hamilton_jacobi

# hamilton_jacobi - HAMILTON-JACOBI THEORY G OAL Find a...

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Unformatted text preview: HAMILTON-JACOBI THEORY G OAL : Find a particular canonical transformation such that the “new” Hamiltonian is a function only of the “new” momenta. M ATHEMATICAL PRELIMINARIES A canonical transformation may be derived from a generating function . Arguments of a generating function mix “old” and “new” variables. e.g., old momenta, new displacements S(q*,p) Differentiation yields “old” displacements and “new” momenta. p* = – ¡ S/ ¡ q* q = – ¡ S/ ¡ p Mod. Sim. Dyn. Sys. Hamilton-Jacobi theory page 1 There are three other possible generating functions: S(p*,q) q* = ¡ S/ ¡ p* p = ¡ S/ ¡ q S(q*,q) p* = – ¡ S/ ¡ q* p = ¡ S/ ¡ q S(p*,p) q* = ¡ S/ ¡ p* q = – ¡ S/ ¡ p Mod. Sim. Dyn. Sys. Hamilton-Jacobi theory page 2 “O LD ” H AMILTONIAN H(q 1 , ..., q n , p 1 , ... , p n ) To find the required transformation to the “new” variables, use a generating function S(q 1 , ..., q n , p* 1 , ... , p* n ) from which p j = ¡ S/ ¡ q j “N EW ” H AMILTONIAN...
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hamilton_jacobi - HAMILTON-JACOBI THEORY G OAL Find a...

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