Structure of dynamical systems: a symplectic view of physics
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Boston [u.a.]
Birkhäuser
1997
|
| Series: | Progress in mathematics
149 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXXIV, 406 S. graph. Darst. |
| ISBN: | 0817636951 3764336951 |
Staff View
MARC
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| 100 | 1 | |a Souriau, Jean-Marie |d 1922-2012 |e Verfasser |0 (DE-588)172380499 |4 aut | |
| 240 | 1 | 0 | |a Structure des systèmes dynamiques |
| 245 | 1 | 0 | |a Structure of dynamical systems |b a symplectic view of physics |c J.-M. Souriau |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1997 | |
| 300 | |a XXXIV, 406 S. |b graph. Darst. | ||
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| 490 | 1 | |a Progress in mathematics |v 149 | |
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| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 4 | |a Mechanics | |
| 650 | 4 | |a Quantum theory | |
| 650 | 4 | |a Statistical mechanics | |
| 650 | 4 | |a Symplectic manifolds | |
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Record in the Search Index
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|---|---|
| adam_text | Table of Contents
Introduction xvii
I. Differential Geometry
§1. Manifolds
The definition of a manifold 3
Open sets 6
Differentiable maps 6
The tangent space 7
Submanifolds 10
Manifolds defined by an equation 12
Covering spaces 13
Quotient manifolds 14
Connectedness 14
Homotopy 16
§2. Derivations
Variables 18
Vector fields and derivations 19
Derivations of linear operators 22
The image of a vector field 23
Lie brackets 25
§3. Differential equations
The exponential of a vector field 27
The image of a differential equation 28
The derivative of the exponential map 29
x Table of Contents
§4. Differential forms
Covariant fields 31
The inverse image of a covariant field . . • 31
The Lie derivative 33
Covariant tensor fields 34
p Forms 35
The exterior derivative 36
§5. Foliated manifolds
Foliations 38
The quotient of a manifold by a foliation 42
Integral invariants 43
The characteristic foliation of a form 44
§6. Lie groups
Actions of a Lie group on a manifold 46
The Lie algebra of a Lie group 48
Orbits 49
. The adjoint representation 49
Lie subalgebras and Lie subgroups 51
The stabilizer 53
Classical examples of Lie groups 53
Euclidean spaces 55
Matrix realizations 59
§7. The calculus of variations
Classical variational problems 62
Canonical variables 63
The Hamiltonian formalism 65
A geometrical interpretation of the canonical equations 66
Transformations of a variational problem 68
Noether s theorem 68
Table of Contents xi
II. Symplectic Geometry
§8. 2 Forms
Orthogonality 73
Canonical bases 74
The symplectic group 77
§9. Symplectic manifolds
Symplectic and presymplectic manifolds 81
Symplectic structures arising from a 1 form 84
Poisson brackets 85
Induced symplectic structures 87
§10. Canonical transformations
Canonical charts 90
Canonical transformations 93
Canonical similitudes 94
Covering spaces of symplectic manifolds 96
Infinitesimal canonical transformations 97
§11. Dynamical Groups
The definition of a dynamical group 100
The cohomology of a dynamical group 104
The cohomology of a Lie group 107
The cohomology of a Lie algebra 108
Symplectic manifolds defined by a Lie group Ill
III. Mechanics
§12. The geometric structure of classical mechanics
Material points 121
Systems of material points 122
Constraints 122
Describing forces 125
xii Table of Contents
The evolution space 126
Phase spaces and the space of motions 127
The Lagrange 2 form 129
The Lagrange form for constrained systems 131
Changing the reference frame 133
The principle of Galilean relativity 135
Maxwell s principle 137
Potentials and the variational formalism 139
Geometric consequences of Maxwell s principle 141
An application: variation of constants 142
Galilean moments 144
Remarks 148
Examples of dynamical groups 149
§13. The principles of symplectic mechanics
Nonrelativistic symplectic mechanics 154
Moments, mass, and the center of mass 155
The center of mass decomposition 157
Minkowski space and the Poincare group 163
Relativistic mechanics 166
§14. A mechanistic description of elementary particles
Elementary systems 173
A particle with spin 174
Remarks 179
A particle without spin 180
A massless particle 182
Remarks 184
Nonrelativistic particles 185
Mass and barycenter of a relativistic system 188
Inversions of space and time 189
A particle with nonzero mass 191
A massless particle 192
§15. Particle dynamics
A material point in an electromagnetic field 194
A particle with spin in an electromagnetic field 197
Systems of particles without interactions 200
Interactions 202
Table of Contents xiii
Scattering theory 205
Bounded scattering sources 210
Geometrical optics 212
Planar mirrors 214
Collisions of free particles 218
IV. Statistical Mechanics
§16. Measures on a manifold
Composite manifolds 223
Compact sets 224
Riesz spaces 227
Measures 229
The tensor product of measures 232
Examples of measures 234
Completely continuous measures 235
Examples of completely continuous measures 237
The support of a measure 241
Bounded measures 241
Integrable functions 246
The image of a measure 249
Examples 253
Random variables 256
Average values 258
Entropy and Gibbs measures 260
