Hamiltonian Mechanical Systems and Geometric Quantization

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.