Hamiltonian Mechanical Systems and Geometric Quantization

This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden--Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton--Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault--Kostant complex, and their applications. A discussion of the relation between geometric quantization and the Marsden--Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids. Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. For graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics. Front Matter....Pages i-viii Symplectic Geometry....Pages 1-27 Hamiltonian Mechanics....Pages 28-51 Lie Groups. Momentum Mappings. Reduction.....Pages 52-95 Hamilton-Poisson Mechanics....Pages 96-133 Hamiltonian Mechanical Systems and Stability....Pages 134-156 Geometric Prequantization....Pages 157-181 Geometric Quantization....Pages 182-206 Foliated Cohomology and Geometric Quantization....Pages 207-235 Symplectic Reduction. Geometric Quantization. Constrained Mechanical Systems....Pages 236-249 Poisson Manifolds and Geometric Prequantization....Pages 250-261 Back Matter....Pages 262-280