1 / 7

Hamilton-Jacobi

Hamilton-Jacobi. 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. The easiest integration for H is if H is independent of all variables.

benard
Download Presentation

Hamilton-Jacobi

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related