Hamiltonian Reduction by Stages

Jerrold E. Marsden, Gerard Misiolek, Juan-Pablo Ortega, Matthew Perlmutter, Tudor S. Ratiu (auth.)3540724699, 9783540724704, 9783540724698

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.

Table of contents :
Front Matter….Pages I-XV
Front Matter….Pages 1-1
Symplectic Reduction….Pages 3-42
Cotangent Bundle Reduction….Pages 43-99
The Problem Setting….Pages 101-109
Front Matter….Pages 111-111
Commuting Reduction and Semidirect Product Theory….Pages 113-142
Regular Reduction by Stages….Pages 143-175
Group Extensions and the Stages Hypothesis….Pages 177-210
Magnetic Cotangent Bundle Reduction….Pages 211-237
Stages and Coadjoint Orbits of Central Extensions….Pages 239-250
Examples….Pages 251-283
Stages and Semidirect Products with Cocycles….Pages 285-396
Reduction by Stages via Symplectic Distributions….Pages 397-407
Reduction by Stages with Topological Conditions….Pages 409-420
Front Matter….Pages 421-422
The Optimal Momentum Map and Point Reduction….Pages 423-436
Optimal Orbit Reduction….Pages 437-459
Optimal Reduction by Stages….Pages 461-481
Back Matter….Pages 483-523