Fooling around with Hamilton-Jacobi theory: can this be right?

[pellman]

A system with one particle in one dimension x, momentum p, and hamiltonin H(x,p). Hamilton's principal function S(x,t) is a function satisfying.

$$H(x,\frac{\partial S}{\partial x})+\frac{\partial S}{\partial t}=0$$

Now when we say that

$$p=\frac{\partial S}{\partial x}$$

this is somewhat nonsensical. The LHS, p, is a dynamical variable, independent of x, whereas the RHS is a function of x and t. It makes sense to refer to x(t) and p(t) separately, but not generally of p(x,t).


What $$p=\frac{\partial S}{\partial x}$$ must mean is that along any particular solution (x(t), p(t)) in the flow of solutions in phase space, it is the case that


$$p(t)=\frac{\partial S}{\partial x}|_{x=x(t)}$$


That is, we can speak of a function $$\hat{p}(x,t)\equiv\frac{\partial S}{\partial x}$$ and then it is true that if x(t),p(t) are solutions to the equations of motions then for all t


$$p(t)=\hat{p}(x(t),t)$$.


Let me pause here before proceeding to the question proper. Is what I say so far correct?

 

[confinement]

You are correct that partial derivative notation is ambiguous (cf. the additional notation which accompanies thermodynamic identities), although it should not normally cause confusion.

 

[pellman]

Thanks, confinement.

Actually, I don't need to post the actual question now. I figured out where I was going wrong. However, I would be interested in any comments about the OP.