Symplectic Geometry [2 ed.] 9782881249013, 2881249019

Table of contents :
Contents
Preface
Notation
Chapter 1. Symplectic geometry in Euclidean space
1.1 Some information from matrix group theory
1.1.1 Lie groups and algebras
1.1.2 The complete linear groups GL(n, R) and GL(n, C) and their Lie algebras
1.1.3 The special linear groups SL(n, R) and SL(n, C)
1.1.4 The orthogonal group O(n) and the special orthogonal group SO(n)
1.1.5 The unitary group U(n) and the special unitary group SU(n)
1.1.6 Connected components of matrix groups
1.1.7 The realification operation and complex structures
1.2 Groups of symplectic transformations of a linear space
1.2.1 Symplectic linear transformations
1.2.2 The noncompact groups Sp(n, R) and Sp(n, C)
1.2.3 The compact group Sp(n)
1.2.4 The relation between symplectic groups and other matrix groups
1.3 Lagrangian manifolds
1.3.1 Real Lagrangian manifolds in a symplectic linear space
1.3.2 Complex Lagrangian Grassmann manifolds
1.3.3 Real Lagrangian Grassmann manifolds
Chapter 2. Symplectic geometry on smooth manifolds
2.1 Local structure of symplectic manifolds
2.1.1 Local symplectic coordinates
2.1.2 Hamiltonian vector fields
2.1.3 The Poisson bracket
2.1.4 Darboux' theorem
2.2 Embeddings of symplectic manifolds
2.2.1 Embeddings of symplectic manifolds in R^{2N}
2.2.2 Embeddings of symplectic manifolds in CP^N
2.2.3 Examples of symplectic manifolds
Chapter 3. Hamiltonian systems with symmetries on symplectic manifolds
3.1 Liouville's theorem
3.1.1 Integrals of Hamiltonian systems
3.1.2 Complete involutive sets of functions
3.2 Hamiltonian systems with noncommutative symmetries
3.2.1 Finite-dimensional Lie subalgebras in a space of functions on a symplectic manifold
3.2.2 A theorem on integration of systems with noncommutative symmetries
3.2.3 Connections between systems with commutative and noncommutative symmetries
3.2.4 Noncommutative integration in those cases when the sets of integrals do not form an algebra
3.2.5 Integration in quadratures of systems with noncommutative integrals
3.2.6 The canonical form of the Poisson bracket in a neighbourhood of a singular point. The case of degenerate Poisson brackets
3.2.7 Noncommutative integrability and its connection with canonical submanifolds and isotropic tori
3.2.8 Solvable Lie algebras of functions on symplectic manifolds and integration of mechanical systems corresponding to them
3.3 Dynamical systems generated by sectional operators
3.3.1 General plan of construction of sectional operators
3.3.2 Construction of a many-parameter family of exterior 2-forms on orbits of stationary groups of symmetric spaces
Chapter 4. Geodesic flows on two~dimensional Riemann surfaces
4.1 Completely integrable geodesic flows on a sphere and a torus
4.1.1 Geodesic flow of a two-dimensional Riemannian metric
4.1.2 A necessary and sufficient condition for the existence of an additional polynomial integral quadratic in the momenta
4.1.3 Description of Riemannian metrics on a sphere and a torus that admit an additional integral
4.1.4 Geometric properties of metrics on a sphere that admits an additional integral
4.2 Nonintegrability of analytic geodesic flows on surfaces of genus g>1
4.3 Nonintegrability of the problem of n centres for n>2
4.4 Morse-type theory of integrable Hamiltonian systems. Connections between integrability of systems, existence of stable periodic solutions and the one-dimensional homology group of surfaces of constant energy
Chapter 5. Effective methods of constructing completely integrable systems on Lie algebras. Dynamics of multi-dimensional rigid body
5.1 Left-invariant Hamiltonian systems on Lie groups and the Euler equations on Lie algebras
5.1.1 Symplectic structure and left-invariant Hamiltonians
5.1.2 Quadratic Hamiltonians associated with the displacement of the argument on Lie algebras
5.1.3 Properties of the general Euler equations
5.2 A brief summary of classical results on the root decomposition of complex semisimple Lie algebras
5.3 Analogs of multidimensional rigid body motion for semisimple Lie algebras
5.3.1 The sectional decomposition of an algebra coincides with Cartan's decomposition
5.3.2 Various types of sectional operators. Complex metrics. Normal nilpotent metrics. Normal solvable metrics
5.3.3 Compact series of metrics
5.3.4 Normal series of metrics
5.4 Construction of integrals of the Euler equations corresponding to complex, compact and normal dynamics of multi-dimensional rigid body
5.4.1 Integrals of a complex left-invariant metrics
5.4.2 Integrals of a compact left-invariant metrics
5.4.3 Integrals of a normal left-invariant metrics
5.4.4 Involutoriness of integrals
5.5 Complete integrability of the Euler equations for "symmetrical" multi-dimensional rigid body
5.5.1 Complex integrable cases
5.5.2 Compact integrable cases
5.5.3 Normal integrable cases
5.5.4 Integrability of the Euler equations on singular orbits
5.6 Quadratic integrals of the Euler equations
5.7 Integrability of geodesic flows of left-invariant metrics of the form \varphi_{abD} on semisimple groups and geodesic flows on symmetric spaces
5.7.1 Geodesic flow on T*G
5.7.2 G-invariant geodesic flows on T*(G/H)
5.7.3 Geodesic flows of general form on symmetric surfaces
Chapter 6. A brief review of the theory of topological classification of integrable nondegenerate Hamiltonian equations with two degrees of freedom
6.1 Formulation of the problem
6.1.1 Example: classical Hamiltonian equations of the motion of a rigid body
6.1.2 Integrability or nonintegrability as a manifestation of symmetry or randomness in system evolution
6.1.3 Examples of physical and mechanical systems integrable in the Liouville sense
6.1.4 Classification of all integrable nondegenerate Hamiltonian systems (integrable Hamiltonians) with two degrees of freedom
6.2 Smooth functions typical on smooth manifolds
6.2.1 Morse simple functions
6.2.2 Simple atoms and simple molecules
6.2.3 Complex Morse functions
6.2.4 Complex atoms and complex molecules
6.3 Bott's functions as "typical" int~grals of integrable systems
6.3.1 Bott's functions
6.3.2 Integrals which are "typical" in the Hamiltonian physics
6.4 Rough and fine topological equivalence of integrable systems
6.5 Theorem of rough and fine classification of integrable Hamiltonian systems with two degrees of freedom. Applications in physics and mechanics
6.5.1 Formulation of the main theorem
6.5.2 Relation between invariants W, W* and the topology of an integrable system. Substantial interpretation of atoms and molecules
6.6 Method of computing topological invariants for specific physical integrable Hamiltonians
6.7 A brief historical commentary
6.8 Class (H) of isoenergy three-dimensional integrable manifolds. "Five faces" of this class
6.8.1 Class (H) of the isoenergy 3-surfaces
6.8.2 Class (Q) of three-dimensional manifolds glued from two types of blocks
6.8.3 Class (W) of Waldhausen manifolds (graph-manifolds)
6.8.4 The class (S)
6.8.5 The class (T) of isointegrable manifolds corresponding to Hamiltonians with tame integrals
6.8.6 The class (R) of manifolds glued from round handles
6.8.7 Theorem on the coincidence of five classes
6.9 Application of the topological classification theory of integrable systems to geodesic flows on a 2-sphere and 2-torus
6.9.1 Hypothesis on geodesic flows
6.9.2 Integrable geodesic flows on a 2-sphere and a 2-torus
6.9.3 Complexity of integrable geodesic flows on a 2-sphere and a 2-torus
6.9.4 Hypothesis: linearly-quadratically integrable metrics "approximate" any nondegenerate integrable Riemannian metric on a 2-torus
6.10 Topological classification of classical cases of integrability in the dynamics of a heavy rigid body
References
1-18
19-36
37-52
53-68
69-88
89-107
108-125
126-138
139-151
152-166
Subject Index

Citation preview

SYMPLECTIC GEOMETRY

Advanced Studies in Contemporary Mathematics A series of books and monographs edited by R.V. Gamkrelidze, V.A. Steklov Institute of Mathematics, Russian Academy of Sciences; Moscow, Russia

Volume 1 Geometry of Jet Spaces and Nonlinear Partial Differential Equations I.S. Krasil'sh~hik, V.V. Lychagin, and A.M. Vinogradov Volume 2 Integrable Systems on Lie Algebras and Symmetric Spaces A.T. Fomenko and V.V. Trofimov Volume 3 Lagrange and Legendre Characteristic Classes V.A. Vassilyev Volume 4 Some Classes of Partial Differential Equations A.V. Bitsadze Volume 5 Symplectic Geometry A.T. Fomenko Volume 6 Mathematical Foundations of Classical Statistical Mechanics D.Ya. Petrina, V.I. Gerasimenko, and P.V. Malyshev Volume 7 Representation of Lie Groups and Related Topics A.M. Vershik and D~P. Zhelobenko This book is part of a series. The publishers will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details.

