Hamiltonian reduction by stages:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2007
|
| Schriftenreihe: | Lecture Notes in Mathematics
1913 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 519 S. graph. Darst. |
| ISBN: | 9783540724698 |
Internformat
MARC
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| 100 | 1 | |a Marsden, Jerrold E. |d 1942-2010 |e Verfasser |0 (DE-588)124171141 |4 aut | |
| 245 | 1 | 0 | |a Hamiltonian reduction by stages |c Jerrold E. Marsden ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2007 | |
| 300 | |a XV, 519 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Lecture Notes in Mathematics |v 1913 | |
| 650 | 7 | |a Dynamische systemen |2 gtt | |
| 650 | 7 | |a Hamilton-vergelijkingen |2 gtt | |
| 650 | 7 | |a Sistemas hamiltonianos |2 larpcal | |
| 650 | 7 | |a Symplectische ruimten |2 gtt | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 4 | |a Équations différentielles | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 0 | 7 | |a Ordnungsreduktion |0 (DE-588)4136085-0 |2 gnd |9 rswk-swf |
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| 830 | 0 | |a Lecture Notes in Mathematics |v 1913 |w (DE-604)BV000676446 |9 1913 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-015739767 | ||
Datensatz im Suchindex
| _version_ | 1804136643719856128 |
|---|---|
| adam_text | Contents
Part I: Background and the Problem Setting
1
Symplectic Reduction
3
1.1
Introduction to Symplectic Reduction
........... 3
1.2
Symplectic Reduction
-
Proofs and Further Details
.... 12
1.3
Reduction Theory: Historical Overview
........... 24
1.4
Overview of Singular Symplectic Reduction
........ 36
Cotangent Bundle Reduction
43
2.1
Principal Bundles and Connections
............. 43
2.2
Cotangent Bundle Reduction: Embedding Version
..... 59
2.3
Cotangent Bundle Reduction: Bundle Version
....... 71
2.4
Singular Cotangent Bundle Reduction
........... 88
The Problem Setting
101
3.1
The Setting for Reduction by Stages
............ 101
3.2
Applications and Infinite Dimensional Problems
...... 106
Part II: Regular Symplectic Reduction by Stages 111
4
Commuting Reduction and Semidirect Product Theory
113
4.1
Commuting Reduction
....................113
4.2
Semidirect Products
.....................119
XIV Contents
4.3
Cotangent Bundle Reduction and Semidirect Products
. . 132
4.4
Example: The Euclidean Group
............... 137
5
Regular Reduction by Stages
143
5.1
Motivating Example: The
Heisenberg
Group
........ 144
5.2
Point Reduction by Stages
.................. 149
5.3
Poisson
and Orbit Reduction by Stages
........... 171
6
Group Extensions and the Stages Hypothesis
177
6.1
Lie Group and Lie Algebra Extensions
........... 178
6.2
Central Extensions
...................... 198
6.3
Group Extensions Satisfy the Stages Hypotheses
..... 201
6.4
The Semidirect Product of Two Groups
.......... 204
7
Magnetic Cotangent Bundle Reduction
211
7.1
Embedding Magnetic Cotangent Bundle Reduction
.... 212
7.2
Magnetic Lie-Poisson and Orbit Reduction
......... 225
8
Stages and Coadjoint Orbits of Central Extensions
239
8.1
Stage One Reduction for Central Extensions
........ 240
8.2
Reduction by Stages for Central Extensions
........ 245
9
Examples
251
9.1
The
Heisenberg
Group Revisited
.............. 252
9.2
A Central Extension of L(Sil)
................ 253
9.3
The Oscillator Group
..................... 259
9.4
Bott-Virasoro Group
..................... 267
9.5
Fluids with a Spatial Symmetry
............... 279
10
Stages and Semidirect Products with Cocycles
285
10.1
Abelian Semidirect Product Extensions:
First Reduction
........................ 286
10.2
Abelian Semidirect Product Extensions:
Coadjoint Orbits
....................... 295
10.3
Coupling to a Lie Group
................... 304
10.4
Poisson
Reduction by Stages:
General Semidirect Products
................ 309
10.5
First Stage Reduction: General Semidirect Products
. . . 315
10.6
Second Stage Reduction: General Semidirect Products
. . 321
10.7
Example: The Group T®U
................. 347
11
Reduction by Stages via Symplectic Distributions
397
11.1
Reduction by Stages of Connected Components
...... 398
11.2
Momentum Level Sets and Distributions
.......... 401
11.3
Proof: Reduction by Stages II
................ 406
Contents
XV
12
Reduction by Stages with Topological Conditions
409
12.1
Reduction by Stages III
................... 409
12.2
Relation Between Stages II and III
