Author | Libermann, Paulette. author |
---|---|
Title | Symplectic Geometry and Analytical Mechanics [electronic resource] / by Paulette Libermann, Charles-Michel Marle |
Imprint | Dordrecht : Springer Netherlands, 1987 |
Connect to | http://dx.doi.org/10.1007/978-94-009-3807-6 |
Descript | XVI, 526 p. online resource |
I. Symplectic vector spaces and symplectic vector bundles -- 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 -- 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 -- 3: Remarks concerning the operator ? and Lepageโs decomposition theorem -- 15. The decomposition theorem in a symplectic vector space -- 16. Decomposition theorem for exterior differential forms -- 17. A first approach to Darbouxโs theorem -- 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 Poincarรฉ-Cartan integral invariant -- 13. Mechanical systems with time dependent Hamiltonian or Lagrangian functions -- 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 -- 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. Eulerโs 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 -- 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. Darbouxโs theorem for Pfaffian forms and equations -- 6. Strictly contact structures and Pfaffian structures -- 7. Protectable 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 conformai infinitesimal transformations -- 5. Time-dependent vector fields and forms -- 6. Tubular neighborhoods -- 7. Generalizations of Poincarรฉโs 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. Jacobiโs theorem -- 5. The Hamilton-Jacobi equation for autonomous systems -- 6. The Hamilton-Jacobi equation for non autonomous systems