SYMPLECTIC GEOMETRY (second edition) A.T. Fomenko

Chair of Differential Geometry and Applications Faculty of Mechanics and Mathematics Moscow State University Moscow, Russia Translated from the Russian by R.S. Wadhwa

GORDON AND BREACH PUBLISHERS Australia • Austria • China • France • Germany • India • Japan Luxembourg • Malaysia • Netherlands • Russia • Singapore Switzerland • Thailand • United Kingdom • United States

© 1995 by OPA (Overseas Publishers Association) Amsterdam B.V. Published under licence by Gordon and Breach Science Publishers SA

All rights reserved. 3 Boulevard Royal L-2449 Luxembourg First published 1988 Second edition published 1995 Originally published in Russian in 1988 as CJ.1MrmeKTJ.1qecKa51 reoMeTpJ.151 by Nauka Publishers, Moscow © 1988 by Nauka Publishers, Moscow British Library Cataloguing in Publication Data Fomenko, A.T. Symplectic Geometry. - 2Rev.ed. (Advanced Studies in Contemporary Mathematics, ISSN 0884-0016; Vol. 5) I. Title II. Wadhwa, R.S. UL Series 516.36 ISBN 2-88124-901-9

No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by· any information storage or retrieval system, without permission in writing from the publisher.

CONTENTS Preface

xi

Notation

xii

Chapter 1. Symplectic geometry in Euclidean space 1.1

Some information from matrix group theory 1. 1.1 Lie groups and algebras 1.1.2 The complete linear groups GL(n, IR) and GL(n, C) and their Lie algebras 1.1.3 The special linear groups SL(n, IR) and SL(n, C) 1.1.4 The orthogonal group O(n) and the special orthogonal group SO(n) 1.1.5 The unitary group U(n) and the special unitary group SU(n) 1.1.6 Connected components of matrix groups 1.1.7 The realification operation and complex structures 1.2 Groups of symplectic transformations of a linear space 1.2.1 Symplectic linear transformations 1.2.2 The noncompact groups Sp(n, IR} and Sp(n; C) 1.2.3 The compact group Sp(n) 1.2.4 The relation between symplectic groups and other matrix groups 1.3 Lagrangian manifolds 1.3.1 Real Lagrangian manifolds in a symplectic linear space 1.3.2 Complex Lagrangian Grassmann manifolds 1.3.3 Real Lagrangian Grassmann manifolds

I I 3 4 6 10 13 16 24 24 29 39 46 51 51 59 68

Chapter 2. Symplectic geometry on smooth manifolds 2.1

Local 2.1.1 2.1.2 2.1.3 2.1.4

structure of symplectic manifolds Local symplectic coordinates Hamiltonian vector fields The Poisson bracket Darboux' theorem

74 74 77 81 87

vi

2.2

CONTENTS

Embeddings of symplectic manifolds 2.2.1 Embeddings of symplectic manifolds in IR2N 2.2.2 Embeddings of symplectic manifolds in CPN 2.2.3 Examples of symplectic manifolds

91 91 93 94

Chapter 3. Hamiltonian systems with symmetries on symplectic manifolds Liouville's theorem 3.1.1 Integrals of Hamiltonian systems 3.1.2 Complete involutive sets of functions 3.2 Hamiltonian systems with noncommutative symmetries 3.2.1 Finite-dimensional Lie subalgebras in a space of , functions on a symplectic manifold 3.2.2 A theorem on integration of systems with noncommutative symmetries 3.2.3 Connections between systems with commutative and noncommutative symmetries 3.2.4 Noncommutative integration in those cases when the sets of integrals do not form an algebra Integration in quadratures of systems with 3.2.5 noncommutative integrals 3.2.6 The canonical form of the Poisson bracket in a neighbourhood of a singular point. The case of degenerate Poisson brackets 3.2.7 Noncommutative integrability and its connection with canonical submanifolds and isotropic tori 3.2.8 Solvable Lie algebras of functions on symplectic manifolds and integration of mechanical systems corresponding to them 3.3 Dynamical systems generated by sectional operators 3.3.1 General plan of construction of sectional operators 3.3.2 Construction of a many-parameter family of exterior 2-forms on orbits of stationary groups of 3.1

symmetric spaces

99 99 101 107 107 110 114 120 124

131 138

154 160 160

163

CONTENTS

vii

Chapter 4. Geodesic flows on two~dimensional Riemann surfaces 4.1

Completely integrable geodesic flows on a sphere and a torus 4.1.1 Geodesic flow of a two-dimensional Riemannian metric 4.1.2 A necessary and sufficient condition for the existence of an additional polynomial integral quadratic in the momenta 4.1.3 Description of Riemannian metrics on a sphere and a torus that admit an additional integral 4.1.4 Geometric properties of metrics on a sphere that admits an additional integral 4.2 Nonintegrability of analytic geodesic flows on surfaces of genus g> 1 4.3 Nonintegrability of the problem of n centres for n > 2 4.4 Morse-type theory of integrable Hamiltonian systems. Connections between integrability of systems, existence of stable periodic solutions and the one-dimensional homology group of surfaces of constant energy

175 175

176 182 187 191 194

199

Chapter 5. Effective methods of constructing completely integrable systems on Lie algebras. Dynamics of multi-dimensional rigid body 5.1

Left-invariant Hamiltonian systems on Lie groups and the Euler equations on Lie algebras 5.1.1 Symplectic structure and left-invariant Hamiltonians 5.1.2 Quadratic Hamiltonians associated with the displacement of the argument on Lie algebras 5.1.3 Properties of the general Euler equations 5.2 A brief summary of classical results on the root decomposition of complex semisimple Lie algebras 5.3 Analogs of multidimensional rigid body motion for semisimple Lie algebras 5.3.1 The sectional decomposition of an algebra coincides with Cartan's decomposition

226 226 235 240 245 251 251

viii

5.4

5.5

5.6 5.7

CONTENTS

5.3.2 Various types of sectional operators. Complex metrics. Normal nilpotent metrics. Normal solvable metrics 5.3.3 Compact series of metrics 5.3.4 Normal series of metrics Construction of integrals of the Euler equations corresponding to complex, compact and normal dynamics of multi-dimensional rigid body 5.4.1 Integrals of a complex left-invariant metrics 5.4.2 Integrals of a compact left-invariant metrics 5.4.3 Integrals of a normal left-invariant metrics 5.4.4 Involutoriness of integrals Complete integrability of the Euler equations for "symmetrical" multi-dimensional rigid body 5.5.1 Complex integrable cases 5.5.2 Compact integrable cases 5.5.3 Normal integrable cases 5.5.4 Integrability of the Euler equations on singular orbits Quadratic integrals of the .Euler equations Integrability of geodesic flows of left-invariant metrics of the form 'PabD on semisimple groups and geodesic flows on symmetric spaces 5.7.1 Geodesic flow on T*ff, 5.7.2 ff,-invariant geodesic flows on T*(CfJ/fJ) 5.7.3 Geodesic flows of general form on symmetric surfaces