............. 416
12.3
Connected Components of Reduced Spaces
......... 419
Conclusions for Part 1
......................... 420
Part III: Optimal Reduction and Singular Reduction
by Stages, by Juan-Pablo Ortega
421
13
The Optimal Momentum Map and Point Reduction
423
13.1
Optimal Momentum Map and Space
............ 423
13.2
Momentum Level Sets and Associated Isotropies
..... 426
13.3
Optimal Momentum Map Dual Pair
............ 427
13.4
Dual Pairs, Reduced Spaces, and Symplectic Leaves
. . . 430
13.5
Optimal Point Reduction
.................. 432
13.6
The Symplectic Case and Sjamaar s Principle
....... 435
14
Optimal Orbit Reduction
437
14.1
The Space for Optimal Orbit Reduction
.......... 437
14.2
The Symplectic Orbit Reduction Quotient
......... 443
14.3
The Polar Reduced Spaces
.................. 446
14.4
Symplectic Leaves and the Reduction Diagram
...... 454
14.5
Orbit Reduction: Beyond Compact Groups
........ 455
14.6
Examples: Polar Reduction of the Coadjoint Action
.... 457
15
Optimal Reduction by Stages
461
15.1
The Polar Distribution of a Normal Subgroup
....... 461
15.2
Isotropy Subgroups and Quotient Groups
......... 464
15.3
The Optimal Reduction by Stages Theorem
........ 466
15.4
Optimal Orbit Reduction by Stages
............. 470
15.5
Reduction by Stages of Globally Hamiltonian Actions
. . 475
Acknowledgments for Part III
.................... 481
Bibliography
............................................... 483
Index
...................................................... 509
|
| adam_txt |
Contents
Part I: Background and the Problem Setting
1
Symplectic Reduction
3
1.1
Introduction to Symplectic Reduction
. 3
1.2
Symplectic Reduction
-
Proofs and Further Details
. 12
1.3
Reduction Theory: Historical Overview
. 24
1.4
Overview of Singular Symplectic Reduction
. 36
Cotangent Bundle Reduction
43
2.1
Principal Bundles and Connections
. 43
2.2
Cotangent Bundle Reduction: Embedding Version
. 59
2.3
Cotangent Bundle Reduction: Bundle Version
. 71
2.4
Singular Cotangent Bundle Reduction
. 88
The Problem Setting
101
3.1
The Setting for Reduction by Stages
. 101
3.2
Applications and Infinite Dimensional Problems
. 106
Part II: Regular Symplectic Reduction by Stages 111
4
Commuting Reduction and Semidirect Product Theory
113
4.1
Commuting Reduction
.113
4.2
Semidirect Products
.119
XIV Contents
4.3
Cotangent Bundle Reduction and Semidirect Products
. . 132
4.4
Example: The Euclidean Group
. 137
5
Regular Reduction by Stages
143
5.1
Motivating Example: The
Heisenberg
Group
. 144
5.2
Point Reduction by Stages
. 149
5.3
Poisson
and Orbit Reduction by Stages
. 171
6
Group Extensions and the Stages Hypothesis
177
6.1
Lie Group and Lie Algebra Extensions
. 178
6.2
Central Extensions
. 198
6.3
Group Extensions Satisfy the Stages Hypotheses
. 201
6.4
The Semidirect Product of Two Groups
. 204
7
Magnetic Cotangent Bundle Reduction
211
7.1
Embedding Magnetic Cotangent Bundle Reduction
. 212
7.2
Magnetic Lie-Poisson and Orbit Reduction
. 225
8
Stages and Coadjoint Orbits of Central Extensions
239
8.1
Stage One Reduction for Central Extensions
. 240
8.2
Reduction by Stages for Central Extensions
. 245
9
Examples
251
9.1
The
Heisenberg
Group Revisited
. 252
9.2
A Central Extension of L(Sil)
. 253
9.3
The Oscillator Group
. 259
9.4
Bott-Virasoro Group
. 267
9.5
Fluids with a Spatial Symmetry
. 279
10
Stages and Semidirect Products with Cocycles
285
10.1
Abelian Semidirect Product Extensions:
First Reduction
. 286
10.2
Abelian Semidirect Product Extensions:
Coadjoint Orbits
. 295
10.3
Coupling to a Lie Group
. 304
10.4
Poisson
Reduction by Stages:
General Semidirect Products
. 309
10.5
First Stage Reduction: General Semidirect Products
. . . 315
10.6
Second Stage Reduction: General Semidirect Products
. . 321
10.7
Example: The Group T®U
. 347
11
Reduction by Stages via Symplectic Distributions
397
11.1
Reduction by Stages of Connected Components
. 398
11.2
Momentum Level Sets and Distributions
. 401
11.3
Proof: Reduction by Stages II
. 406
Contents
XV
12
Reduction by Stages with Topological Conditions
409
12.1
Reduction by Stages III
. 409
12.2
Relation Between Stages II and III
. 416
12.3
Connected Components of Reduced Spaces
. 419
Conclusions for Part 1
. 420
Part III: Optimal Reduction and Singular Reduction
by Stages, by Juan-Pablo Ortega
421
13
The Optimal Momentum Map and Point Reduction
423
13.1
Optimal Momentum Map and Space
