Hamilton-Jacobi Problem

1. The problem statement, all variables and given/known data



I'm given the time-dependent potential,



[tex] V(x,t) = -mAxe^{-\gamma t} [/tex]



and asked to find the solution to the Hamilton-Jacobi equation,



[tex] H(x,\frac{\partial S}{\partial x}) + \frac{ \partial S}{\partial t} = 0 [/tex]





3. The attempt at a solution



Without any additional information, I'm assuming the correct Hamiltonian is given simply by,



[tex] H = \frac{p^2}{2m} -mAxe^{-\gamma t} [/tex]



which gives me,



[tex] \frac{1}{2m}\bigg ( \frac{\partial S}{\partial x} \bigg )^2 - mAxe^{-\gamma t} + \frac{ \partial S}{\partial t} = 0 [/tex]



but I'm having troule separating the variables in order to solve this equation. Normally, when [itex] V = V(x) [/itex] you can use the form [itex] S(x,\alpha,t) = W(x,\alpha) - Et [/itex], but here this won't work.



Have I somehow used the wrong Hamiltonian, or do I just need to guess correctly the right form of [itex] S [/itex]?