Part I: Background and the Problem Setting .................... 1
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
2. 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
3. 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
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  (S1) ........................ 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  ............................ 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
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 Isotropics ....... 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
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