Symplectic geometry and analytical mechanics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Dordrecht u.a.
Reidel
1987
|
| Series: | Mathematics and its applications
35 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Franz. übers. |
| Physical Description: | XVI, 526 S. |
| ISBN: | 9027724385 9027724393 |
Staff View
MARC
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Record in the Search Index
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| adam_text | 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 differentia! forms 56
4. The Liouville form on the cotangent bundle 58
viii Contents
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 III. 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 Daxboux Weinstein theorems 153
16. Completely integrable Hamiltonian systems 160
17. Exercises 181
Chapter TV. 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
Contents be
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. Differentiate 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 Poincare 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. Differentiate distributions of constant rank 393
x Contents
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
|
| any_adam_object | 1 |
| author | Libermann, Paulette Marle, Charles-Michel |
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| ctrlnum | (OCoLC)15054078 (DE-599)BVBBV000690128 |
| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV000690128 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:17:51Z |
| institution | BVB |
| isbn | 9027724385 9027724393 |
| language | English French |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000430472 |
| oclc_num | 15054078 |
| open_access_boolean | |
| owner | DE-12 DE-91G DE-BY-TUM DE-384 DE-355 DE-BY-UBR DE-29T DE-19 DE-BY-UBM DE-706 DE-634 DE-83 DE-11 DE-188 DE-20 |
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| physical | XVI, 526 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Reidel |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Libermann, Paulette Verfasser aut Symplectic geometry and analytical mechanics Paulette Libermann and Charles-Michel Marle Dordrecht u.a. Reidel 1987 XVI, 526 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 35 Aus d. Franz. übers. Géométrie différentielle Mechanica gtt Mécanique analytique Symplectische ruimten gtt Variétés symplectiques Mechanics, Analytic Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Mechanik (DE-588)4038168-7 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Theoretische Mechanik (DE-588)4185100-6 s DE-604 Symplektische Geometrie (DE-588)4194232-2 s Mechanik (DE-588)4038168-7 s Physik (DE-588)4045956-1 s Marle, Charles-Michel Verfasser aut Mathematics and its applications 35 (DE-604)BV008163334 35 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000430472&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Libermann, Paulette Marle, Charles-Michel Symplectic geometry and analytical mechanics Mathematics and its applications Géométrie différentielle Mechanica gtt Mécanique analytique Symplectische ruimten gtt Variétés symplectiques Mechanics, Analytic Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd Differentialgeometrie (DE-588)4012248-7 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Mechanik (DE-588)4038168-7 gnd Physik (DE-588)4045956-1 gnd |
| subject_GND | (DE-588)4194232-2 (DE-588)4012248-7 (DE-588)4185100-6 (DE-588)4038168-7 (DE-588)4045956-1 |
| title | Symplectic geometry and analytical mechanics |
| title_auth | Symplectic geometry and analytical mechanics |
| title_exact_search | Symplectic geometry and analytical mechanics |
| title_full | Symplectic geometry and analytical mechanics Paulette Libermann and Charles-Michel Marle |
| title_fullStr | Symplectic geometry and analytical mechanics Paulette Libermann and Charles-Michel Marle |
| title_full_unstemmed | Symplectic geometry and analytical mechanics Paulette Libermann and Charles-Michel Marle |
| title_short | Symplectic geometry and analytical mechanics |
| title_sort | symplectic geometry and analytical mechanics |
| topic | Géométrie différentielle Mechanica gtt Mécanique analytique Symplectische ruimten gtt Variétés symplectiques Mechanics, Analytic Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd Differentialgeometrie (DE-588)4012248-7 gnd Theoretische Mechanik (DE-588)4185100-6 gnd Mechanik (DE-588)4038168-7 gnd Physik (DE-588)4045956-1 gnd |
| topic_facet | Géométrie différentielle Mechanica Mécanique analytique Symplectische ruimten Variétés symplectiques Mechanics, Analytic Symplectic geometry Symplektische Geometrie Differentialgeometrie Theoretische Mechanik Mechanik Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000430472&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT libermannpaulette symplecticgeometryandanalyticalmechanics AT marlecharlesmichel symplecticgeometryandanalyticalmechanics |