This is a problem from my Classical Mechanics course, and it is: Suppose the potential in a problem of one degree of freedom is linearly dependent upon time, such that the Hamiltonian has the form

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where $A$ is a constant. Solve the dynamical problem by means of the Hamiltonian-Jacobi equation, under the initial conditions $q=0$, $p=mv_0$ at $t=0$.

I substituted the Hamiltonian into the Hamilton-Jacobi's equation:

enter image description here

However I do not know how to solve for S.

  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/109598/2451 $\endgroup$
    – Qmechanic
    Apr 19 '17 at 19:04
  • 2
    $\begingroup$ Thank you Qmechanics. The answer you presented requires known solution to Hamiltonian, which seems to have defeated the purpose of the question, which is to use the H-J equation. Is there are way of solving for S if I do not know the solution to Hamiltonian? $\endgroup$ Apr 19 '17 at 19:23

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