Symplectic geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Gordon and Breach
1995
|
| Edition: | 2. ed. |
| Series: | Advanced studies in contemporary mathematics
5 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 467 S. graph. Darst. |
| ISBN: | 2881249019 |
Staff View
MARC
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| 240 | 1 | 0 | |a Simplektičeskaja geometrija |
| 245 | 1 | 0 | |a Symplectic geometry |c A. T. Fomenko |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York |b Gordon and Breach |c 1995 | |
| 300 | |a XVI, 467 S. |b graph. Darst. | ||
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| 337 | |b n |2 rdamedia | ||
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Record in the Search Index
| _version_ | 1804125359305654272 |
|---|---|
| adam_text | CONTENTS
Preface xi
Notation xii
Chapter 1. Symplectic geometry in Euclidean space
1.1 Some information from matrix group theory 1
1.1.1 Lie groups and algebras 1
1.1.2 The complete linear groups GL(«, R) and GL(n, C)
and their Lie algebras 3
1.1.3 The special linear groups SL(«, R) and SL(n, C) 4
1.1.4 The orthogonal group O(/i) and the special
orthogonal group SO(n) 6
1.1.5 The unitary group U(«) and the special unitary
group SU(n) 10
1.1.6 Connected components of matrix groups 13
1.1.7 The realification operation and complex structures 16
1.2 Groups of symplectic transformations of a linear space 24
1.2.1 Symplectic linear transformations 24
1.2.2 The noncompact groups Sp(n, R) and Sp(/i, C) 29
1.2.3 The compact group Sp(n) 39
1.2.4 The relation between symplectic groups and other
matrix groups 46
1.3 Lagrangian manifolds 51
1.3.1 Real Lagrangian manifolds in a symplectic linear
space 51
1.3.2 Complex Lagrangian Grassmann manifolds 59
1.3.3 Real Lagrangian Grassmann manifolds 68
Chapter 2. Symplectic geometry on smooth manifolds
2.1 Local structure of symplectic manifolds 74
2.1.1 Local symplectic coordinates 74
2.1.2 Hamiltonian vector fields 77
2.1.3 The Poisson bracket 81
2.1.4 Darboux theorem 87
v
vi CONTENTS
2.2 Embeddings of symplectic manifolds 91
2.2.1 Embeddings of symplectic manifolds in Uw 91
2.2.2 Embeddings of symplectic manifolds in CPN 93
2.2.3 Examples of symplectic manifolds 94
Chapter 3. Hamiltonian systems with symmetries on
symplectic manifolds
3.1 Liouville s theorem 99
3.1.1 Integrals of Hamiltonian systems 99
3.1.2 Complete involutive sets of functions 101
3.2 Hamiltonian systems with noncommutative symmetries 107
3.2.1 Finite dimensional Lie subalgebras in a space of
functions on a symplectic manifold 107
3.2.2 A theorem on integration of systems with
noncommutative symmetries 110
3.2.3 Connections between systems with commutative
and noncommutative symmetries 114
3.2.4 Noncommutative integration in those cases when
the sets of integrals do not form an algebra 120
3.2.5 Integration in quadratures of systems with
noncommutative integrals 124
3.2.6 The canonical form of the Poisson bracket in a
neighbourhood of a singular point. The case of
degenerate Poisson brackets 131
3.2.7 Noncommutative integrability and its connection
with canonical submanifolds and isotropic tori 138
3.2.8 Solvable Lie algebras of functions on symplectic
manifolds and integration of mechanical systems
corresponding to them 154
3.3 Dynamical systems generated by sectional operators 160
3.3.1 General plan of construction of sectional operators 160
3.3.2 Construction of a many parameter family of
exterior 2 forms on orbits of stationary groups of
symmetric spaces 163
CONTENTS vii
Chapter 4. Geodesic flows on two dimensional Riemann
surfaces
4.1 Completely integrable geodesic flows on a sphere and a
torus 175
4.1.1 Geodesic flow of a two dimensional Riemannian
