici, M., Variétés totalement extrémales et variétés totalement géodésiques dans les espaces de Finsler, Ann. Sci. Univ. Jassy, 29, 1939, 559–644. [102] Hasegawa, I., Yamaguchi, K., Shimada, H., Sasakian structures on Finsler manifolds, in P.L. Antonelli and R. Miron (eds.), Lagrange and Finsler Geometry, Kluwer Academic Publishers, 1996, p. 75–80. [103] Hashiguchi, M., On conformal transformation of Finsler metrics, J. Math. Kyoto. Univ., 16 (1976), 25–50. [104] Hashiguchi, M., On the generalized Finsler spaces, An. St. Univ. “Al.I.Cuza”, Iasi, XXX, 1984, 69–73.

BIBLIOGRAPHY

255

[105] Hedric Stevanovici, K. [106] Hermann, R., Differential geometry and the calculus of variations, Review Math Sci Press, 1977, Mathematics. [107] Hermann, R., Vector Bundles in Mathematical Physics, W. A. Benjamin, Inc. 1970. [108] Horváth, J.I., New geometrical methods of the theory of physical fields, Nouv. Cimento 9 (1958), pp. 444–496. [109] Hrimiuc, D., Shimada, H., On the L−duality between Lagrange and Hamilton manifolds, Nonlinear World 3 (1996), 613–641. [110] Ianus, S., Differential Geometry and Applications to Theory of Relativity (in Romanian), Ed. Acad. Rom., 1980. [111] Ichijyo, Y., Miron, R., On some structures defined on dominant vector bundles, J. Math. Tokushirna Univ. vol. 20, 1986. [112] Ichijyo, Y., Miron, R., On some G−structures defined on tangent bundles, Tensor N.S. 42, 1985, 179–190. [113] Igarashi, T., Cartan Spaces and Applications, “Al. I. Cuza” Univ. Iasi, Ph. D. Thesis (1994). [114] Ingarden, R.S., Differential Geometry and Physics, Tensor N.S. 30 (1976), pp. 201–209. [115] Ingarden, R.S.,On physical interpretations of Finsler and Kawaguchi Geometries, Tensor, N.S. 46 (1987), pp. 354–360. [116] Ikeda, S., Advanced Study in Applied Geometry, Seizansha, Kanagawa, 1995. [117] Ionescu-Pallas, N., General Relativity and Cosmology, Ed. Stiintifica, Bucharest, 1980. [118] Ishikawa, H., Einstein equation in lifted Finsler spaces, Nuovo Cimento B, Serie 11, vol. 56B, 1980, p. 252–262. [119] Kawaguchi, M. An introduction to the theory of higher-order spaces I: the theory of Kawaguchi spaces, in RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of

256

BIBLIOGRAPHY

Geometry, vol. III, Edited by Kondo, K., 718–734, Gakujutsu Bunken Fukyu-Kai, Tokyo, 1962. [120] Kawaguchi, A., On the vectors of higher order and the extended aﬀine connections, Ann. di Mat. pura ed Appl., Volume 55, Number 1, 1961, 105–117. [121] Kern, J., Lagrange Geometry, Arch. Math., 25, 1974, 438–443. [122] Kirkovits, M., Variational calculus in Lagrangian spaces, Thesis, Iasi, 1989. [123] Klein, J., Espaces Variationnels et Mécanique, Ann. Inst. Fourier, Grenoble, 12, p. 1–124, (1962). [124] Klepp, F.C., Remarkable Finsler structures and the Finsler geometry of vector bundles, Ph. D. Thesis. Univ. “Al. I. Cuza”, Iasi, 1982. [125] Kobayashi, S., Nomizu, K., Foundations of differential geometry, vol. I-1963, vol. II-1973, Wiley Interscience, New York. [126] Kolár, I., Canonical forms on the prolongations of principal fiber bundles, Rev. Roumaine Math. Pures Appl. 16 (1971), 1091–1106. [127] Kolár, I., On the prolongations of geometric object fields, Analele stiintifice ale Universitatii “Al.I.Cuza”, XVII, 1971, 437–446. [128] Kondo, K., Construction of Kawaguchi space-time by statistical observation of monads, and the origin of the quantum, physical fields and particles, in RAAG Memoirs of the Unifying Study of Basic Problems in Engineering and Physical Sciences by Means of Geometry, vol. III, Edited by Kondo, K., 263–306, Gakujutsu Bunken Fukyu-Kai, Tokyo, 1962. [129] Kondo, K. On the physical meaning of the Zermelo conditions of Kawaguchi space, Tensor N.S., 14 (1963), 191–215. [130] Koszul, L., Lectures on fibre bundles and differential geometry, Tata Institute, Bombay, 1960. [131] Kozma, L., On some classical conditions for Finsler type connections, Proc. Nat. Sem. on Finsler and Lagrange Spaces, Brasov, 1988, 207–216. [132] Krupka, D., Lepagean forms in higher order variational theory, Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, June 1982.