. 423
13.2
Momentum Level Sets and Associated Isotropies
. 426
13.3
Optimal Momentum Map Dual Pair
. 427
13.4
Dual Pairs, Reduced Spaces, and Symplectic Leaves
. . . 430
13.5
Optimal Point Reduction
. 432
13.6
The Symplectic Case and Sjamaar's Principle
. 435
14
Optimal Orbit Reduction
437
14.1
The Space for Optimal Orbit Reduction
. 437
14.2
The Symplectic Orbit Reduction Quotient
. 443
14.3
The Polar Reduced Spaces
. 446
14.4
Symplectic Leaves and the Reduction Diagram
. 454
14.5
Orbit Reduction: Beyond Compact Groups
. 455
14.6
Examples: Polar Reduction of the Coadjoint Action
. 457
15
Optimal Reduction by Stages
461
15.1
The Polar Distribution of a Normal Subgroup
. 461
15.2
Isotropy Subgroups and Quotient Groups
. 464
15.3
The Optimal Reduction by Stages Theorem
. 466
15.4
Optimal Orbit Reduction by Stages
. 470
15.5
Reduction by Stages of Globally Hamiltonian Actions
. . 475
Acknowledgments for Part III
. 481
Bibliography
. 483
Index
. 509 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Marsden, Jerrold E. 1942-2010 |
| author_GND | (DE-588)124171141 |
| author_facet | Marsden, Jerrold E. 1942-2010 |
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| author_sort | Marsden, Jerrold E. 1942-2010 |
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 344f |
| ctrlnum | (OCoLC)154712316 (DE-599)DNB983896836 |
| dewey-full | 514/.74 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology 515 - Analysis |
| dewey-raw | 514/.74 515.39 |
| dewey-search | 514/.74 515.39 |
| dewey-sort | 3514 274 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:07:24Z |
| indexdate | 2024-07-09T20:59:40Z |
| institution | BVB |
| isbn | 9783540724698 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015739767 |
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| series2 | Lecture Notes in Mathematics |
| spelling | Marsden, Jerrold E. 1942-2010 Verfasser (DE-588)124171141 aut Hamiltonian reduction by stages Jerrold E. Marsden ... Berlin [u.a.] Springer 2007 XV, 519 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Mathematics 1913 Dynamische systemen gtt Hamilton-vergelijkingen gtt Sistemas hamiltonianos larpcal Symplectische ruimten gtt Systèmes hamiltoniens Équations différentielles Differential equations Hamiltonian systems Ordnungsreduktion (DE-588)4136085-0 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Ordnungsreduktion (DE-588)4136085-0 s Symplektische Geometrie (DE-588)4194232-2 s DE-604 Lecture Notes in Mathematics 1913 (DE-604)BV000676446 1913 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739767&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marsden, Jerrold E. 1942-2010 Hamiltonian reduction by stages Lecture Notes in Mathematics Dynamische systemen gtt Hamilton-vergelijkingen gtt Sistemas hamiltonianos larpcal Symplectische ruimten gtt Systèmes hamiltoniens Équations différentielles Differential equations Hamiltonian systems Ordnungsreduktion (DE-588)4136085-0 gnd Symplektische Geometrie (DE-588)4194232-2 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| subject_GND | (DE-588)4136085-0 (DE-588)4194232-2 (DE-588)4139943-2 |
| title | Hamiltonian reduction by stages |
| title_auth | Hamiltonian reduction by stages |
| title_exact_search | Hamiltonian reduction by stages |
| title_exact_search_txtP | Hamiltonian reduction by stages |
| title_full | Hamiltonian reduction by stages Jerrold E. Marsden ... |
| title_fullStr | Hamiltonian reduction by stages Jerrold E. Marsden ... |
| title_full_unstemmed | Hamiltonian reduction by stages Jerrold E. Marsden ... |
| title_short | Hamiltonian reduction by stages |
| title_sort | hamiltonian reduction by stages |
| topic | Dynamische systemen gtt Hamilton-vergelijkingen gtt Sistemas hamiltonianos larpcal Symplectische ruimten gtt Systèmes hamiltoniens Équations différentielles Differential equations Hamiltonian systems Ordnungsreduktion (DE-588)4136085-0 gnd Symplektische Geometrie (DE-588)4194232-2 gnd Hamiltonsches System (DE-588)4139943-2 gnd |
| topic_facet | Dynamische systemen Hamilton-vergelijkingen Sistemas hamiltonianos Symplectische ruimten Systèmes hamiltoniens Équations différentielles Differential equations Hamiltonian systems Ordnungsreduktion Symplektische Geometrie Hamiltonsches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015739767&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT marsdenjerrolde hamiltonianreductionbystages |