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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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.141 taught by Professor Nevillehogan during the Fall '06 term at MIT.

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

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