BIBLIOGRAPHY

257

[133] Krupka, D., Local Invariants of a Linear Connection, in Coll. Math. Soc, Janos Bolyai, 31. Differential Geometry, Budapest 1979. North Holland, 1982. [134] Krupka, D., Janyska, J., Lectures on Differential Invariants, Brno University, Brno, Czech Republic, 1990. [135] Krupková, O., The Geometry of Ordinary Variational Equations, SpringerVerlag, (1997). [136] Krupková, O., Variational Analysis on Fibered Manifolds over Onedimensional Bases, PhD Thesis, Silesian University, Opava & Charles University, Prague, 1992, 67 pp. [137] Kupershmidt, B. A., Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms, Lect. Notes Math. 775, 1980, pp. 162–218. [138] de Leon, M., Rodrigues, P. R., Almost tangent geometry and higher order mechanical systems, Proceedings of the Conference on Di erential Geometry and Its Applications, August, 24-30, 1986, Brno, Reidel and J. E. Purkyne, Brno (1987) 179–195. [139] Lovelock D., Rund H., Tensors, differential forms, and variational principles, John Wiley & Sons, New York, 1975. [140] Mangeron, D., Irimiciuc, N., Mecanica rigidelor cu aplicatii in inginerie, Editura Tehnica, Bucuresti, 1980. [141] de Leon, M., Rodrigues, P.R., Methods of Differential Geometry in Analytical Mechanics, North-Holland (1989). [142] de Leon, M., Rodrigues, P.R., Generalized Classical Mechanics and field Theory, North-Holland, 1985, 73–124. [143] Libermann, P., Marle, Ch.M., Symplectic Geometry and Analytical Mechanics, D.Reidel Publ. Comp. (1987). [144] Matsumoto, M., Foundations of Finsler Geometry and Special Finsler Spaces, Kaisheisha Press, Otsu, 1986. [145] Matsumoto, K., Differential geometry of locally conformal Kähler manifolds and their submanifolds, PhD Thesis, “Al. I. Cuza” Univ., Iasi, 1994.

258

BIBLIOGRAPHY

[146] Matsumoto, K., Mihai, I., On a certain transformation in a Lorentzian para Sasakian manifold, Tensor, N.S. 47 (1988), 189–197. [147] Matsumoto, K., Mihai, I., Rosca, R., Certain real hypersurfaces of a locally conformal Kähler space form, Bull. Yamagata Univ. 14, 1992, 1–11. [148] Mayer, O., Opera Matematică, vol. I, Ed. Acad., Bucuresti, 1974. [149] Mihai, I., Capitole Speciale de Geometria Varietatilor Complexe, Ed. Univ. Bucuresti, 1995. [150] Mihai, I., Generic submanifolds of some almost paracontact Riemannian manifolds, Topics in Differential Geometry, Coll. Math. Soc. J. Bolyai, 46, 1988, 777–791, North-Holland. [151] Miron, R., Lagrange Geometry, Math. Comput. Modelling 20 (4/5), 1994. [152] Miron, R., General Randers Spaces, Lagrange and Finsler Geometry, in P.L. Antonelli and R. Miron, Editors, FTPH no. 59, Kluwer Acad. Publ., Otsu (1996). [153] Miron, R., Geometry of vector subbundles in a vector bundle, Tensor N.S., vol. 53, 1993, 126–139. [154] Miron, R., Hamilton geometry, Univ. Timisoara Sem. Mecanica 3(1987). [155] Miron, R., Noether Theorems in Higher–Order Lagrangian Mechanics, Int. Journ. of Theoretical Phys. 34, 7(1995), 1123–1146. [156] Miron, R., Spaces with Higher Order Metric Structures, Tensor N.S., 53(1993), 1–23. [157] Miron, R., Techniques of Finsler geometry in the theory of vector bundles, Acta Sci. Math. 49 (1985), 119–129. [158] Miron, R., The higher–order Lagrange spaces: theory of subspaces, The Proc. of Workshop in Diff. Geom., dec. 1995, “Aristoteles” Univ. Thessaloniki. [159] Miron, R., Metrical Finsler structures and metrical Finsler connections, J. Math. Kyoto Univ. 23 (1983), pp. 219–224. [160] Miron, R., A Lagrangian theory of relativity, I, II, An. Şt. Univ. “Al. I. Cuza” Iaşi, 32(1986), 37–62, 7–16.

