Symplectic Geometry and Analytical Mechanics Mathematics and Its Applications Managing Editor: M. HAZE WINKEL Centre for Mathematics and Computer Science. Amsterdam. The Netherlands Editorial Board: F. CALOGERO. Universita deg/i Studi di Roma. Italy Yu. I. MANIN. Stekiov Institute of Mathematics. Moscow. U.S.S.R. A. H. G. RINNOOY KAN. Erasmus Ullh·ersity. Rotterdam. The Netherlands G.-c. ROTA. M.l. T.. Cambridge. Mass .. U.S.A. Symplectic Geometry and Analytical Mechanics Paulette Libermann Universite de Paris VIL Paris. France' and Charles-Michel Marle Universite Pierre et Marie Curie (Paris VI). Paris. France Translated by Bertram Eugene Schwarzbach D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP Dordrecht / Boston / Lancaster / Tokyo Library of Congress Cataloging in PubHcation Data Libermann, Paulette, 1919- Symplectic geometry and analytical mechanics. (Mathematics and its applications) Translated from the French. Bibliography: p. Includes index. 1. Geometry, Differential. 2. Symplectic manifolds. 3. Mechanics, An­ alytic. I. Marle, Charles. II. Title. III. Series: Mathematics and its applica­ tions (D. Reidel Publishing Company) QA649.L465 1987 516.3'6 86-31568 ISBN-13: 978-90-277-2439-7 e-ISBN-13: 978-94-009-3807-6 DOl: 10.1007/978-94-009-3807-6 Published by D. Reidel Publishing Company, P.O. Box 17,3300 AA Dordrecht, Holland. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322,3300 AH Dordrecht, Holland. All Rights Reserved Softcover reprint of the hardcover 1st edition 1987 © 1987 by D. Reidel Publishing Company, Dordrecht, Holland No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner ala memoire d'Elie Cartan et de Charles Ehresmann en hommage it Andre Lichnerowicz Contents Series Editor's Preface xi Preface .. xiii Chapter I. Symplectic vector spaces and symplectic vector bundles 1 Part 1: Symplectic vector spaces . 2 1. Properties of exterior forms of arbitrary degree 2 2. Properties of exterior 2-forms ....... 3 3. Symplectic forms and their automorphism groups 6 4. The contravariant approach . 8 5. Orthogonality in a symplectic vector space 10 6. Forms induced on a vector subspace of a symplectic vector space 12 7. Additional properties of Lagrangian subspaces 16 8. Reduction of a symplectic vector space. Generalizations 20 9. Decomposition of a symplectic form ....... 23 10. Complex structures adapted to a symplectic structure 26 11. Additional properties of the symplectic group 33 Part 2: Symplectic vector bundles 36 12. Properties of symplectic vector bundles 36 13. Orthogonality and the reduction of a symplectic vector bundle 38 14. Complex structures on symplectic vector bundles . 40 Part 3: Remarks concerning the operator A and Lepage's decomposition theorem 43 15. The decomposition theorem in a symplectic vector space 43 16. Decomposition theorem for exterior differential forms 48 17. A first approach to Darboux's theorem .. 51 Chapter II. Semi-basic and vertical differential forms in mechanics 53 1. Definitions and notations 54 2. Vector bundles associated with a surjective submersion 54 3. Semi-basic and vertical differentia! forms 56 4. The Liouville form on the cotangent bundle 58 viii Contents 5. Symplectic structure on the cotangent bundle 63 6. Semi-basic differential forms of arbitrary degree 67 7. Vector fields and second-order differential equations 72 8. The Legendre transformation on a vector bundle . 73 9. The Legendre transformation on the tangent and cotangent bundles 75 10. Applications to mechanics: Lagrange and Hamilton equations 77 11. Lagrange equations and the calculus of variations 81 12. The Poincare-Cartan integral invariant 83 13. Mechanical systems with time dependent Hamiltonian or Lagrangian functions 86 Chapter m. Symplectic manifolds and Poisson manifolds 89 1. Symplectic manifolds; definition and examples 90 2. .special submanifolds of a symplectic manifold 91 3. Symplectomorphisms 94 4. Hamiltonian vector fields 96 5. The Poisson bracket 99 6. Hamiltonian systems 102 7. Presymplectic manifolds 105 8. Poisson manifolds 107 9. Poisson morphisms 114 10. Infinitesimal automorphisms of a Poisson structure 121 11. The local structure of Poisson m~ifolds 125 12. The symplectic foliation of a Poisson manifold 130 13. The local structure of symplectic manifolds 134 14. Reduction of a symplectic manifold . 