256 259 266

271 271 274 282 282 287 287 303 305 334 341

351 351 355 365

Chapter 6. A brief review of the theory of topological classification of integrable nondegenerate Hamiltonian equations with two degrees of freedom 6.1

Formulation of the problem 6.1.1 Example: classical Hamiltonian equations of the motion of a rigid body 6.1.2 Integrability or nonintegrability as a manifestation of symmetry or randomness in system evolution 6.1.3 Examples of physical and mechanical systems integrable in the Liouville sense

368 368 372 373

CONTENTS

6.1.4 Classification of all integrable nondegenerate Hamiltonian systems (integrable Hamiltonians) with two degrees of freedom 6.2 Smooth functions typical on smooth manifolds 6.2.1 Morse simple functions 6.2.2 Simple atoms and simple molecules 6.2.3 Complex Morse functions 6.2.4 Complex atoms and complex molecules 6.3 Bott's functions as "typical" int~grals of integrable systems 6.3.1 Bott's functions 6.3.2 Integrals which are "typical" in the Hamiltonian physics 6.4 Rough and fine topological equivalence of integrable systems 6.5 Theorem of rough and fine classification of integrable Hamiltonian systems with two degrees of freedom. Applications in physics and mechanics 6.5.1 Formulation of the main theorem 6.5.2 Relation between invariants W, W* and the topology of an integrable system. Substantial interpretation of atoms and molecules 6.6 Method of computing topological invariants for specific physical integrable Hamiltonians 6.7 A brief historical commentary 6.8 Class (H) of isoenergy three-dimensional integrable manifolds. "Five faces" of this class 6.8.1 Class (H) of the isoenergy 3-surfaces 6.8.2 Class (Q) of three-dimensional manifolds glued from two types of blocks 6.8.3 Class (W) of Waldhausen manifolds (graph-manifolds) 6.8.4 The class (S) 6.8.5 The class (T) of isointegrable manifolds corresponding to Hamiltonians with tame integrals 6.8.6 The class (R) of manifolds glued from round handles 6.8.7 Theorem on the coincidence of five classes

ix

376 377 377 379 386 387 391 391 393 395

398 398

416 426 431 434 434 435 436 437 437 438 439

X

CONTENTS

6.9

Application of the topological classification theory of integrable systems to geodesic flows on a 2-sphere and 2-torus 6.9.1 Hypothesis on geodesic flows 6.9.2 Integrable geodesic flows on a 2-sphere and a 2-torus 6.9.3 Complexity of integrable geodesic flows on a 2-sphere and a 2-torus 6.9.4 Hypothesis: linearly-quadratically integrable metrics "approximate" any nondegenerate integrable Rieman11ian metric on a 2-torus 6.10 Topological classification of classical cases of integrability in the dynamics of a heavy rigid body

441 441 442 445

447 449

References

454

Subject Index

464

PREFACE TO SECOND EDITION The interest in symplectic geometry and its various applications .has been on the increase in recent years. The second edition of this book contains some new topics of recently developed modern symplectic geometry. In particular, the reader can find here a new theory of rough and fine topological classification for integrable Hamiltonian systems with two degrees of freedom. This is the basis of a new theory for the orbital classification of dynamical systems, which was developed by the author together with A. V. Bolsinov in 1994-95. At present, symplectic geometry has been transformed into a discipline with many. branches. This book has been conceived so as to acquaint the reader with some fundamental aspects of symplectic geometry, and then quickly take him over to the more important and rapidly developing branches. In our opinion, applications of the methods of symplectic geometry to the modern mathematical physics and theoretical mechanics are of special interest. Along with the text constituting a significant part of the book (Part One is essentially a textbook on symplectic geometry), we also present the latest results, which are mainly concentrated in the last chapters. As well as the results obtained by the author and his pupils, we also present results obtained by some of the participant$ in the seminars "Modern Geometric Methods" (conducted by the author) and "Geometry and Mechanics" (conducted by V.V. Kozlov and the author) at the Mechanics and Mathematics Faculty of the Moscow State University. Nearly all the original results in this book were presented and discussed at these seminars. Proofs are given for most of the results included in the "text" part of the book. After this, complete proofs are given only for the results connected with the theory of noncommutative integration of Hamiltonian systems and with theory of complete integration (in Liouville's sense) of Euler equations describing the "most symmetric" multidimensional analogs of the rigid body dynamics on semi:-simple Lie groups. However, we also want to give the reader an idea about many other modern theories that have been developed in recent years in the fields xi

xii

PREFACE

of symplectic geometry and topology. Here, we are obliged to confine ourselves to just a brief enumeration of the main results and demonstration of the most fundamental ideas, omitting the proofs for want of space. Among other things, this applies to the new theory of topological classification of integrable Hamiltonian systems of differential equations with two degrees of freedom having nondegenerate integrals of motion (Chapter 6). However, we have included a fairly extensive list of references at the end of the book so that the interested reader can refer to the journals for details. Most of the sections can be read independently and do not require any special extensive background. At the same time, knowledge of the foundations of a modern course on differential geometry is desirable. References to some textbooks have been included in the bibliography for this purpose. The bibliography, especially the part dealing with more specialized topics, is. by no means exhaustive. The book is intended for undergraduate and graduate students of physics, mechanics and mathematics departments of universities, and also for a wide circle of readers interested in the modern applications of symplectic geometry. A. T. FOMENKO

NOTATION

m G

GL(n, ~) gl(n, ~)

GL(n, C) gl(n, C)

SL(n, ~) sl(n, ~) O(n) SO(n) so(n) U(n) SU(n) su(n)

/=(~ -:) Sp(n, ~) sp(n, ~) Sp(n, C) sp(n, C) Sp(n) sp(n)

Pi•···•Pn,ql•···•qn

Lie group Lie algebra (corresponding to Lie group G>), G* - dual space Euclidean space complex Hermitian space quaternion space Euclidean scalar product skew-symmetric nondegenerate scalar product in symplectic space real complete linear group its Lie algebra complex complete linear group its Lie algebra special real linear group its Lie algebra orthogonal group special orthogonal group its Lie algebra unitary group special unitary group its Lie algebra canonical symplectic structure real non-compact symplectic group its Lie algebra complex non-compact symplectic group its Lie algebra real compact symplectic group its Lie algebra canonical symplectic coordinates in ~Zn

xiii

xiv

NOTATION

canonical symplectic structure in IR2n i=l

LG!= U(n)/O(n)

real Lagrangian Grassmann manifold

LG~= Sp(n)/U(n)

complex Lagrangian Grassmann manifold symplectic structure on symplectic manifold

co= I,cou(x)dx; Adxi i:

U(n)--+ S0(2n)

that arises under the realification

SOME INFORMATION FROM MATRIX GROUP THEORY

21

can be written in the form

g=C+iB ~ _!__,_ (BC

-B) C

eS0(2n),

where the matrices C and Bare real and of order n. In addition,

cpU(n) = S0(2n) n cpGL(n, C). Proof Let e1 , ••• , en be a Hermitian basis in en. Under realification, it goes into the orthonormal basis e 1 , ... , en, ie 1 , •.. , ien. Hence, ,

arid

Consequently, we obtain cpg = (~

-B) C .

Direct calculation shows that det 0. We now prove. that cpU(n) = S0(2n) n cpGL(n, C). Suppose that cpgecpU(n). Then, on the one hand, cpg e S0(2n), while on the other hand the operator cpg is obtained by realification of a complex nonsingular operator, that is, cpU(n) c S0(2n) n cpGL(n, C).

22

SYMPLECTIC GEOMETRY IN EUCLIDEAN SPACE

Conversely, suppose that and

gES0(2n)

gEa:H--+ V,

Also,

Proof According to the definition of (!>a, the intersection of (l>.H and II consists of all elements of the form [h, a] that commute with a, where h EH. We prove that the planes cl>.H and T are orthogonal. Let us consider the scalar product ( (!>. h, t), where h EH and t E T. Then ((f).h, t) = ([h,a], t) = (h, [a, t]) = 0, since t ET, that is,

[a,t]=O. We have also used the following property of the operation ad:

(adx Y,Z) = -(Y,adxZ). In other words, .the operators adx are skew symmetric with respect to the Killing form. Thus, T ..l (f)aH and T ..l ((f)aH n II). Consequently,

This proves the lemma.