BIBLIOGRAPHY

259

[161] Miron, R., The Geometry of Higher Order Lagrange Spaces. Applications to Mechanics and Physics, Kluwer Acad. Publ. FTPH no. 82, 1997. [162] Miron, R., The Geometry of Higher-Order Finsler Spaces, Handronic Press, Inc., USA, 1998. [163] Miron, R., The Geometry of Higher-Order Hamilton Spaces. Applications to Hamiltonian Mechanics, Kluwer Acad. Publ. FTPH, no. 132, 2003. [164] Miron, R., Anastasiei, M., The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Acad. Publ., FTPH, no.59, 1994. [165] Miron, R., Anastasiei, M., Vector Bundles and Lagrange Spaces with Applications to Relativity, Geometry Balkan Press, Bucureşti, no.1, 1997. [166] Miron, R., Anastasiei, M., Bucătaru, I., The Geometry of Lagrange Spaces in Handbook of Finsler Geometry edited by Antonelli, P.L., Kluwer, 2003. [167] Miron, R., Atanasiu, Gh., Compendium on the higher-order Lagrange spaces: The geometry of k−osculator bundles. Prolongation of the Riemannian, Finslerian and Lagrangian structures. Lagrange spaces, Tensor N.S. 53 (1993), 39-57. [168] Miron, R., Atanasiu, Gh., Differential geometry of the k−osculator bundle, Rev. Roumaine Math. Pures et Appl. 41, 3/4 (1996), 205-236. [169] Miron, R., Atanasiu, Gh., Geometrical theory of gravitational and electromagnetic fields in Higher Order Lagrange Spaces, Tsukuba J. of Math. 20, 1 (1996), 137–149. [170] Miron, R., Atanasiu, Gh., Higher-order Lagrange spaces, Rev. Roumaine Math. Pures et Appl., 41, 3/4 (1996), 251–262. [171] Miron, R., Atanasiu, Gh., Lagrange geometry of second order, Math. Comput Modelling, 20, 4 (1994), 41–56. [172] Miron, R., Atanasiu, Gh., Prolongations of the Riemannian, Finslerian and Lagrangian structures, Rev. Roumaine Math. Pures et Appl., 41, 3/4 (1996), 237–249.

