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Hamilton-Jacobi

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  1. Hamilton-Jacobi

  2. Time-Dependent Generator • A generator determines a canonical transformation. • The transform generally changes the form of H. • If time-dependent, the value of H also changes.

  3. The easiest integration for H is if H is independent of all variables. Select f to give that result Coordinates and momenta are constants of the motion. 2fconstants Independent Hamiltonian

  4. Hamilton-Jacobi Equation • Hamilton-Jacobi is a partial differential equation. • First order, generally second degree • f+1 independent variables: qj, t • f+1 constants, one additive, others are ak.

  5. Principal Function • The Lagrangian is directly related to the generator. • Generator f is Hamilton’s principal function since setting one set at t and the other at t1:

  6. The action is defined when H does not involve time. f is the principal function Additive constant is possible Subsitute to simplify HJ equation. Time-independent H Time variable separates E is not independent Principal Function and Action

  7. Path Equations • Choose energy symmetrically • Simplifies action relations • Gives f parameterized equations • One can pick one for E. • Path described without time • One equation for location on the path next