HAMILTONIAN SYSTEMS

168

In particular, the notation B that we have used for the plane orthogonal to Tin Tl is in complete accord with the notation of §3.3.1, where the construction of sectional operators is described. Since we are considering the case of a symmetric operator M = G'>/f,, the decomposition of the algebra G = H + V has important properties. It is easy to see that

[Ii,H]cH,

[H, V] CV,

[V, V] cH.

Let T' be the orthogonal complement of T in V, and let K' be the orthonormal complement of Kin H. LEMMA 3.9

For an element a E T of the genera/position the mapping

where .h= [h,a], establishes a linear isomorphism between the planes K' and T'. Also,

Proof We note that the planes Hand V are orthogonal in G. Since

we have

By Lemma 3.8 the plane .H = .K' is orthogonal to T m V. Consequently,

The mapping . is a monomorphism on the plane K', since Kn K' = Ker. nK' = 0.

Let us consider the action of the stationary subgroup f, on the plane V. Obviously, from each point t ET there originates an orbit of this action.

STRUCTURES GENERATED BY SECTIONAL OPERATORS

169

Let us consider the orbits of general position O(t) originating from points of general position in T. Then dim V = dim T

+ dim O(t).

Obviously, dim O(t) = dim T'. At the same time, the tangent plane to the orbit O(t) at the point t E T is naturally identified.with the quotient space

H/Kera = H/K ~ K'. Consequently, dim K' = dim O(t) = dim T'. But since the mapping a monomorphically embeds K' in T', we have

as required. LEMMA

3.10

The following relations are satisfied in the space V:

(a) V= lmpb + Ker pb, where

Im pb n Ker pb = O;

and

in particular, R = R' in the notation of §3.3.1, and R

+ B = T'.

Proof The operator pb, being restricted to the plane V, is skewsymmetric with respect to the Killing form. We can therefore write it as a skew-symmetric matrix. For a suitable choice of basis it assumes a canonical block-diagonal form and the corresponding decomposition of Vin to a sum of planes on one of which the operatorpb is trivial, while on the other it reduces to rotations in two-dimensional planes, coincides with the decomposition into the sum of the image and the kernel of the

HAMILTONIAN SYSTEMS

170

operator pb. Furthermore, the image of pb is orthogonal to the kernel of pb, and since Kerpb = II, we have Impb_i_II

and

Impb_l_ T,

since Tc II. Consequently, Im pb c T' = aK', that is, Since R'

= aK' nimpb,

we have R'=Impb.

Hence, aK' == Im pb + Ker pb n aK' = Im pb + II n aK' = Im pb + B.

Thus, Since R'=Impb·,

we have R = R', that is,

This proves the lemma. Thus, in our case the canonical sectional operator C: V--+ H takes the form C: T

+ B + R--+ K + B + R.,

where aET,

K = K(a) = Ann(a) c H,

II= B + T = Ker pb,

bEK,

R=Impb,

R. = ; 1 (pb)R = ; 1 R';

STRUCTURES GENERATED BY SECTIONAL OPERATORS

171

and D

C= ( 0

0

0

; 1 0

The first statements of Theorem 3.12 are thus completely proved. It remains to determine the exterior 2-form F. Let C be a sectional operator. We consider an arbitrary point XE V, and let ~ and 17 be arbitrary vectors of V, applied at the point X. Then

and CXEH. Thus, we have defined the scalar product and the involution u: ffi--+ ffi is defined as u(x, y) = (y, x). The corresponding decomposition in the Lie algebra H form V= (X, -X),

XEH,

H

+H

has the

= (X,X),

(the same letter has been used for H and its realization in G = H 0, kE"lL; (d) the functions f and hare positive everywhere except at the points ka and kb respectively, and are at least twice continuously differentiable. We note that the vanishing of the expression f(u) + h(v) at the umbilical points just expresses the fact that the Liouville system of coordinates degenerates at these points. Conversely, if f and h satisfy the conditions mentioned above, then after gluing the infinite cross, as described above, we obtain a smooth Riemannian metric on the sphere. Transition to the global system of coordinates is accomplished by the Weierstrass p-function. In future, we shall assume that the conditions (a)-(d) are satisfied. Thus, we have arrived at the problem of describing the behaviour of geodesics in a Lagrangian system with Lagrangian L

= ½U(u) + h(v))(u 2 + v2 ),

to which, after Legendre transformation, there corresponds a Hamiltonian system with Hamiltonian

H =~

p; +p; .

2 f(u)

+ h(v)

The straight lines u = const and v = const in the infinite cross and the curves corresponding to them on the sphere will be called horizontal and

190

GEODESIC FLOWS ON TWO-DIMENSIONAL RIEMANN SURFACES

vertical parallels respectively. Let us now study the qualitative behaviour of geodesics. We use the method of separation of variables to write down the equations of the geodesics in the variables u and v. A motion with unit velocity is realized for H = ½. The Hamilton-Jacobi equation has the form

(ai:;-as) + (as) av" 2

f(u)

2

=const=c 2 =2H.

+ h(v)

Now putting c1 =

(8S)

2

ou

- f(u)c 2 = h(v)c 2

-

(8S)

2

ov

,

we obtain the following two equations, which give a geodesic in the coordinates u, v:

~ (f dt

-d

dt

du ±JJ(u)c2

+ c1

-I· I

dv ) ±Jh(v)c2 -c 1

(f--====+ f(u) du ±JJ(v)c2 +c 1

h(v) dv

=0 '

)

- , = = = = =1

±Jh(v)c 2 +c 1

·

Here, the plus and minus signs are determined by the signs of Pu and Pv· Next we put c 2 = 1 everywhere (this corresponds to a unit velocity), and c 1 = c. Now from the formulae

p; = h(v)-

p; = f(u) +c,

C

we see that for c > 0 the motion takes place in the band h(v) ~ c, while for c < 0 the motion takes place in the band f(u) ~ On the other hand, geodesics passing through the umbilical points have the value c = 0. Let us now investigate geodesics corresponding to values c #- 0. It is easy to see that the equations of such a geodesic passing through the point (u 0 , v0 ) can be written as follows:

-c.

f"

d~ Ju0 ±JJ(~)+c

-f" . v0

d~ =0· ±Jh(~)-c '

NONINTEGRABILITY OF ANALYTIC GEODESIC FLOWS

f

u

Uo

f(() d(

------,c===+

± jf(() + C

191

IV ------,c====t h(() d( Vo

± jh(() -

C

.

Writing the closure equations for geodesics corresponding to values c < 0 (and replacing c by -c), we obtain the following basic system of equations, where fo = f(u 0 ) = max f:

I f

vz(c)

v 1 (c)

uz(c)

ui(c)

(a

dx jh(x) -

C

= Jo Jf(x) + c'

(b

dx jf(x) -

dx

C -

E(O, h0 ), (4.8)

dx

Jo jh(x) +

u;(c)j(u;) = c;

c

c E(0,f0 ), C '

v;(c):h(v;) = c.

