Symplectic Geometry and Analytical Mechanics

Cover
Title page
Series Editor's Preface
Preface
Chapter 1. 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 form
10. Complex structures adapted to a symplectic structure
11. Additional properties of the symplectic group
Part 2: Symplectic vector bundles
12. Properties of symplectic vector bundles
13. Orthogonality and the reduction of a symplectic vector bundle
14. Complex structures on symplectic vector bundles
Part 3: Remarks concerning the operator Λ and Lepage's decomposition theorem
15. The decomposition theorem in a symplectic vector space
16. Decomposition theorem for exterior differential forms
17. 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 differentia! forms
4. The Liouville form on the cotangent bundle
5. 13. 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 cotan gent 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 matlifolds
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 S0(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 foliation
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
Bibliography
Index