The Gibbs canonical ensemble of a dynamical group 265
§17. The principles of statistical mechanics
Statistical states 271
Hypotheses of the kinetic theory of gases 271
Equilibria of a conservative system 273
Ideal gases 279
A monatomic ideal gas 280
An arbitrary ideal gas 280
An ideal gas thermometer 281
Heat and work 283
Specific heat 284
Covariant statistical mechanics 287
xiv 7 aWe of ( outents
Examples 291
The statistical equilibrium of an isolated system 293
Relativistic statistical mechanics 295
A relativistic ideal gas 296
Statistical equilibria of photons 298
V. A Method of Quantization
§18. Geometric quantization
Prequantum manifolds 307
Prequantization of a symplectic manifold 309
Prequantization of a symplectic manifold admitting
a potential 312
Prequantization of a sphere .S 2 313
Prequantization by fusion 315
Prequantization of a direct product 316
Prequantization of a relativistic particle with spin | 318
Prequantization of a massless particle 321
Massless particle with spin ~ 322
Massless particle with spin 1 321
Planck manifolds 325
Quantoinorphi.srns 326
Homotopy and prequantization 328
Systems of elementary particles 331
Infinitesimal quantomorphisrns 331
Quantization of dynamical groups 331
The Hilbert space of a prequantum manifold 339
§19. Quantization of dynamical systems
The correspondence principle 346
State vectors and observables 347
The formulation of Planck s condition 350
Stationary states 351
The formation of wave equations 352
The nonrelativistic material point 354
I he relativistic material point 358
The nonrelativistic particle with spin ^ 360
The relativistic particle with spin | 362
Table of Contents xv
The massless particle with spin | 364
The massless particle with spin 1 366
Assemblies of particles 370
Creation and annihilation operators 373
Quantum states 378
Bibliography 387
Index 391
List of notation 405
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:13:09Z |
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| isbn | 0817636951 3764336951 |
| language | English French |
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| physical | XXXIV, 406 S. graph. Darst. |
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| series2 | Progress in mathematics |
| spelling | Souriau, Jean-Marie 1922-2012 Verfasser (DE-588)172380499 aut Structure des systèmes dynamiques Structure of dynamical systems a symplectic view of physics J.-M. Souriau Boston [u.a.] Birkhäuser 1997 XXXIV, 406 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 149 Dynamische systemen gtt Mécanique statistique ram Mécanique ram Physique mathématique ram Symplectische ruimten gtt Théorie quantique ram Variétés symplectiques ram Mathematische Physik Quantentheorie Mathematical physics Mechanics Quantum theory Statistical mechanics Symplectic manifolds Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Theoretische Physik (DE-588)4117202-4 gnd rswk-swf Theoretische Physik (DE-588)4117202-4 s Symplektische Geometrie (DE-588)4194232-2 s DE-604 Progress in mathematics 149 (DE-604)BV000004120 149 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007841484&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Souriau, Jean-Marie 1922-2012 Structure of dynamical systems a symplectic view of physics Progress in mathematics Dynamische systemen gtt Mécanique statistique ram Mécanique ram Physique mathématique ram Symplectische ruimten gtt Théorie quantique ram Variétés symplectiques ram Mathematische Physik Quantentheorie Mathematical physics Mechanics Quantum theory Statistical mechanics Symplectic manifolds Symplektische Geometrie (DE-588)4194232-2 gnd Theoretische Physik (DE-588)4117202-4 gnd |
| subject_GND | (DE-588)4194232-2 (DE-588)4117202-4 |
| title | Structure of dynamical systems a symplectic view of physics |
| title_alt | Structure des systèmes dynamiques |
| title_auth | Structure of dynamical systems a symplectic view of physics |
| title_exact_search | Structure of dynamical systems a symplectic view of physics |
| title_full | Structure of dynamical systems a symplectic view of physics J.-M. Souriau |
| title_fullStr | Structure of dynamical systems a symplectic view of physics J.-M. Souriau |
| title_full_unstemmed | Structure of dynamical systems a symplectic view of physics J.-M. Souriau |
| title_short | Structure of dynamical systems |
| title_sort | structure of dynamical systems a symplectic view of physics |
| title_sub | a symplectic view of physics |
| topic | Dynamische systemen gtt Mécanique statistique ram Mécanique ram Physique mathématique ram Symplectische ruimten gtt Théorie quantique ram Variétés symplectiques ram Mathematische Physik Quantentheorie Mathematical physics Mechanics Quantum theory Statistical mechanics Symplectic manifolds Symplektische Geometrie (DE-588)4194232-2 gnd Theoretische Physik (DE-588)4117202-4 gnd |
| topic_facet | Dynamische systemen Mécanique statistique Mécanique Physique mathématique Symplectische ruimten Théorie quantique Variétés symplectiques Mathematische Physik Quantentheorie Mathematical physics Mechanics Quantum theory Statistical mechanics Symplectic manifolds Symplektische Geometrie Theoretische Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007841484&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
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