Theorems 4.3 and 4.4 were proved by V. N. Kolokoltsov. 4.3 A necessary and sufficient condition for the closure of all geodesics of a geodesic flow on a sphere having an additional first integral quadratic in the momenta is that the system of equations (4.8) is satisfied for the functions defining it in the natural system of coordinates described above. All single geodesics are found to be simple, that is, nonselfintersecting curves. THEOREM

4.4 Let a Riemannian metric be given on a sphere S 2 , such that the geodesic flow has an additional integral quadratic in the momenta. The following conditions are equivalent: (a) all geodesics are closed; (b) the corresponding Riemarznian manifold is an SC-manifold, that is, all single geodesics are simple closed curves of the same length; (c) the functions defining the metric in a special system of coordinates on the sphere satisfy the system of equations (4.8). · THEOREM

4.2. Nonintegrability of analytic geodesic flows on surfaces of genus g> 1 In this section we shall briefly describe the results obtained by V. V. Kozlov. We consider a natural mechanical system with two degrees of freedom. We shall assume that its position space M 2 is a compact

192

GEODESIC FLOWS ON TWO-DIMENSIONAL RIEMANN SURFACES

orientable analytic surface. The motion of a natural system is described by the Hamilton equations in the cotangent bundle T* M, which is its phase space. The bundle T* M has the natural structure of a fourdimensional analytic manifold. We shall assume that the Hamiltonian

H: T*M-+ IR is analytic everywhere. Since H

= T(p, q) + U(q)

and T(p, q) is a quadratic form in p ET/ M for all q EM, the functions T(p, q) (kinetic energy) and U(q) (potential energy) are analytic on T* M and M respectively. The solutions of the canonical equations

. oH op

q=-

are analytic mappings from a straight line to T* M. The total energy

H=T+U is constant on the trajectories of the system. THEOREM 4.5 (see (35]) If the genus of the surface M 2 is not O or 1, that is, if the closed two-dimensional surface M 2 is not homeomorphic to a sphere S2 or a torus T 2 , then the canonical equations do not have a first integral that is analytic on T* M and independent of the energy integral.

The analytic functions are assumed to be independent if they are independent at any point (then they are independent almost everywhere). In the infinitely differentiable case Theorem 4.5 is not valid, generally speaking: for any smooth surface M 2 we can indicate a "natural" Hamiltonian H = T + U such that the canonical Hamilton equations on T*M have an additional infinitely differentiable integral that is independent (to be more precise, not dependent everywhere) and has Hamiltonian H. This theorem is a consequence of a stronger assertion, which establishes the nonintegrability of the equations of motion for fixed sufficiently large values of the total energy. The exact formulation of this

NONINTEGRABILITY OF ANALYTIC GEODESIC FLOWS

193

statement is as follows. For all values h > maxM Uthe total energy level Qh

= {x

E T* M : T

+ U = h}

is a three-dimensional analytic manifold having the natural structure of a fibre space with base Mand fibre 8 1 . Local coordinates on Qh are q and max:M U the phase flow on Qh does not have an infinitely differentiable first integral f(q, cp) : Qh

-+

JR

that: (a) has finitely many critical values;

(b) is such that the points q EM for which the sets (f (q, µ if J(H;) = µ(H;) for i = l, 2, ... , k and J(Hk+ i) > µ(Hk + i). We recall that if A andµ are roots, then J(h') and µ(h') are real numbers for any h' E T0 • Thus, a linear ordering arises in the set Li. A root IX E Li is called positive if IX> 0, that is, 1X(H;) = 0 for i = l, 2, ... , k and 1X(Hk + 1 ) > 0. The fact that the root IX is positive means that its first nonzero coordinate is positive. Linear ordering is not uniquely introduced; henceforth we shall assume that the basis H 1 , . . . , H, (r is the rank of G) is fixed. Denoting the set of positive roots by Li+, we can write

where

and there is a one-to-one correspondence between Li+ and Li-, established by the involution IX --+ - IX. Clear! y, if IX E Li+, then - IX E Li - . A positive root IX is called simple if it cannot be represented as the sum of two positive roots. If r

= rank G = dime T,

then there exist exactly r simple roots 1X 1 , ... , IX,, which form a basis in T over IC and a basis in T0 over IQ. Also, every root /3 Eli can be represented in the form

where m; E Z are integers with the same sign. If m;? 0, then /J E Li; if m; ~ 0,. then /3 E Li - . The system of simple roots IX 1 , ..• , IX, is usually denoted by Il. The system Li+ is uniquely reconstructed from the system Il. We put v+=

~ L, a>O

Ga,

v- =

~ L,

Ga,·

at~·Un:rkit'

cqs-e

Fig. 6.42

By varying the energy monotonically (for example, increasing it), we obtain a discrete family of molecules. Depicting them by points on the complexity lattice (each of whose cells corresponds to a certain complexity (m, n)), and connecting them consecutively by segments, we obtain a polygonal trajectory. Moving along it, we can visualize the evolution of the molecule upon an increase in energy. Figure 6.42 shows polygonal trajectories calculated for various physical systems. Evolution curves for. the Kovalevskaya and Goryachev-Chaplygin-Sretenskii integrability cases are presented in the figure. Why do several evolution polygons correspond to the Kovalevskaya case instead of one? As a matter of fact, the initial phase space in the dynamics of a heavy rigid body is a six-dimensional manifold. A reduction to a four-dimensional symplectic manifold is carried out by fixing the values of two classical integrals. Fixing different values, we obtain different manifolds M 4 • Consequently, the Kovalevskaya case, for example, is realized on different symplectic 4manifolds. Each of them corresponds to its own evolution curve. In the Kovalevskaya case, different polygonal lines correspond to different values of so-called area integral, while in the Goryachev-ChaplyginSretenskii case they correspond to different values of hyrostatic moment. Thus, a polygonal line indicates the topological behaviour of a given system on the entire phase space M 4 (if M 4 is fixed). The experimental result described above gives rise to many new problems and hypotheses since we have discovered extremely interesting

416

CLASSIFICATJON OF HAMILTONIAN EQUATIONS

regularities in the distribution of physical systems in the table of all "mathematically feasible" systems. 6.5.2. Relation between invariants W, W* and the topology of an integrable system. Substantial interpretation of atoms and molecules

The main objects, viz., a molecule Wand a marked molecule W*, were introduced above formally. It would be appropriate here to explain their actual meaning from the point of view of an integrable system. The invariant W* is closely connected with the Liouville foliation, this connection being quite simple. This allows us, among other things, to calculate effectively the topological invariants of Hamiltonian systems. Let Q~ be a constant-energy 3-manifold with an integrable system v, i.e., supplied with Liouville's foliation. For the sake of simplicity, we fix an additional integral f (it is immaterial which integral is fixed since the theory does not depend on the choice of the specific form off in view of the nonresonance nature of the system). Let c by the critical value for f, and fc denote the connected component of the complete inverse image 1-1 (c) (Fig. 6.43), containing the singular fiber (layer) of the Liouville foliation. The dimension of fc in this case is either one (when critical circle is minimax), or two (when critical circle is saddle). We shall denote by .t.Q~ the connected component of the total iverse image 1-1 ( c - t:, c + c:), containing f c (Fig. 6.43). The three-dimensional manifold .t.Q~ has a boundary consisting of a few Liouville's tori.

C

Fig. 6.43

CLASSIFICATION

417

It is well known that the Liouville foliation is defined on the manifold AQ~. Its fibers are defined by the equation f = const. This foliation has only one singular fiber (namely, Jc). All the remaining fibers are nonsingular Liouville's tori. As the value of the constant varies, these tori are somehow transformed and rearranged at the moment of intersection of the critical level f = c. However, along with this obvious foliation, there exists on the 3manifold AQ~ one more remarkable foliation which is not so easy to see. But it is this foliation that plays the leading role in the theory of topological classification of integrable systems. The Liouville foliation is two-dimensional, i.e., almost all its fibers (with the exception of isolated minimax circles) are two-dimensional. The fibers of the other foliation are one-dimensional (they are simply circles) and lie on the fibers of the former foliation. the first projection (the mapping of f) projects AQ~ onto the one-dimensional segment (c + E:, c - e:), the fiber of the projection f being two-dimensional. The following important theorem forms the basis of the entire theory of classification. THEOREM 6.9. (Substantial interpretation of atoms in terms of a Hamiltonian system) (see [103]; A.Fomenko). · (a) Each 3-manifold AQ~ is a Seifert foliation with a circular fiber above a certain two-dimensional base which we shall denote by P';. Singular fibers of Seifert foliation can only be of the type (2,1) (it should be recalled that in the general Seifert foliation, singular fibers of any type (p, g) can be encountered) ( see Fig. 6 .44). (b) As the manifold AQ~ is projected on the base P';, the singular fiber f c is projected onto a certain graph Kc. It turns out that the surface P'; with the graph Kc lying on it coincides with a certain atom. Conversely, any atom can be obtained by the method described above.

