Classical Dynamics

1. 1. the Hamilton-Jacobi equation: derivation, properties of solutions, relation to the action
2. 2. separation of variables: the method of separation, example: charged particle in a magnetic field
3. 4. geometry and the HJ equation
4. 5. the analogy between optics and the HJ method

1. 1. action-angle variables: invariant tori, the canonical transformation to AA variables, example: a particle on a vertical cylinder
2. 2. Liouville's integrability theorem: complete integrability, the tori, the J's, example: the Neumann problem
3. 3. motion on the tori: rational and irrational winding lines, Fourier series

1. 1. example: the Quartic oscillator
2. 2. Hamiltonian perturbation theory: perturbation via canonical transformations, averaging, canonical perturbation theory in one freedom, canonical perturbation theory in many freedoms, the Lie transformation method, example: the quartic oscillator

1. 1. adiabatic theorem: oscillator with time-dependent frequency, the theorem, remarks on N >1
2. 2. higher approximations
3. 3. the Hannay angle
4. 4. motion of a charged particle in a magnetic field: the action integral, three magnetic adiabatic invariants

Problems