141 15. The Darboux-Weinstein theorems 153 16. Completely integrable Hamiltonian systems 160 17. Exercises .............. 18] Chapter IV. Action of a Lie group on a symplectic manifold 185 1. Symplectic and Hamiltonian actions 186 2. Elementary properties of the momentum map 195 3. The equivariance of the momentum map 200 4. Actions of a Lie group on its cotangent bundle 204 5. Momentum maps and Poisson morphisms 213 6. Reduction of a symplectic manifold by the action of a Lie group 217 7. Mutually orthogonal actions and reduction 228 8. Stationary motions of a Hamiltonian system . 238 9. The motion of a rigid body about a fixed point 246 10. Euler's equations . 253 11. Special formulae for the group SO(3) . 256 12. ,The Euler-Poinsot problem ........ 260 13. The Euler-Lagrange and Kowalevska problems 265 14. Additional remarks and comments 267 15. Exercises 269 Contents ix Chapter V. Contact manifolds 275 1. Background and notations . 276 2. Pfaffian equations 277 3. Principal bundles and projective bundles 279 4. The class of Pfaffian equations and forms 284 5. Darboux's theorem for Pfaffian forms and equations 286 6. Strictly contact structures and Pfaffian structures 289 7. Project able Pfaffian equations 299 8. Homogeneous Pfaffian equations 302 9. Liouville structures . 306 10. Fibered Liouville structures 307 11. The automorphisms of Liouville structures 313 12. The infinitesimal automorphisms of Liouville structures 315 13. The automorphisms of strictly contact structures . 318 14. Some contact geometry formulae in local coordinates 324 15. Homogeneous Hamiltonian systems . 327 16. Time-dependent Hamiltonian systems . 328 17. The Legendre involution in contact geometry 332 18. The contravariant point of view ..... 336 Appendix 1. Basic notions of differential geometry 341 1. Differentiable maps, immersions, submersions 341 2. The flow of a vector field ....... 346 3. Lie derivatives 349 4. Infinitesimal automorphisms and conformal infinitesimal transformations 352 5. Time-dependent vector fields and forms 354 6. Tubular neighborhoods . 358 7. Generalizations of Poincare's lemma 359 Appendix 2. Infinitesimal jets 365 1. Generalities . 365 2. Velocity spaces . 367 3. Second-order differential equations 371 4. Sprays and the exponential mapping 373 5. Covelocity spaces . 376 6. Liouville forms on jet spaces . 379 Appendix 3. Distributions, Pfaffian systems and foliations 382 1. Distributions and Pfaffian systems ........ 382 2. Completely integrable distributions . 384 3. Generalized foliations defined by families of vector fields 387 4. Differentiable distributions of constant rank . 393 x Contents Appendix 4. Integral invariants 395 1. Integral invariants of a vector field 395 2. Integral invariants of a foliation 401 3. The characteristic distribution of a differential form 404 Appendix 5. Lie groups and Lie algebras 409 1. Lie groups and Lie algebras; generalities 409 2. The exponential map . 415 3. Action of a Lie group on a manifold 419 4. The adjoint and coadjoint representations 425 5. Semi-direct products ........ 429 6. Notions regarding the cohomology of Lie groups and Lie algebras 433 7. Affine actions of Lie groups and Lie algebras . 439 Appendix 6. The Lagrange-Grassmann manifold 448 1. The structure of the Lagrange-Grassmann manifold 448 2. The signature of a Lagrangian triplet . 454 3. The fundamental groups of the symplectic group and of the Lagrange-Grassmann manifold 458 Appendix 7. Morse families and Lagrangian submanifolds 461 1. Lagrangian submanifolds of a cotangent bundle 461 2. Hamiltonian systems and first-order partial differential equations 473 3. Contact manifolds and first-order partial differential equations 477 4. Jacobi's theorem . 484 5. The Hamilton-Jacobi equation for autonomous systems 490 6. The Hamilton-Jacobi equation for nonautonomous systems 492 Bibliography 497 Index .... 519 SERIES EDITOR'S PREFACE Approach your problems from the right end It isn't that they can't see the solution. and begin with the answers. Then one day, It is that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' Brown 'The point of a Pin'. in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thouglit to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci­ ences has changed drastically in recent years: measure theory is used (non-trivially) in re­ gional