metric 175
4.1.2 A necessary and sufficient condition for the
existence of an additional polynomial integral
quadratic in the momenta 176
4.1.3 Description of Riemannian metrics on a sphere
and a torus that admit an additional integral 182
4.1.4 Geometric properties of metrics on a sphere that
admits an additional integral 187
4.2 Nonintegrability of analytic geodesic flows on surfaces of
genus g 191
4.3 Nonintegrability of the problem of n centres for n 2 194
4.4 Morse type theory of integrable Hamiltonian systems.
Connections between integrability of systems, existence
of stable periodic solutions and the one dimensional
homology group of surfaces of constant energy 199
Chapter 5. Effective methods of constructing completely
integrable systems on Lie algebras.
Dynamics of multi dimensional rigid body
5.1 Left invariant Hamiltonian systems on Lie groups and
the Euler equations on Lie algebras 226
5.1.1 Symplectic structure and left invariant
Hamiltonians 226
5.1.2 Quadratic Hamiltonians associated with the
displacement of the argument on Lie algebras 235
5.1.3 Properties of the general Euler equations 240
5.2 A brief summary of classical results on the root
decomposition of complex semisimple Lie algebras 245
5.3 Analogs of multidimensional rigid body motion
for semisimple Lie algebras 251
5.3.1 The sectional decomposition of an algebra
coincides with Carton s decomposition 251
viii CONTENTS
5.3.2 Various types of sectional operators.
Complex metrics. Normal nilpotent metrics.
Normal solvable metrics 256
5.3.3 Compact series of metrics 259
5.3.4 Normal series of metrics 266
5.4 Construction of integrals of the Euler equations
corresponding to complex, compact and normal
dynamics of multi dimensional rigid body 271
5.4.1 Integrals of a complex left invariant metrics 271
5.4.2 Integrals of a compact left invariant metrics 274
5.4.3 Integrals of a normal left invariant metrics 282
5.4.4 Involutoriness of integrals 282
5.5 Complete integrability of the Euler equations for
symmetrical multi dimensional rigid body 287
5.5.1 Complex integrable cases 287
5.5.2 Compact integrable cases 303
5.5.3 Normal integrable cases 305
5.5.4 Integrability of the Euler equations
on singular orbits 334
5.6 Quadratic integrals of the Euler equations 341
5.7 Integrability of geodesic flows of left invariant metrics
of the form (pabD on semisimple groups and
geodesic flows on symmetric spaces 351
5.7.1 Geodesic flow on 7 *© 351
5.7.2 © invariant geodesic flows on 7 (©/§) 355
5.7.3 Geodesic flows of general form
on symmetric surfaces 365
Chapter 6. A brief review of the theory of topological
classification of integrable nondegenerate
Hamiltonian equations with two degrees of freedom
6.1 Formulation of the problem 368
6.1.1 Example: classical Hamiltonian equations
of the motion of a rigid body 368
6.1.2 Integrability or nonintegrability as a manifestation
of symmetry or randomness in system evolution 372
6.1.3 Examples of physical and mechanical systems
integiable in the Liouville sense 373
CONTENTS ix
6.1.4 Classification of all integrable nondegenerate
Hamiltonian systems (integrable Hamiltonians)
with two degrees of freedom 376
6.2 Smooth functions typical on smooth manifolds 377
6.2.1 Morse simple functions 377
6.2.2 Simple atoms and simple molecules 379
6.2.3 Complex Morse functions 386
6.2.4 Complex atoms and complex molecules 387
6.3 Bolt s functions as typical integrals of integrable systems 391
6.3.1 Bott s functions 391
6.3.2 Integrals which are typical
in the Hamiltonian physics 393
6.4 Rough and fine topological equivalence
of integrable systems 395
6.5 Theorem of rough and fine classification of integrable
Hamiltonian systems with two degrees of freedom.
