From Newton–Euler to Hamilton–Jacobi

Let’s say that we want to reformulate “Newtonian” particle mechanics so that it looks analogous to wave motion, in order to make our knowledge of one math subject applicable to another. A light ray, minding its own business, propagates in the direction perpendicular to its wavefronts. So, the particle momentum should be perpendicular to lines of something, meaning that it should be the gradient of something:
\vec{p} = \vec{\nabla} S \, .
We don’t know what $S$ is, except that it’s a field whose value presumably depends upon position and time, and it ought to satisfy some equation. Can we find an equation for it?
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