For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as
$$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$, and finally find $S$, then find the positions or whatever needed by differentiating $W$ with respect to the constant.
But, when the equation is not separable we need to solve the equation numerically and for that initial conditions shall be needed. What are the initial conditions for Hamilton-Jacobi equation, say, for a simple kinetic energy plus potential energy type hamiltonian, and how to find it?
And from the boundary condition, after finding $S$, how can we find the characteristics of the motion, say position as a function of time?
For example, I wish to solve the HJ equation in harmonic oscillator numerically to get position and momentum as functions of time under the initial condition:
At $$t=0,$$ $$x=1$$ $$p=0$$
Now how shall I proceed numerically, not asking the numerical method of solving the pde, but how to impose the initial conditions on action, and after getting action as a function of position and momentum, how to get $$x$$ and $$p$$ as functions of time.