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. TOC:Preface.- Part I: Background and the Problem Setting.- 1. Symplectic Reduction.- 2. Cotangent Bundle Reduction.- 3. The Problem Setting.- Part II: Regular Symplectic Reduction by Stages.- 4. Commuting Reduction and Semidirect Product Theory.- 5. Regular Reduction by Stages.- 6. Group Extensions and the Stages Hypothesis.- 7. Magnetic Cotangent Bundle Reduction.- 8. Stages and Coadjoint Orbits of Central Extensions.- 9. Examples.- 10. Stages and Semidirect Products with Cocycles.- 11. Reduction by Stages via Symplectic Distributions.- 12. Reduction by Stages with Topological Conditions.- Part III: Optimal Reduction and Singular Reduction by Stages, by J.-P. Ortega.- 13. The Optimal Momentum Map and Point Reduction.- 14. Optimal Orbit Reduction.- 15. Optimal Reduction by Stages.- Bibliography.- Index. 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. 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 L(S ) 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 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 I 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