Symplectic geometry and analytical mechanics / Paulette Libermann and Charles-Michel Marle ; translated by Bertram Eugene Schwarzbach.
Idioma: Inglés Lenguaje original: Francés Series Mathematics and its applications (D. Reidel Publishing Company): Editor: Dordrecht : D. Reidel, c1987Descripción: xvi, 526 p. ; 25 cmISBN: 9027724385Tema(s): Symplectic geometry | Mechanics, AnalyticOtra clasificación: 53Dxx (70Hxx 70G45)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 differential forms [56] 4. The Liouville form on the cotangent bundle [58] 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 HI. 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 manifolds [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 [181] 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] 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. Projectable 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 Poincaré’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] 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]
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Bibliografía: p. 497-517.
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 differential forms [56] --
4. The Liouville form on the cotangent bundle [58] --
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 HI. 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 manifolds [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 [181] --
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] --
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. Projectable 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 Poincaré’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] --
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] --
MR, 88c:58016
Traducido del francés.
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