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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.

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The Hamilton–Jacobi (HJ) equation is a non-linear 1st order PDE. The flow is typically found using the method of characteristics starting from some initial conditions on a suitable (codimension-1) Cauchy hypersurface. Keep in mind that the flow parameter might not be physical time, and the word "initial" therefore should be interpreted accordingly.

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  • $\begingroup$ Could you please recommend or show any articke where I can find a concrete example of how this is done ? Not method of solving, I am asking how to choose the initial conditions . $\endgroup$ – user157588 Jun 25 at 11:21
  • $\begingroup$ FWIW, there is a huge literature on non-linear 1st order PDE. $\endgroup$ – Qmechanic Jun 26 at 8:13
  • $\begingroup$ I did not ask for general methods for initial conditions . Suppose a ball is in a harmonic potential, and i wish to solve the h-j equation numerically, if I know the position and momentum of the ball initially, how shall I impose the initial condition in H-J equation, in terms of the action . I am asking that . $\endgroup$ – user157588 Jun 28 at 6:13

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