260

BIBLIOGRAPHY

[173] Miron, R., Anastasiei, M., Ianus, S., The Geometry of the dual of a Vector Bundle, Publ. de l’Inst. Math., 46, 60, 145–162 (1989). [174] Miron, R., Hrimiuc, D., Shimada, H., Sabău, S.V., The Geometry of Hamilton and Lagrange Spaces, Kluwer Academic Publ., FTPH nr. 118, 2001. [175] Miron, R., Dynamical Sytems of Lagrangian and Hamiltonian Mechanical Systems, Advance Studies in Pure Mathematics 48, p.309–340, (2007). [176] Miron, R., Kawaguchi, T., Relativistic Geometrical Optics, Int. Journal of Theoretical Physics, 30, p.1521–1543, (1991). [177] Miron, R., Kawaguchi, T., Higher–order relativistic geometrical Optics, Tensor N.S., vol. 53, 1993. [178] Miron, R., Kirkowits, M., A Geometrical Model for Variational Problem of Multiple Integrals, Proc. Int. Conf. Diff. Geom. Appl. Dubrovnik, 1988, 8–25. [179] Miron R., Pop I., Topologie algebrică, Omologie, Omotopie, Spaţii de acoperire, Editura Academiei, Bucuresti, 1974. [180] Miron, R., Roman, M., Sabău, V.S., Shimada, H., Geometrization of the Riemannian nonconservative mechanical systems, J. Balkan Geometry [181] Miron, R., Rosca, R., Anastasiei, M., Buchner, K., New aspects of Lagrangian Relativity, Found. of Physics Letters, 5 (2), 141–171 (1992). [182] Miron, R., Radivoiovici-Tatoiu, M., A Lagrangian Theory of Electromagnetism, Sem. Mecanica, Timisoara, 1988, 1–55. [183] Miron, R., Tavakol, R., Geometry of Space-Time and Generalized Lagrange Spaces, Publicationes Mathematicae, 44 (1–2), 167–174 (1994). [184] Miron, R., Tavakol, R., Balan, V., Roxburg, I., Geometry of Space-Time and Generalized Lagrange Gauge Theory, Publicationes Mathematicae, 42 (3–4), 215–224 (1993). [185] Miron R., Zet G., Post-Newtonian approximation in relativistic optics, Tensor N.S. Vol. 53 (1993) 92–95. [186] Miron R., Zet G., Relativistic optics of nondispersive media, Found. Physics Vol. 25, No. 9 (1995) 1371–1382.

BIBLIOGRAPHY

261

[187] Miron, R., Watanabe, S., Ikeda, S., Some connections on tangent bundle and their applications to the general relativity, Tensor N. S., 46, 8–22, (1987). [188] Moisil, G.C., Opera matematica, Editura Academiei Republicii Socialiste Romania, 1976. [189] Morimoto, A., Prolongations of geometric structures, Lect. Notes, Math. Inst. Nagoya Univ. 1969. [190] Motreanu, D., Popa, C., Hamilton-Jacobi equations on infinite-dimensional Riemannian manifolds, Nonlinear Analysis TMA 9 (1985), 739-761. [191] Munoz-Lecanda, M.C., Yaniz-Fernandez, F.J., Dissipative Control of Mechanical Systems. A Geometric Approach, SIAM J. Control Optim., 40(5), p. 1505–1516, (2002). [192] Munteanu, Gh., Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Academic Publisher (2004). [193] Munteanu, G., Lagrangian Gauge Theory of Superior Order, in Proc. Math. Seminar, Univ. Brasov, 1996. [194] Munteanu G., Atanasiu G., On Miron connections in Lagrange second order geometry, Tensor NS 50 (1991), 248–255. [195] Munteanu G., Ikeda S., On the gauge theory of the second order, Tensor NS 56(1995), 166–174. [196] Myller, A., Scrieri matematice, Buc. Ed. Acad. R.P.R, 1959. [197] Neagu, A., Borcea, V., Some considerations about regular mappings of Finsler spaces, Revue Roumaine de Mathematiques Pures et Appliquees, vol. 40, no.2, pp. 195–212, 1995. [198] Niminet, V., Structure equations for distributions on manifolds, Analele Ştiinţifice ale Univ. “Al. I. Cuza” Iasi, Seria Matematica, Tom LIV, fasc. 1, 2008, 51–64. [199] Niminet, V., New geometrical properties of generalized Lagranges spaces of relativistic optics, Tensor N.S., Japan, Vol. 68, No. 1(2007), 66–70. [200] Obadeanu, V., Sisteme Dinamice Diferenţiale, Timişoara, Editura Mirton, (2003).

