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Preliminary syllabus
0. Introduction
- From Aristotle to Newton to Lagrange to Hamilton to Poisson to Cartan
- Dynamics: Do we need Lagrangian or Hamiltonian?
1. Differential Geometry
- Basic manifolds: configuration and phase spaces, fiber bundles, jets, Aristotelian, Galilean and Minkowski space-time, etc
- Objects: differential forms and vector fields, geometric structures on manifold, etc.
- Natural operations: exterior derivative, Lie derivative, Schouten-Nijenhuis bracket, etc.
2. Dynamical systems and symplectic geometry (the Hamilton equations)
- Geometry of cotangent bundle; symplectic manifolds and Hamilton eqns
- Poisson manifolds and the Schouten-Nijenhuis bracket
- Liouville and Arnold on integrability; Noether theorem(1), Poincare invariants
- Other applications: Optics, Thermodynamics, Berry's phase, Schroedinger eqn
3. Tangent geometry
- Tangent bundle, tangent endomorphism, Euler-Lagrange equation:
LXq = dL
- Noether theorem reconsidered (2); geometry of Legendre map
- The inverse problem and ambiguity of Lagrangian desciption
4. Cartan's description -- generalized
- Pre-symplectic geometry, contact structure, Reeb's Theorem
- Particle-wave duality; the meaning of the Lagrangian and Hamiltonian
- Lagrange and Hamilton eqns unified:
iXda = 0
- Noether theorem (3); Lagrangian submanifolds and Jacobi-Hamilton equations
- Seven myths concerning CM
5. New ideas and alternative analytic descriptions
- Nambu Mechanics
- Bi-Hamiltonian systems
- Lax equation; Yang-Baxter eqns
- Generalizations: statistical physics; classical field theory
- More geometry: Nijenhuis and Schouten brackets, super-algebras
6. The geometry of a Lie algebra and dynamical systems
- Lie algebra as a manifold, orbits of (co-)adjoint action, Lie-Poisson structure
- Symmetries, momentum map; integrabe systems
- (Example: spinning top -- from Euler eqns to Lax eqns and beyond)
- Gauge theory and fiber bundles (Wong equation)
7. What is "quantization"
- Geometric quantization; Moyal bracket (deformation of Poisson brackets)
- Other alternative approaches
- Quantum groups; perspectives
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