Time Dependant Hamiltonian Jacob question

[Liquidxlax]

Homework Statement

I have a question given to me by my prof that is a time dependent Hamiltonian

H(q,p,t) = g(t)(p²/(2m) + kq²/2)

where f(t) has 2 different forms i need to solve

1) e^at 2) cos(gt)

problem is goldstein only covers conserved hamiltonians in chapter 10 for the H-J equations and we have no notes in the class about them. My prof doesn't like to answer questions?

Can someone give me some insight how to deal with the time dependent function?

Homework Equations

H(q, ∂S/∂q , t) + ∂S/∂t = 0 i think... but like i said no material on the time dependance

The Attempt at a Solution

 

[Liquidxlax]

it's all right i figured it out