Hamiltonian Mechanical Systems and Geometric Quantization

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.

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.