| 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 |
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