Agnes Scott College

Paulette Libermann and Charles-Michel Marle

Symplectic Geometry and Analytical Mechanics
D. Reidel Publishing Company, 1987


(Translated by Bertram Eugene Schwarzbach)

Preface (Excerpts)

During the last two centuries, analytical mechanics have occupied a prominent place among scientists' interests. The work in this field by such mathematicians as Euler, Lagrange, Laplace, Hamilton, Jacobi, Poisson, Liouville, Poincaré, Carathéodory, Birkhoff, Lie and E. Cartan has played a major role in the development of several important branches of mathematics: differential geometry, the calculus of variations, the theory of Lie groups and Lie algebras, and the theory of ordinary and partial differential equations. During the last thirty years, the study of the geometric structures which form the basis of mechanics (symplectic, Poisson and contact structures) has enjoyed renewed vigor. The introduction of modern methods of differential geometry is one of the reasons for this renewal; it has permitted a formulation of global problems and furnished tools with which to solve them.

Even though there are already a number of books that treat this subject, the authors believe that it is of value to provide readers with an approach to these methods and to permit them to familiarize themselves with certain recent developments which are not mentioned in the other textbooks in this field, and to acquire the information necessary in order to pursue current research. They have also expounded and employed the methods of exterior algebra which were introduced by E. Cartan.

This work, which is in large part based on lectures given by the authors at the Universities of Paris VI and VII, incorporates numerous points recalled to facilitate study. It was written for students at the end of the "second Cycle" program, or at the beginning of their "Third Cycle"—these are the French designations that correspond, approximately, to American Masters' and Doctoral programs, respectively. It is mainly directed at readers interested in mathematics, but it may be of interest too for physicists, engineers, and for anyone who may be interested in differential geometry and the foundations of mechanics.

This work is composed of five chapters and seven appendices. Chapters I, II and V, as well as appendix 2, were written by the first author (P.L.), while chapters III and IV, as well as the remaining appendices, were written by the second author (C.-M. M.)

Table of Contents

Chapter I. Symplectic vector spaces and symplectic vector bundles

Part 1: Symplectic vector spaces

  1. Properties of exterior forms of arbitrary degree
  2. Properties of exterior 2-forms
  3. Symplectic forms and their automorphism groups
  4. The contravariant approach
  5. Orthogonality in a symplectic vector space
  6. Forms induced on a vector subspace of a symplectic vector space
  7. Additional properties of Lagrangian subspaces
  8. Reduction of a symplectic vector space. Generalizations
  9. Decomposition of a symplectic forms
  10. Complex structures adapted to a symplectic structure
  11. Additional properties of the symplectic group

Part 2: Symplectic vector bundles

  1. Properties of symplectic vector bundles
  2. Orthogonality and the reduction of a symplectic vector bundle
  3. Complex structures on symplectic vector bundles

Part 3:Remarks concerning the operator Λ and Lepage's decomposition theorem

  1. The decomposition theorem in a symplectic vector space
  2. Decomposition theorem for exterior differential forms
  3. A first approach to Darboux's theorem

Chapter II. Semi-basic and vertical differential forms in mechanics

  1. Definitions and notations
  2. Vector bundles associated with a surjective submersion
  3. Semi-basic and vertical differential forms
  4. The Liouville form on the cotangent bundle
  5. Symplectic structure on the cotangent bundle
  6. Semi-basic differential forms of arbitrary degree
  7. Vector fields and second-order differential equations
  8. The Legendre transformation on a vector bundle
  9. The Legendre transformation on the tangent and cotangent bundles
  10. Applications to mechanics: Lagrange and Hamilton equations
  11. Lagrange equations and the calculus of variations
  12. The Poincaré-Cartan integral invariant
  13. Mechanical systems with time dependent Hamiltonian or Lagrangian functions

