Hamilton Jacobi equation for time dependent potential

[DanielA]

Homework Statement

Suppose the potential in a problem of one degree of freedom is linearly dependent upon time such that
$$H = \frac{p^2}{2m} - mAtx $$ where A is a constant. Solve the dynamical problem by means of Hamilton's principal function under the initial conditions t = 0, x = 0, $p = mv_0$

Homework Equations

Hamilton's Principal function is of from S(x, P, t) or $S(x,\alpha,t)$ where alpha is a constant of motion.

Hamilton Jacobi equation:
$$\frac{1}{2m}\frac{\partial S}{\partial x}^2 - mAtx + \frac{\partial S}{\partial t} = 0$$

The Attempt at a Solution

In class today my professor admitted he just trial and errored to find S. I've been trying to do it, but I can't find any working solution. I recognize S will be some polynomial in form, with multiple mixed terms, but the squared term messed up any simpler solution I've tried. I've only attempted to get a similar functional form and iterate it and mess with constants out front to get it exact.
I've tried quadratic forms like $(x-t)^2$ and similar with the goal of crossing out the $x^2$ and $t^2$ terms to leave only the cross terms. I briefly considered attempting a Fourier Transform style solution like we were learning in Math Methods before realizing it wouldn't work.

I found a solution on Stack Exchange, but they use the solution obtained by hamilton's equations to obtain the principal function which just defeats the purpose it doing this. I know hamilton jacobi theory heavily favors time independent hamiltonians, but doing the above just makes me think that the whole question was pointless. https://physics.stackexchange.com/q...cobi-equation-with-time-dependent-hamiltonian

I'm just trying to find a method that can deal with the square or maybe a discussion on how to make educated guesses on a solution. All my searching of my textbooks and google have only given me solutions to linear partial differential equations which this one isn't because of the squared term.

 

[TSny]

This is probably not going to be too satisfying. But, if you can motivate the ansatz $S(x,t) = xf(t) + g(t)$, then it is not hard to determine functions $f(t)$ and $g(t)$ that will work.

Note that with this ansatz, $\frac {\partial S}{\partial t} = x f'(t) + g'(t)$

$f'(t)$ can then be chosen so that the $xf'(t)$ term cancels the middle term of the Hamilton-Jacobi equation. Also, $\frac {\partial S}{\partial x} = f(t)$ and will therefore be a function of $t$ alone. So, the $g'(t)$ term in $\frac {\partial S}{\partial t}$ can be chosen to cancel the first term in the H-J equation.

Hopefully, someone else can provide a better approach.