Fig. 6.44

418

CLASSIFICATION OF HAMILTONIAN EQUATIONS

(c) If an atom has no asterisks as vertices, the Seifert foliation corresponding to it is the direct product of the base and a circle, i.e. Q~ = P; x 8 1 . If, however, an atom has asterisks as vertices, then AQ~ is no longer a direct product and is a nontrivial Seifert foliation. In this case, the asterisk vertices are in one-to-one correspondence with singular fibers of a foliation of the type ( 2 ,1). (d) Each atom ( P;, Kc) exactly describes the pattern of interaction and bifurcation of Liouvi/le's tori in the constant-energy 3-manifold Q~ during their transition through the critical fiber f = c. Events happening to the tori are completely described by the evolution of the atom corresponding to level curves of the Morse function determining the foliation of the atom into circles and a singular curve. In a certain exact sense ( see below), these level curves are the "traces" of the Liouvill tori, which are cut by the latter on a two-dimensional atom during their evolution within the constant-energy 3-surface. (e) Therefore, the classification of atoms is exactly the classification of all possible types of bifurcations of the Liouvi/le tori in the vicinity of a critical value. Ultimately, this gives us the description of all possible bifurcations of the solutions ( their closures) of an integrable system. Denoting the projection of the Seifert foliation by Pc, we can write the statement of the theorem in the form of the following diagram:

AQ~ Pc

l

p2 C

Thus, atoms are the images of 3-manifolds AQ~ foliated in Seifert sense. The definition of Seifert's foliations and the description of their most important properties can be found in [126], [136]. Here we shall confine ourselves only to the following three graphic examples, viz., Seifert's foliations corresponding to three simplest atoms A, B, and A*. Realization of atom A. Figure 6.45 shows the solid torus, viz., the direct product of the disk D 2 by a circle. The solid torus here plays the role of the 3-manifold AQ~. It is foliated into concentric Liouville's tori whose common axis is the circle, viz., the axis of the solid torus. The construction of Seifert's foliation is very simple. Its fibers are parallels on Liouville's tori, viz.,

CLASSIFICATION

419

ii

A= Fig. 6.45

the cycles parallel to the axis of the solid torus. Seifert's foliation has no singular fibers and is obviously a direct product. Projection Pc projects the solid torus onto a disk. Clearly, the projection of the axial circle of the solid torus is the center of the disk. We consider atom A. Such a manifold .6.Q~ always emerges in physical systems when we consider a tubular normal neighbourhood of the minimax critical circle (for the integral f). Realization of atom B. Figure 6.46 shows one of typical 3-manifolds .6.Q~ which are also frequently encountered in real physical systems. This is the direct product of a disk with· two holes and a circle. Figure 6.46 shows the foliation of this 3-manifold into Liouville's tori. One fiber is singular. It is homeomorphism to two tori glued along the parallel. Obviously, the manifold .6.Q~ can be presented in the form of a solid torus from which two thin "parallel" solid tori are removed. The normal section of this manifold by a plane gives a disk with two holes. This manifold .6.Q~ emerges practically in all cases of integrability of a heavy rigid body. Seifert's foliation in this case has a very simple structure: all fibers are parallel to one another and to the axis of the solid torus, viz., the cycle which is the axis of the singular fiber f c- The projection Pc is also shown in Fig. 6.46 as the projection on the tubular neighbourhood of figure of eight embedded into a plane. Obviously, this situation corresponds to the atom B.

420

CLASSIFICATION OF HAMILTONIAN EQUATIONS

''

... - - .. _.

>BII

t Pc

pz

~ C

.B=~ Fig. 6.46

Realization of atom A*. The third example is not less "physical', although it is more sophisticated. It emerges, for example, in the Kovalevskaya integrable case. To begin with, let us consider Klein bottle immersed in a threedimensional Euclidean space as shown in Fig. 6.47. We advise the reader is not familiar with such an immersion to stop here for a few minutes to make sure that he understands indeed its construction. For better understanding, you must take a circle and draw a small figure of eight orthogonal to it at an arbitrary point of the circle. Now move the figure eight along the circle so that its center always slides along the circle and the figure of eight rotates uniformly in a plane orthogonal to the circle. The speed of rotation is chosen so that the figure of eight returns at its initial point having completed an 180° tum. The twodimensional surface (with self-intersection) swept by the figure of eight during its motion is the immersed Klein bottle. Let us now consider a normal tubular neighbourhood of this surface. We obtain 3-manifold shown in Fig. 6.47, It can be described as follows. We must remove from the solid torus a "thin' solid torus passing twice along the axis of the former (large) solid torus. This 3-manifold b.Q~ is foliated into Liouville's tori and has a single singular fiber, viz., an immersed Klein's bottle to which nearest tori are contracted. On the other hand, the manifold b.Q~ is a Seifert foliation

CLASSIFICATION

421

Fig. 6.47

whose fibers are circles. A nonsingular fiber is shown in Fig. 6.48 in the form of a circle passing twice along the axis of the large solid torus. The normal section of the manifold AQ~ by a plane (orthogonal to the axis of the solid torus) is a disk with two holes. A nonsingular circular fiber pierces this disk exactly at two points. If, however, the point of intersection of the circle with the normal section tends to the center of the figure of eight, the nonsingular circular fiber tends to the singular fiber. The singular one-dimensional fiber of Siefert's foliation is a circle along which the center of the figure of eight slided (see above). Clearly, close nonsingular fibers pass along it twice. In the limiting transition, the nonsingular circle is wound twice on the singular circle. What is the structure of the base of this Seifert foliation? It is clear from Fig. 6.48 that each fiber can be put in correspondence with the point of its meeting with half the normal section lying, for example, on the left of the vertical segment mn. This is "nearly always" a one-to-one correspondence. The one-to-one correspondence is violated only for circles meeting the half at points of the segment mn. In order to restore the complete one-to-one correspondence, we must identify the segments sm and sn as shown in Fig. 6.48. As a result, we obtain a ring with a circle having a vertex s lying on its axis. This vertex must be marked by

CLASSIFICATION OF HAMILTONIAN EQUATIONS

422

_... -------- ..........,

,,

"\

.,.,,.,"""

,, ......

~--.

\

I

I

I

I

I I

I

I

I

I

I \

I \

\

, \

\ '\

',, ........

... __ -- -

I

....

__ _. "'

-_,..,,,

/

I

, J ~

~

.,"

-jfq-t-om Fig. 6.48

P;

"asterisk". Thus, we obtained the 2-surface which is the base of the Seifert foliation. We have described in detail the realization of three simplest atoms. The structure of other atoms is more complex. For example, Fig. 6.49 shows the atom F 1 describing the transformation of the Liouville's torus into two tori (and vice versa). The trace cut on the atom by the "lower" Liouville torus (i.e., the one ascending from the domain f < c) is depicted here by a solid circle (marked by number 1). After passing through the critical level, the Liouville torus is transformed into two tori (numbers 2 and 3 in Fig. 6.49). Their traces on the atom are presented in the form of two dashed circles. The bifurcation pattern is completely reflected by the evolution of level lines on a two-dimensional atom.

....-~

,

,, .,.,:., .

......

I

I I I I

\ \

'

...

"' ',~, ~~~ '' ', .. .,.,,

2.--F. 3,,.......,.. 1

Fig. 6.49

_1.