Chapter III. Symplectic manifolds and Poisson manifolds

  1. Symplectic manifolds; definition and examples
  2. Special submanifolds of a symplectic manifold
  3. Symplectomorphisms
  4. Hamiltonian vector fields
  5. The Poisson bracket
  6. Hamiltonian systems
  7. Presymplectic manifolds
  8. Poisson manifolds
  9. Poisson morphisms
  10. Infinitesimal automorphisms of a Poisson structure
  11. The local structure of Poisson manifolds
  12. The symplectic foliation of a Poisson manifold
  13. The local structure of symplectic manifolds
  14. Reduction of a symplectic manifold
  15. The Darboux-Weinstein theorems
  16. Completely integrable Hamiltonian systems
  17. Exercises

Chapter IV. Action of a Lie group on a symplectic manifold

  1. Symplectic and Hamiltonian actions
  2. Elementary properties of the momentum map
  3. The equivariance of the momentum map
  4. Actions of a Lie group on its cotangent bundle
  5. Momentum maps and Poisson morphisms
  6. Reduction of a symplectic manifold by the action of a Lie group
  7. Mutually orthogonal actions and reduction
  8. Stationary motions of a Hamiltonian system
  9. The motion of a rigid body about a fixed point
  10. Euler's equations
  11. Special formulae for the group SO(3)
  12. The Euler-Poinsot problem
  13. The Euler-Lagrange and Kowalevska problems
  14. Additional remarks and comments
  15. Exercises

Chapter V. Contact manifolds

  1. Background and notations
  2. Pfaffian equations
  3. Principal bundles and projective bundles
  4. The class of Pfaffian equations and forms
  5. Darboux's theorem for Pfaffian forms and equations
  6. Strictly contact structures and Pfaffian structures
  7. Projectable Pfaffian equations
  8. Homogeneous Pfaffian equations
  9. Liouville structures
  10. Fibered Liouville structures
  11. The automorphisms of Liouville structures
  12. The infinitesimal automorphisms of Liouville structures
  13. The automorphisms of strictly contact structures
  14. Some contact geometry formulae in local coordinates
  15. Homogeneous Hamiltonian systems
  16. Time-dependent Hamiltonian systems
  17. The Legendre involution in contact geometry
  18. The contravariant point of view

Appendix 1. Basic notions of differential geometry

  1. Differentiable maps, immersions, submersions
  2. The flow of a vector field
  3. Lie derivatives
  4. Infinitesimal automorphisms and conformal infinitesimal transformations
  5. Time-dependent vector fields and forms
  6. Tubular neighborhoods
  7. Generalizations of Poincaré's lemma

Appendix 2. Infinitesimal jets

  1. Generalities
  2. Velocity spaces
  3. Second-order differential equations
  4. Sprays and the exponential mapping
  5. Covelocity spaces
  6. Liouville forms on jet spaces

Appendix 3. Distributions, Pfaffian systems and foliations

  1. Distributions and Pfaffian systems
  2. Completely integrable distributions
  3. Generalized foliations defined by families of vector fields
  4. Differentiable distributions of constant rank

Appendix 4. integral invariants

  1. Integral invariants of a vector field
  2. Integral invariants of a foliations
  3. The characteristic distribution of a differential form

Appendix 5. Lie groups and Lie algebras

  1. Lie groups and Lie algebras; generalities
  2. The exponential map
  3. Action of a Lie group on a manifold
  4. The adjoint and coadjoint representations
  5. Semi-direct products
  6. Notions regarding the cohomology of Lie groups and Lie algebras
  7. Affine actions of Lie groups and Lie algebras

Appendix 6: The Lagrange-Grassmann manifold

  1. The structure of the Lagrange-Grassmann manifold
  2. The signature of a Lagrangian triplet
  3. The fundamental groups of the symplectic group and of the Lagrange-Grassmann manifold

Appendix 7: Morse families and Lagrangian submanifolds

  1. Lagrangian submanifolds of a cotangent bundle
  2. Hamiltonian systems and first-order partial differential equations
  3. Contact manifolds and first-order partial differential equations
  4. Jacobi's theorem
  5. The Hamilton-Jacobi equation for autonomous systems
  6. The Hamilton-Jacobi equation for nonautonomous systems