Classical Dynamics

A Contemporary Approach

by Jorge V. José and Eugene Saletan
Classical Dynamics Book

Chapter Five:

Hamiltonian Formulation of Mechanics






Fig 4.10


Hamilton's canonical equations
  1. 1. local considerations: from the Lagrangian to the Hamiltonian, a brief review of special relativity, the relativistic Kepler problem
  2. 2. the Legendre transform
  3. 3. unified coordinates on T*Q and Poisson brackets: the xi notation, variational derivation of Hamilton's equations, Poisson brackets, Poisson brackets and Hamiltonian dynamics

Symplectic geometry
  1. 1. the cotangent manifold
  2. 2. two-forms
  3. 3. the symplectic form omega

Canonical transformations
  1. 1. local considerations: reduction on T*Q by constants of the motion, definition of canonical transformations, changes induced by canonical transformations, two examples
  2. 2. intrinsic approach
  3. 3. generating functions of canonical transformations: generating functions, the generating function gives the new Hamiltonian, generating functions of type
  4. 4. one-parameter groups of canonical transformations: infinitesimal generators of one-parameter groups; Hamiltonian flows, the Hamiltonian Noether theorem, flows and Poisson brackets

Two theorems: Liouville and Darboux
  1. 1. Liouville's volume theorem: volume, integration on T*Q; the Liouville theorem, Poincaré invariants, density of states
  2. 2. Darboux's theorem: the theorem, reduction

Appendix: canonicity implies PB preservation
Problems
:: back to main :: previous chapter . . . . . . . . . . . . . next chapter ::
ter,