Hamilton-Jacobi. Time-Dependent Generator  A generator determines a canonical transformation. The transform generally changes the form of H.The transform.

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Presentation transcript:

Hamilton-Jacobi

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

Independent Hamiltonian  The easiest integration for H is if H is independent of all variables. Select  to give that resultSelect  to give that result  Coordinates and momenta are constants of the motion. 2f constants2f constants

Hamilton-Jacobi Equation  Hamilton-Jacobi is a partial differential equation. First order, generally second degreeFirst order, generally second degree f+1 independent variables: q j, tf+1 independent variables: q j, t f+1 constants, one additive, others are  k.f+1 constants, one additive, others are  k.

Principal Function  The Lagrangian is directly related to the generator. Generator  is Hamilton’s principal functionGenerator  is Hamilton’s principal function since setting one set at t and the other at t 1 :

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

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