CLASSIFICATION

423

The .more complex the atom, the more muddled the bifurcation pattern for Liouville's tori. This pattern cannot be simplified (in the general case) by a small perturbation of the system. It was noted above that the integrals of a Hamiltonian system generally cannot be "perturbated". Herein lies line significant difference between the present theory and the Morse theory in which we can always "move" the critical points to different levels. If we could do something of this kind in the theory of Hamiltonian systems, we would not require any complex atoms and could do with the three simplest atoms A, B, and A*. However, as was mentioned above, the Hamiltonian function in the Hamiltonian mechanics (as well as the corresponding symplectic structure) is assumed to be defined and fixed (the expressi~n "a Hamiltonian system is defined" implies just this fact). A change (perturbation) of a Hamiltonian involves a transition to another system (and to a different physical problem). Therefore, we cannot "perturbate" the Hamiltonian (and the symplectic structure on the phase space) and have to dwell deeply into a nontrivial classification of bifurcations of an unperturbed Hamiltonian. It is expedient to introduce here the concepts of simple and complex Hamiltonian. DEFINITION 6.U. A Hamiltonian H (or the flow corresponding to it) will be referred to as simple if there is exactly one connected critical manifold on each critical level f = c. All the remaining Hamiltonians will be called complex. The molecule of a simple Hamiltonian (system) clearly may include only the simplest atoms A, B, and A*. If a molecule contains at least one atom differing from them, its means that the Hamiltonian is complex. In spite of the physical impossibility to "perturbate" the Hamiltonian of an integrable system, it would be interesting to answer the following interesting theoretical question: can a small perturbation transform a complex Hamiltonian into a simple one? In other words, is the set of simple Bott Hamiltonians dense everywhere in the set of all Bott Hamiltonians? (It should be recalled that here we assume the Hamiltonians to be of nonresonance type everywhere).

6.1 (T. Nguen [146]). Any complex Bott Hamiltonian can always be approximated as closely as desired ( even through a smooth homotopy) by a simple Bott Hamiltonian in the class of perturbations "involving" both H and w.

PROPOSITION

424

CLASSIFICATION OF HAMILTONIAN EQUATIONS

In this case, we naturally "spoil" the initial integrable Hamiltonian system sgrad H by replacing it by some other (however close) new integrable Hamiltonian system sgrad H. This new system generally will have a different physical meaning. . The interpretation of atoms presented above can be refined. Let us consider the Seifert foliation Pc: LlQ~ - t

P;.

6.2. A two-dimensional base P; can always be embedded into the space of foliation LlQ~ as a section of Seifert's foliation (in general, ambiguously). In other words, there always exists a map i: - t LlQ~ such that the composition Pei is the identical mapping of the base into itself (Fig. 6.50). Therefore, we can arrange within LlQ~ both the atom 2PcR2 and Liouville's tori transversal to it. These tori cut on it level lines whose evolution indicates the bifurcations of the tori themselves within the constant-energy 3-surface. Thus, atoms can be given the following interpretation: an atom is a two-dimensional surface with a boundary which is embedded into LlQ~ transversally to the tori of the Liouville foliation. Considering the we obtain a family of curves, intersection of the tori with the surface viz., foliation on the atom. These curves are actually the level curves of the function (integral) f. Let us now go over to the interpretation of molecules. PROPOSITION

P;

P;

3

Ll

l

QC

t Pc .E2 C

Fig. 6.50

CLASSIFICATION

425

Fig. 6.51

We shall consider all critical levels on the manifold Q~ and all its "pieces" AQ~ (Fig. 6.51). It is quite clear that the entire manifold Q~ can be presented in the form of the union of its pieces, namely,

Q~ = LAQ~,, where the sum (union) is taken over all critical levels. Let AQ~ and AQ~, be two adjacent pieces. Then they are glued (within Qt along some of their boundary tori. Let T2 be any of them. There are obviously many· possible versions of identification (gluing) of the two boundary tori. Consequently, we must indicate the gluing which is realized in the given case (for the given system). If a basis consisting of two cycles is chosen (and fixed) on each of the tori, then all possible diffeomorphisms of one torus on the other are uniquely defined relative to each other (to within isotopy) by the set of integral-valued matrices (

~

~) , where

a8 - (3,

= 1.

Thus, in order to specify the diffeomorphism gluing the two boundary tori (Fig. 6.51), it is sufficient to define the integral-valued matrix. However, the basis is chosen on the tori ambiguously, and hence the matrix determining the gluing is defined ambiguously. Nevertheless, it turns out (we shall not dwell into details and refer the reader to more detailed works contained in the References) that we can always choose unambiguously on each torus "half the basis", namely, one cycle (one

-426

CLASSIFICATION OF HAMILTONIAN EQUATIONS

circle). Consequently, the arbitrariness in the choice of the gluing matrix is reduced (although it does not disappear completely). Further, it was found that the rational number r = (a/(3) mod l is actually defined uniquely. It indicates the "extent of twisting" characterizing the gluing of two boundary tori of the adjacent pieces. The number r can vary between O and I in the case when /3 =f O. If, however, (3 = 0, we put (by definition) r = oo. Thus, we have generally explained the meaning of the rational parameter r which participates in the formal definition of molecule. According to this definition, the number r should be placed on the cylinder T2 x D 1 connecting two adjacent pieces, i.e., two adjacent atoms. In the molecular form of notation, this cylinder is depicted by an .edge connecting two adjacent atoms. Therefore, each edge of a molecule is assigned a rational mark. In the definition of molecule, there appear two more numerical parameters, viz., marks ei and marks nk. We shall omit their description in order to simplify the analysis. Important remark. In spite of the fact. that each three-dimensional piece foliates (in Seifert's sense) over the base P; the entire threedimensional manifold Qf need not necessarily foliate over the 2-surface. The obstacle to such a foliation is of topological nature and is "measured" just by numerical marks introduced above (primarily, by the rational marks ri). Let us sum up the results obtained. Substantial Interpretation of Molecules in Terms of a Hamiltonian System (a) A molecule W* depicts the Liouvillefoliation on a constant-energy 3-manifold. Numerical marks characterize the extent of "twisting" of the foliation between its adjacent singular fibers. (b) Moving along the edges of a molecule, we see in reality the evolution of the Liouville tori within the constant-energy manifold. In this sense, the molecule gives a clear visualization of global topology of the given system ( on the entire Q).

6.6. Method of computing topological invariants for specific physical integrable Hamiltonians Let us now consider the natural question concerning the computation of this invariant W*. It. may turn out to be so complicated that its

PHYSICAL INTEGRABLE HAMILTONIANS

427

computation is not possible in practice. However, the very first calculations revealed that the invariant can be computed easily, although certain efforts are required in each particular case. We shall now describe an informal algorithm which was used to evaluate the invariants of many specific physical integrable Hamiltonians. Step I. Computations can be started right away if the Hamiltonian H and some of its supplementary (additional) integrals f are defined in an explicit form (i.e., by simple analytical formulas) in some suitable coordinates. It should be recalled that f is defined ambiguously, but the final result is independent of the arbitrariness in its choice. Thus, 1et us fix a specific integral f. We write down the image of the moment (momentum mapping) µ : M4

--+

IR2

defined by the formula µ(x) = (H(x), f(x)). In other words, we put each point x in the phase space in correspondence with two numbers, viz., the values of the Hamiltonian and the integral at this point. It can be assumed that two Cartesian coordinates are introduced on the Euclidean space R 2 and can be denoted by the same letters H and f (Fig. 6.52). The image of the manifold M 4 obtained as a result of the mapping µ is a certain closed set (compact if M 4 is compact). The function H is found to be quadratic in many. cases.

h. Fig. 6.52

CLASSIFICATION OF HAMILTONIAN EQUATIONS

428

Step 2. DEFINITION 6.12. Consider the differential dµ of the momentum mapping. A point x 0 in M 4 is called singular ( critical) for the mappingµ if the differential at this point has a rank smaller than two. The image of such a point, i.e. µ(x 0 ), is called the critical value of the momentum mapping. The set of all critical values is denoted by Land is called the bifurcation diagram of the momentum mapping.

