Hamiltonian Reduction by Stages

In this volume readers will find for the first time a detailed account of the theory of symplectic reduction by stages, along with numerous illustrations of the theory. Special emphasis is given to group extensions, including a detailed discussion of the Euclidean group, the oscillator group, the Bott-Virasoro group and other groups of matrices. Ample background theory on symplectic reduction and cotangent bundle reduction in particular is provided. Novel features of the book are the inclusion of a systematic treatment of the cotangent bundle case, including the identification of cocycles with magnetic terms, as well as the general theory of singular reduction by stages.

Background and the Problem Setting	p. 1
Symplectic Reduction	p. 3
Introduction to Symplectic Reduction	p. 3
Symplectic Reduction - Proofs and Further Details	p. 12
Reduction Theory: Historical Overview	p. 24
Overview of Singular Symplectic Reduction	p. 36
Cotangent Bundle Reduction	p. 43
Principal Bundles and Connections	p. 43
Cotangent Bundle Reduction: Embedding Version	p. 59
Cotangent Bundle Reduction: Bundle Version	p. 71
Singular Cotangent Bundle Reduction	p. 88
The Problem Setting	p. 101
The Setting for Reduction by Stages	p. 101
Applications and Infinite Dimensional Problems	p. 106
Regular Symplectic Reduction by Stages	p. 111
Commuting Reduction and Semidirect Product Theory	p. 113
Commuting Reduction	p. 113
Semidirect Products	p. 119
Cotangent Bundle Reduction and Semidirect Products	p. 132
Example: The Euclidean Group	p. 137
Regular Reduction by Stages	p. 143
Motivating Example: The Heisenberg Group	p. 144
Point Reduction by Stages	p. 149
Poisson and Orbit Reduction by Stages	p. 171
Group Extensions and the Stages Hypothesis	p. 177
Lie Group and Lie Algebra Extensions	p. 178
Central Extensions	p. 198
Group Extensions Satisfy the Stages Hypotheses	p. 201
The Semidirect Product of Two Groups	p. 204
Magnetic Cotangent Bundle Reduction	p. 211
Embedding Magnetic Cotangent Bundle Reduction	p. 212
Magnetic Lie-Poisson and Orbit Reduction	p. 225
Stages and Coadjoint Orbits of Central Extensions	p. 239
Stage One Reduction for Central Extensions	p. 240
Reduction by Stages for Central Extensions	p. 245
Examples	p. 251
The Heisenberg Group Revisited	p. 252
A Central Extension of L(S[superscript 1])	p. 253
The Oscillator Group	p. 259
Bott-Virasoro Group	p. 267
Fluids with a Spatial Symmetry	p. 279
Stages and Semidirect Products with Cocycles	p. 285
Abelian Semidirect Product Extensions: First Reduction	p. 286
Abelian Semidirect Product Extensions: Coadjoint Orbits	p. 295
Coupling to a Lie Group	p. 304
Poisson Reduction by Stages: General Semidirect Products	p. 309
First Stage Reduction: General Semidirect Products	p. 315
Second Stage Reduction: General Semidirect Products	p. 321
Example: The Group T [circledS] U	p. 347
Reduction by Stages via Symplectic Distributions	p. 397
Reduction by Stages of Connected Components	p. 398
Momentum Level Sets and Distributions	p. 401
Proof: Reduction by Stages II	p. 406
Reduction by Stages with Topological Conditions	p. 409
Reduction by Stages III	p. 409
Relation Between Stages II and III	p. 416
Connected Components of Reduced Spaces	p. 419
Conclusions for Part I	p. 420
Optimal Reduction and Singular Reduction by Stages	p. 421
The Optimal Momentum Map and Point Reduction	p. 423
Optimal Momentum Map and Space	p. 423
Momentum Level Sets and Associated Isotropies	p. 426
Optimal Momentum Map Dual Pair	p. 427
Dual Pairs, Reduced Spaces, and Symplectic Leaves	p. 430
Optimal Point Reduction	p. 432
The Symplectic Case and Sjamaar's Principle	p. 435
Optimal Orbit Reduction	p. 437
The Space for Optimal Orbit Reduction	p. 437
The Symplectic Orbit Reduction Quotient	p. 443
The Polar Reduced Spaces	p. 446
Symplectic Leaves and the Reduction Diagram	p. 454
Orbit Reduction: Beyond Compact Groups	p. 455
Examples: Polar Reduction of the Coadjoint Action	p. 457
Optimal Reduction by Stages	p. 461
The Polar Distribution of a Normal Subgroup	p. 461
Isotropy Subgroups and Quotient Groups	p. 464
The Optimal Reduction by Stages Theorem	p. 466
Optimal Orbit Reduction by Stages	p. 470
Reduction by Stages of Globally Hamiltonian Actions	p. 475
Acknowledgments for Part III	p. 481
Bibliography	p. 483
Index	p. 509
Table of Contents provided by Ingram. All Rights Reserved.

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