262

BIBLIOGRAPHY

[201] Okumura, M., On some remarks on special Kawaguchi spaces, Tensor N.S. 11, 1961, 154–160. [202] Onicescu, O., Mecanica, Ed. Tehnica, Bucuresti, 1969. [203] Opris, D., Higher-order Lagrangian and Hamiltonian Mechanics, Univ. Timisoara, Seminarul de Mecanica, no. 26, 1990. [204] Opris, D., Albu, I.D., Geometrical aspects of the covariant dynamics of higher order, Czechoslovak Mathematical Journal, vol. 48 (1998), issue 3, pp. 395– 412. [205] Otsuki, T., Theory of aﬀine connections of the space of tangent directions of a differentiable manifold, I, II, Math. J. Okayama Univ., 7, 1957, 1–74, 95–122. [206] Papaghiuc, N., Semi-invariant products in Sasakian manifolds, An. St. Univ. Al. I. Cuza Iasi, 30 (1984), 75–84. [207] Pham Mau, Q., Induction électromagnetique en rélativité générale, Cahier de Physique no. 96, 1958, 297–307. [208] Pitis, Gh., Foliation and submanifolds of a class of contact manifolds, C.R. Acad. Sci. Paris sér. I math. 310, 1990, 197–202. [209] Postolache, M. [210] Postolache, M. [211] Popovici, I., Anastasiei, M., Sur les bases de géométrie finslerienne, C.R. acad. Sci. Paris 290, S.A., 808–810. [212] Puta, M., Hamiltonian mechanical systems and geometric cuantization, Kluwer Acad. Publ. 260, 1993. [213] Radivoiovici, M.A., Geometria Finsler a fibratelor vectoriale. Aplicatii in relativitate, PhD Thesis, “Al. I. Cuza” University of Iasi, 1990. [214] Raileanu, L., Topological and differential manifolds, Ed. Academiei, Bucuresti. [215] Raileanu, L., Miron, R., Curs de geometrie diferentiala, vol. I, II, Ed. Univ. “Al. I. Cuza”, Iasi, 1989.

BIBLIOGRAPHY

263

[216] Rosca, R., Espace-temps possédant la propriété géodésique, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 4, pp. 305–308. [217] Rosca, R., On pseudo-Sasakian manifolds, Rend. Mat. (7) 4 (1984), no. 3, pp. 393–407. [218] Rund, H., Adjoint connections on group manifolds and gauge transformations, The mathematical heritage of C. F. Gauss, 621–644, World Sci. Publ., River Edge, NJ, 1991. [219] Patriciu, A.-M., Suprafete in spatii Minkowski, PhD Thesis, “Al. I. Cuza” University, Iasi, 2010. [220] Puta, M., Hamiltonian Mechanical Systems and Geometric Quantization, Kluwer Acad. Publishers 260, 1993. [221] Roman, M., Rheonomic Riemannian Mechanical Systems, Tensor N.S., 66(1), p.44-52, (2005). [222] Rund, H., The Differential Geometry of Finsler Spaces, Grundlehr. Math Wiss., Springer Verlag, 101, 1959. [223] Sabau, S., Shimada, H., Finsler Geometry, Sapporo in Memory of Makoto Matsumoto Mathematical Society Of Japan (Japan), 2007. [224] Sakaguchi, T., Subspaces in Lagrange spaces, PhD Thesis, “Al. I. Cuza” University Iasi, 1987. [225] Santilli, R. M., Foundations of theoretical mechanics. I. The inverse problem in Newtonian mechanics, Texts and Monographs in Physics. Springer-Verlag New York, New York-Heidelberg, 1978, xix+266 pp. [226] Santilli, R. M., Foundations of theoretical mechanics. II. Birkhoﬀian generalization of Hamiltonian mechanics, Texts and Monographs in Physics. Springer-Verlag, New York-Berlin, 1983, xix+370 pp. [227] Santilli, R. M., Isotopies of differential calculus and its application to mechanics and geometries, Proceedings of the Workshop on Global Analysis, Differential Geometry and Lie Algebras (Thessaloniki, 1994), 61–104, BSG Proc., 2, Geom. Balkan Press, Bucharest, 1998.

