Hamiltonian mechanics- show that the following are extremals...
The problem statement :
Consider the variational problem with lagrangian function
Show that the extremals are given by the two-parameter family of curves,
, where alpha and beta are the parameters, by means of the following 3 methods:
a) transform L to polar coordinates,
b) using canonical variables,
c)using the Hamilton-Jacobi equation.
Hint for c): Let be a solution of the hamilton-jacobi equation. This will show that is a solution of the hamilton jacobi equation. Now prove that is a solution of the corresponding hamilton's equations.
For a) using and finding i substitute and get L in polar form. That's easy, but now how to check that the given extremal satisfies this polar form? I think i check that the extremal satisfies the Euler - lagrange equation for L polar ? But i'm not sure of some things, like do i first solve for x in the extremal and use this as my general coordinate in the euler-lagrange equation? I also don't know how to get where x would then be the expression for the extremal? Please help me on how to think about this !
for b) really not sure here, but do i just show that for the given extremal, Hamilton's canonical equations are satisfied ? Again should i be solving for x in the given extremal to use in hamilton's equations? If so how do i get the partial with respect to that expression ?
for c) I sub The given form of S into the Hamilton-Jacobi equation to get the 2nd form of S ( in the hints) . Now will me proving somehow yield the given extremal? I would think so, but again the math eludes me.
Any help,especially on the thinking behind all this, is much appreciated.