Applications in physics and mechanics 398
6.5.1 Formulation of the main theorem 398
6.5.2 Relation between invariants W, W* and the topology
of an integrable system. Substantial interpretation
of atoms and molecules 416
6.6 Method of computing topological invariants for
specific physical integrable Hamiltonians 426
6.7 A brief historical commentary 431
6.8 Class (H) of isoenergy three dimensional integrable
manifolds. Five faces of this class 434
6.8.1 Class (H) of the isoenergy 3 surfaces 434
6.8.2 Class (Q) of three dimensional manifolds glued
from two types of blocks 435
6.8.3 Class (W) of Waldhausen manifolds
(graph manifolds) 436
6.8.4 The class (S) 437
6.8.5 The class (T) of isointegrable manifolds
corresponding to Hamiltonians with tame integrals 437
6.8.6 The class (R) of manifolds glued from round handles 438
6.8.7 Theorem on the coincidence of five classes 439
x CONTENTS
6.9 Application of the topological classification theory
of integrable systems to geodesic flows on a 2 sphere
and 2 torus 441
6.9.1 Hypothesis on geodesic flows 441
6.9.2 Integrable geodesic flows
on a 2 sphere and a 2 torus 442
6.9.3 Complexity of integrable geodesic flows
on a 2 sphere and a 2 torus 445
6.9.4 Hypothesis: linearly quadratically integrable metrics
approximate any nondegenerate integrable
Riemannian metric on a 2 torus 447
6.10 Topological classification of classical cases of integrability
in the dynamics of a heavy rigid body 449
References 454
Subject Index 464
|
| any_adam_object | 1 |
| author | Fomenko, Anatolij Timofeevič 1945- |
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| classification_rvk | SK 350 SK 370 |
| classification_tum | MAT 537f |
| ctrlnum | (OCoLC)422033590 (DE-599)BVBBV010871391 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:00:18Z |
| institution | BVB |
| isbn | 2881249019 |
| language | English |
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| physical | XVI, 467 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Gordon and Breach |
| record_format | marc |
| series | Advanced studies in contemporary mathematics |
| series2 | Advanced studies in contemporary mathematics |
| spelling | Fomenko, Anatolij Timofeevič 1945- Verfasser (DE-588)119092689 aut Simplektičeskaja geometrija Symplectic geometry A. T. Fomenko 2. ed. New York Gordon and Breach 1995 XVI, 467 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced studies in contemporary mathematics 5 Géométrie différentielle Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s DE-604 Differentialgeometrie (DE-588)4012248-7 s Advanced studies in contemporary mathematics 5 (DE-604)BV000600809 5 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007267686&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fomenko, Anatolij Timofeevič 1945- Symplectic geometry Advanced studies in contemporary mathematics Géométrie différentielle Differentialgeometrie (DE-588)4012248-7 gnd Symplektische Geometrie (DE-588)4194232-2 gnd |
| subject_GND | (DE-588)4012248-7 (DE-588)4194232-2 |
| title | Symplectic geometry |
| title_alt | Simplektičeskaja geometrija |
| title_auth | Symplectic geometry |
| title_exact_search | Symplectic geometry |
| title_full | Symplectic geometry A. T. Fomenko |
| title_fullStr | Symplectic geometry A. T. Fomenko |
| title_full_unstemmed | Symplectic geometry A. T. Fomenko |
| title_short | Symplectic geometry |
| title_sort | symplectic geometry |
| topic | Géométrie différentielle Differentialgeometrie (DE-588)4012248-7 gnd Symplektische Geometrie (DE-588)4194232-2 gnd |
| topic_facet | Géométrie différentielle Differentialgeometrie Symplektische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007267686&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT fomenkoanatolijtimofeevic simplekticeskajageometrija AT fomenkoanatolijtimofeevic symplecticgeometry |