264

BIBLIOGRAPHY

[228] Sardanashvily, G., Zakharov, O., Gauge gravitation theory, World Scientific Publishing Co., Inc., River Edge, NJ, 1992, viii+122 pp. [229] Sasaki, S., On almost contact manifolds, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) pp. 128–136. [230] Saunders, D. J., The geometry of jet bundles, London Mathematical Society Lecture Note Series, 142, Cambridge University Press, Cambridge, 1989, viii+293 pp. [231] Shen, Z., On a connection in Finsler geometry, Houston J. Math. 20 (1994), no. 4, 591–602. [232] Smaranda, D., Immersion d’une modele d’universe avec champ magnetique dans une espace pseudo–euclidean, Bull. Soc. Sci. Liège, 1967, 23–30. [233] Sofonea, L., Representative Geometries and Physical Theories, Ed. Dacia, Cluj–Napoca, Romania, 1984. [234] Soós, G., Uber Gruppen von Aﬀinitäten und Bewegungen in Finslerschen Räumen, Acta Math. Acad. Sci. Hungar. 5, (1954), 73–84. [235] Souriau, J.-M., Structure des systèmes dynamiques, Maîtrises de mathématiques Dunod, Paris 1970 xxxii+414 pp. [236] Stavre, P., The d−linear connections, Proc. Nat. Sem. on Finsler and Lagrange Spaces, Braşov, Romania, 1988, 375–382. [237] Stavrinos, P.C., Balan, V., Prezas, N., Manouselis, P., Spinor Bundle of Order Two on the Internal Deformed System, An. Şt. Univ. “Al. I. Cuza” Iaşi, 1996. [238] Stoica, E., A geometrical characterization of normal Finsler connections, An. Şt. Univ. “Al. I. Cuza” Iaşi, XXX, mat., 1984-1. [239] Szabó, Z.I., Positive definite Berwald Spaces, Tensor, N.S., vol. 35 (1981). [240] Szenthe, J., On a basic property of Lagrangians, Publ. Math. Debrecen, 42/3-4, 1993, 247–251. [241] Szilagy, J., A Setting for Spray and Finsler Geometry, in: Antonelli, P.L. (ed.), Handbook of Finsler Geometry, KluwerAcademic Publishers, p.11831437, (2003).

BIBLIOGRAPHY

265

[242] Szilagy, J., Kozma, L., Remarks on Finsler-type connections, PRoc. Nat. Sem. on Finsler and Lagrange spaces, Braşov, Romania, 1984, 181–197. [243] Synge, J. L., Relativity: General Theory, North-Holland, 1966. [244] Synge, J.L., Some intrinsic and derived vectors in a Kawaguchi space, Amer. Jour. of Mathematics, vol. LVII, 1935, 679–691. [245] Şandru, O., Hamilton–Lagrange structures associated to some systems of partial differential equations, PhD Thesis, “Al. I. Cuza” Univ. of Iasi, 1993. [246] Takano, Y., On the theory of fields in Finsler spaces, Proc. Int. Symp. on Relativity and Unified Field Theory, 1975-76, 17-26. [247] Tamassy, L., Aﬀine connections inducing a Finsler connection, Coll. Math. Soc. Janos Bolyai 46, North-Holland, 1988, 1185–1193. [248] Tamassy, L., Matsumoto, M., Direct method to characterize conformally Minkowski Finsler space, Tensor N.S. 33, 1979, 380–384. [249] Tamia-Dimopoulou, P., A relationship between CR-structures f −structures satisfying f 3 + f = 0, Tensor N.S. 49, 1990, 250–252.

and

[250] Teodorescu, N., Géométrie finslerienne et propagation des ondes, Bull. Sci. Acad. Roumanie, 23, 1942, 138–144. [251] Vacaru, S., Interactions, Strings and Isotopies in Higher Order Anisotropic Superspaces, Handronic Press, Palm Harbor, USA, 1998. [252] Vacaru, S., Generalized Lagrange transformation: Finsler geometry methods and deformation quantization of gravity, An. Şt. Univ. “Al. I. Cuza” Iaşi, (S.N.) Matematica, Tom LIII, 2007, supliment. [253] Vaisman, I., Lectures on the Geometry of Poisson manifolds, Birkhäuser, Basel, 1994. [254] Vaisman, I., The Isotropic Hamiltonian formalism, “Alexandru Myller” Mathematical Seminar, Proceedings of the Centennial Conference (V. Barbu and O. Carja, eds.), AIP Conf. Proceedings 1329, 2011, 264-280. [255] Vujici, V.A., Hedrih (Stevanovic), K., The Rheonomic constraints change force, Facta Universitatis, 1, p.313-322, (1991).

266

BIBLIOGRAPHY

[256] Yano, K., Ishihara, S., Tangent and Cotangent Bundles. Differential Geometry, M. Dekker Inc., New York, 1973.