This set consists of a certain number of piecewise smooth curves and, perhaps, isolated points (see Fig. 6.52). The boundary of the image of the momentum mapping µ(M4 ) always appears in the bifurcation diagram (all boundary points must be critical values). Figure 6.52 shows the specific bifurcation diagram of the integrable Euler Hamiltonian from the rigid body dynamics. Isolated points in the set L are encountered quite rarely in actual practice. The evaluation of E is quite formal and the only difficulties that are encountered can be associated just with the complexity of the analytic formulas defining H and f. A large portion of computations (in this step) can be algorithmized and carried out on a computer. It should be remembered that the sets µ(M 4 ) and Lare defined ambiguously since they obviously depend on the choice of the second integral f. Step 3. The equation H = h (= const) (see Fig. 6.52) defines on the (H, J) plane a straight line orthogonal to the H axis. It intersects the set µ(M 4 ) along a certain system of segments. Such a segment is denoted by AB in Fig. 6.52. Its complete inverse image (preimage) µ- 1 (AB) obviously coincides with the isoenergy 3-surface Qi. By displacing the straight line H = h to the left and to the right, we obtain a set of all possible isoenergy 3-surfaces of the given integrable system. As this straight line is displaced, it intersects the singularities of the bifurcation diagram. Obviously, Qi and the dynamical system may undergo a topological rearrangement at these moments. For any point y inside µ(M4 ) not lying on the bifurcation diagram, its complete inverse image (for the momentum mapping) consists of a certain number of Liouvelle's 2-tori. The diagram L splits the image µ(M 4 ) into connectivity components which can be called chambers for the sake of convenience. If the point y moves continuously within one chamber, the number of Liouville tori in its inverse image µ- 1 (y) remains unchanged. Changes can occur only if

PHYSICAL INTEGRABLE HAMILTONIANS

429

H h Fig. 6.53

the point y moves from one chamber to another, i.e., if it intersects the bifurcation diagram. In this step, we must try to determine the number of Liouville tori "suspended' over each chamber of the image µ(M4 ). This problem usually boils down to finding the number of solutions of a certain algebraic equation. The difficulties encountered in this case are well known and can be overcome by using methods which are operative in almost all the physical cases of integrability investigated by us. Step 4. We must now consider the points cat which the vertical straight line H = h intersects the curves of the diagram E (Fig. 6.53). Let 'Y be a segment of the straight line H = h passing from a regular point a to another regular point b (these points may lie in the same or different chambers}. A certain number of Liouville tori "hangs" over the regular point a. Let us denote these tori conditionally by points. As the point 7(t)moves along 'Y towards the point c, these tori are smoothly deformed inside Q (through isotopy). The points representing them describe certain smooth arcs which form the edges of the eventual graph-molecule W*. The main events occur at the instant when the point 7(t) merges with the point c, i.e. with the critical value of the momentum mapping. At this instant, the Liouville tori undergo some topological rearrangement, which is described with the help of an atom (or, may be, several atoms). This atom "releases" new Liouville tori on the other side of the point c, which continue their motion as the point 7(t) slides towards the point b (see Fig. 6.54). There may be different numbers of Liouville tori to the right and left of the point c. The main

430

CLASSIFICATION OF HAMILTONIAN EQUATIONS

H Fig. 6.54

task at this stage of investigations is to find the atom that emerge at the instant, of rearrangement over the point c. The solution of this problem requires a certain skill since we do not possess at this instant a universal algorithm for recognizing of the atoms. Specific arguments have to be invoked for each particular dynamic system. Step 5. Thus, we were able to find at the previous stage the atoms "suspended" over all critical values c lying on the segment H = h. Joining the atoms by edges in accordance with the evolution of Liouville tori, we obtain the molecule W. It now remains for us to find the numerical marks. Although a complete algorithm has not been found for this stage also, complete solutions have been obtained for most of the investigated physical systems. The theory for the computation of marks was worked out recently by A. Bolsinov and P. Topalov. In order to find the marks ri, we must know how the critical periodic solutions of the system, which are the "axes" of the adjacent atoms, are "turned" relative to one another. The computation of the marks nk is a quite delicate procedure. However, this problem has been reduced in recent times to a counting of the homology groups of the 3-manifold Q for most cases. Since the topology of actually encountered Q is quite simple, (all their homology groups are known), we directly obtain the marks nk. Hence we obtain a marked molecule at this latest step. In spite of the noninvariance of the momentum mapping and the bifurcation diagram (relative to the choice of the integral f), the final result, viz., the object W*, is free from the above arbitrariness. This completes the investigation.

HISTORICAL COMMENTARY

431

Knowing the invariant for a given Hamiltonian, we can successfully answer the questions concerning the other Hamiltonians to which it is topologically equivalent (or conversely, not equivalent), its complexity, its arrangement (position) in the table of all integrable systems, etc.

6.7. A brief historical commentary Since this book aims at being just a brief review of the topological classification theory of integrable systems (Hamiltonians), the historical review presented here is quite cursory and lays no claim to being comprehensive (like any historical survey, it is also inevitably subjective). The first investigations on the topology of integrable systems were carried out by Poincare and are based on the problems of celestial mechanics. A review of the main stages of this development can be found, for example, in the well-known work "Topology and Mechanics" by Smale [118]. Smale's work, which was published in Inven. Math in 1970, marked a new stage in the development of the symplectic topology. Smale proposed new methods for studying integral manifolds of Hamiltonian systems and used them, among other things, to the n-body problem. Next, he chalked out the investigation plan which determined for a long time the trend of development in this field. A large number of publications devoted to the topological analysis of Hamiltonian equations began to appear since 1970. In particular, Smale's investigations were continued by Katok [156]. The noticeable transition from an analytical approach for integrable Hamiltonians to the geometrical and topological approach is due to several reasons which were described in detail by M.Kharlamov in his book "Topological Analysis of Integrable Problems of the Rigid Body Dynamics" [157]. We shall reproduce some fragments from this book. "The traditional approach towards the solution of problems in mechanics, which consisted in finding the cases of integrability and indicating the methods of reducing the problem to quadratures, essentially led to a stalemate in the rigid body dynamics. In the first place, this is due to the basic nonintegrability of the general problem (Poincare-Hughesson theorem about the nonexistence of algebraic integrals, Kozlov's theorem on the nonexistence of an integral that is analytic in small parameter in the vicinity of the Euler case). Secondly,

432

CLASSIFICATION OF HAMILTONIAN EQUATIONS

the construction of the solution involves a large number of technical difficulties. The method of algebraic relations is no longer acceptable: a complete investigation of even the fourth-order relations requires an enormous amount of work, while analytical computations for higher degrees are practically impossible. Considerable promise is offered by methods of global analysis of mechanical systems employing the differential geometry apparatus, theories of smooth manifolds and smooth mappings, KAM theory, and the Morse theory. The emergence of these techniques is due primarily to the problems of celestial mechanics. The program of topological investigations of classical mechanical systems outlined by Smale [118] and the ways of its realization in natural systems with a symmetry indicated by him also lent an impetus to the phase topology studies in the rigid body dynamics ([157], pp. 3, 4). Liouville's bifurcation tori for the cases of Euler and Lagrange were the first to be investigated (see, for example, the works of A. Iacob [158], Ya. Tatarinov [160], R. Cushman and H. Knorrer [159], etc.). Much progress was made in a series of works by M. Kharlamov and T. Pogosyan [116, 117, 157]. Kharlamov analyzed the phase topology of many mechanical systems with first integrals that are nonlinear in velocity. All these investigations led to a rich body of experimental material showing a vast variety of topological rearrangements of common integral levels and clearly pointing towards the existence of deep-rooted latent regularities which govern these bifurcations. The inadequacy of the earlier methods was also revealed. According to Kharlamov (1987), "the main difficulty lies in that the additional integrals of the EulerPoisson equations in the Euler-Zhukovskii, Kovalevskaya, and Chaplygin-Sretenskii solutions are nonlinear in angular velocity components in mobile axes, and Smale's technique is not applicable for studying such manifolds. . .. The regular level surface of the first integrals of a completely integrable Hamiltonian system is a union of tori filled with quasi-periodic trajectories. The problem about the structure of critical integral surfaces dividing the regions of independence of integrals is more complicated and more interesting from a practical point of view. These surfaces are the carriers of all singular moti