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The book is a revised and updated version of the lectures given by the author at the University of Timi
oara during the academic year 1990-1991. Its goal is to present in detail someold and new aspects ofthe geometry ofsymplectic and Poisson manifolds and to point out some of their applications in Hamiltonian mechanics and geometric quantization. The material is organized as follows. In Chapter 1 we collect some general facts about symplectic vector spaces, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study ofHamiltonian mechanics. We present here the gen eral theory ofHamiltonian mechanicalsystems, the theory ofthe corresponding Pois son bracket and also some examples ofinfinite-dimensional Hamiltonian mechanical systems. Chapter 3 starts with some standard facts concerning the theory of Lie groups and Lie algebras and then continues with the theory ofmomentum mappings and the Marsden-Weinstein reduction. The theory of Hamilton-Poisson mechan ical systems makes the object of Chapter 4. Chapter 5 js dedicated to the study of the stability of the equilibrium solutions of the Hamiltonian and the Hamilton Poisson mechanical systems. We present here some of the remarcable results due to Holm, Marsden, Ra~iu and Weinstein. Next, Chapter 6 and 7 are devoted to the theory of geometric quantization where we try to solve, in a geometrical way, the so called Dirac problem from quantum mechanics. We follow here the construc tion given by Kostant and Souriau around 1964.
Klappentext
This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated.br/ 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.br/ Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems.br/ For graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics.br/
Inhalt
1 Symplectic Geometry.- 1.1 Symplectic Algebra.- 1.2 Symplectic Geometry.- 1.3 Darboux's Theorem.- 1.4 Symplectic Reduction.- 1.5 Problems and Solutions.- 2 Hamiltonian Mechanics.- 2.1 Hamiltonian Mechanical Systems.- 2.2 Poisson Bracket.- 2.3 Infinite Dimensional Hamiltonian Mechanical Systems.- 2.4 Problems and Solutions.- 3 Lie Groups. Momentum Mappings. Reduction.- 3.1 Lie Groups.- 3.2 Actions of Lie Groups.- 3.3 The Momentum Mapping.- 3.4 Reduction of Symplectic Manifolds.- 3.5 Problems and Solutions.- 4 Hamilton-Poisson Mechanics.- 4.1 Poisson Geometry.- 4.2 The Lie-Poisson Structure.- 4.3 Hamilton-Poisson Mechanical Systems.- 4.4 Reduction of Poisson Manifolds.- 4.5 Problems and Solutions.- 5 Hamiltonian Mechanical Systems and Stability.- 5.1 The Meaning of Stability.- 5.2 Hamilton's Equations and Stability.- 5.3 The Energy-Casimir Method.- 5.4 Problems and Solutions.- 6 Geometric Prequantization.- 6.1 Full Quantization and Dirac Problem.- 6.2 Complex Bundles and the Dirac Problem.- 6.3 Geometric Prequantization.- 6.4 Problems and Solutions.- 7 Geometric Quantization.- 7.1 Polarizations and the First Attempts to Quantization.- 7.2 Half-Forms Correction of Geometric Quantization.- 7.3 The Non-Existence Problem.- 7.4 Problems and Solutions.- 8 Foliated Cohomology and Geometric Quantization.- 8.1 Real Foliations and Differential Forms.- 8.2 Complex Foliations and Differential Forms.- 8.3 Complex Elliptic Foliations and Spectral Geometry.- 8.4 Cohomological Correction of Geometric Quantization.- 8.5 Problems and Solutions.- 9 Symplectic Reduction. Geometric Quantization. Constrained Mechanical Systems.- 9.1 Symplectic Reduction and Geometric Prequantization.- 9.2 Symplectic Reduction and Geometric Quantization.- 9.3 Applications to Constrained MechanicalSystems.- 9.4 Problems and Solutions.- 10 Poisson Manifolds and Geometric Prequantization.- 10.1 Groupoids.- 10.2 Symplectic Groupoids.- 10.3 Geometric Prequantization of Poisson Manifolds.- 10.4 Problems and Solutions.- References.