Hamiltonian Mechanical Systems and Geometric Quantization

Inhaltsverzeichnis

• 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.