The Attempt at a Solution
So far I have a solution for a) as
For b) I formulate the equation as
and so far for c) I have
My main idea at the moment is that as the Lagrangian was not time dependent, the Hamiltonian will not be. Following on maybe I can say that when separating variables T is constant so I can separate it so
Any ideas, or are my problems due to a mistake in working?