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.)

**Chapter I. Symplectic vector spaces and symplectic vector bundles**

**Part 1: Symplectic vector spaces**

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

**Part 2: Symplectic vector bundles**

- Properties of symplectic vector bundles
- Orthogonality and the reduction of a symplectic vector bundle
- Complex structures on symplectic vector bundles

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

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

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

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

**Chapter III. Symplectic manifolds and Poisson manifolds**

- Symplectic manifolds; definition and examples
- Special submanifolds of a symplectic manifold
- Symplectomorphisms
- Hamiltonian vector fields
- The Poisson bracket
- Hamiltonian systems
- Presymplectic manifolds
- Poisson manifolds
- Poisson morphisms
- Infinitesimal automorphisms of a Poisson structure
- The local structure of Poisson manifolds
- The symplectic foliation of a Poisson manifold
- The local structure of symplectic manifolds
- Reduction of a symplectic manifold
- The Darboux-Weinstein theorems
- Completely integrable Hamiltonian systems
- Exercises

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

- Symplectic and Hamiltonian actions
- Elementary properties of the momentum map
- The equivariance of the momentum map
- Actions of a Lie group on its cotangent bundle
- Momentum maps and Poisson morphisms
- Reduction of a symplectic manifold by the action of a Lie group
- Mutually orthogonal actions and reduction
- Stationary motions of a Hamiltonian system
- The motion of a rigid body about a fixed point
- Euler's equations
- Special formulae for the group
**SO**(3) - The Euler-Poinsot problem
- The Euler-Lagrange and Kowalevska problems
- Additional remarks and comments
- Exercises

**Chapter V. Contact manifolds**

- Background and notations
- Pfaffian equations
- Principal bundles and projective bundles
- The class of Pfaffian equations and forms
- Darboux's theorem for Pfaffian forms and equations
- Strictly contact structures and Pfaffian structures
- Projectable Pfaffian equations
- Homogeneous Pfaffian equations
- Liouville structures
- Fibered Liouville structures
- The automorphisms of Liouville structures
- The infinitesimal automorphisms of Liouville structures
- The automorphisms of strictly contact structures
- Some contact geometry formulae in local coordinates
- Homogeneous Hamiltonian systems
- Time-dependent Hamiltonian systems
- The Legendre involution in contact geometry
- The contravariant point of view

**Appendix 1. Basic notions of differential geometry**

- Differentiable maps, immersions, submersions
- The flow of a vector field
- Lie derivatives
- Infinitesimal automorphisms and conformal infinitesimal transformations
- Time-dependent vector fields and forms
- Tubular neighborhoods
- Generalizations of Poincaré's lemma

**Appendix 2. Infinitesimal jets**

- Generalities
- Velocity spaces
- Second-order differential equations
- Sprays and the exponential mapping
- Covelocity spaces
- Liouville forms on jet spaces

**Appendix 3. Distributions, Pfaffian systems and foliations**

- Distributions and Pfaffian systems
- Completely integrable distributions
- Generalized foliations defined by families of vector fields
- Differentiable distributions of constant rank

**Appendix 4. integral invariants**

- Integral invariants of a vector field
- Integral invariants of a foliations
- The characteristic distribution of a differential form

**Appendix 5. Lie groups and Lie algebras**

- Lie groups and Lie algebras; generalities
- The exponential map
- Action of a Lie group on a manifold
- The adjoint and coadjoint representations
- Semi-direct products
- Notions regarding the cohomology of Lie groups and Lie algebras
- Affine actions of Lie groups and Lie algebras

**Appendix 6: The Lagrange-Grassmann manifold**

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

**Appendix 7: Morse families and Lagrangian submanifolds**

- Lagrangian submanifolds of a cotangent bundle
- Hamiltonian systems and first-order partial differential equations
- Contact manifolds and first-order partial differential equations
- Jacobi's theorem
- The Hamilton-Jacobi equation for autonomous systems
- The Hamilton-Jacobi equation for nonautonomous systems