Hamilton-Jacobi problem

Question

In analytical mechanics by Fasano and Marmi they consider the Hamilton-Jacobi equation for a conservative autonomous system in one dimension with the following Hamiltonian, \begin{equation} H=\frac{p^2}{2m}+V(x) \end{equation} The Hamilton-Jacobi equation is now, \begin{equation} \frac{1}{2m}\bigg(\frac{\partial W}{\partial x}\bigg)^2+V(x)=E \end{equation} Integrated to be; \begin{equation} W(x,E)=\sqrt{2m}\int ^{x}_{x_0}\sqrt{E-V(\xi)}d\xi \end{equation} With the canonical transformation, \begin{equation} p=\frac{\partial W}{\partial x}=\sqrt{2m(E-V(x))} \end{equation} \begin{equation} \beta =\frac{\partial W}{\partial E}=\sqrt {m/2}\int ^{x_1}_{x_0}\frac{d\xi}{\sqrt{E-V(\xi)}} \end{equation} My question is regarding the use of the dummy variable $\xi$. Why do we use this and why can we not just integrate over $x$? Further, why do we use it for computation of $\beta$ but not $p$?

Answer 1

You should think of the definite integral operation as a function of two arguments: a region over which to integrate (here, $[x_0,x_1]$), and another function $f$ called the integrand (here, $f:\xi \mapsto (E-V(\xi))^{-\frac{1}{2}}$).

So first of all, in my definition of $f$ above, we could have used (almost) any other symbol instead of $\xi$ and the meaning would be the same. But there are two exceptions: if we used either $E$ or $V$ to represent the argument of $f$, the function inside either wouldn't make sense or would be very different than what we meant.

The way I've written $f$ you know that $\xi$ is the argument of $f$, and not something defined externally, because of the "$\xi \mapsto$". These symbols are said to "capture" (in the sense of variable capture) $\xi$.

Now, the integrand in an integral is a function, and it needs to be written down in a way similar to what I've done above. Typically, the "$d\xi$" notation takes the place of the $\xi\mapsto$ notation I used in defining $f$ as a pure function. (Formally, the differential notation can mean something much deeper, but that's irrelvant in our context.)

So, suppose you had written $dx$ instead of $d\xi$. The problem would be that $x$ is already in scope, just like $E$ was already in scope when we defined $f$. It's in scope, as @ACuriousMind commented, because it was introduced by $W(x,E)$. If you think of integration as an operation $\mathtt{Integrate}[f,(x_0,x)]$, you can see that $x$ must have already been in scope outside of the integration, because you passed it to the integration as a parameter.

So, since it's already in scope, you cannot reuse it as the parameter of the integrand $f$. Whenever you used "$x$", it wouldn't be clear whether you meant the upper limit of the integration or the integration variable, becuase $x$ would undergo variable capture twice.

Oftentimes, the notation is abused and people write exactly what you suggested. The convention, when you encounter an $x$ and can't tell what it refers to, is to pick the "closest" variable capture, which is to say the most "local" definition. In our case, that would mean the integration variable. But formally, this is an ambiguous notation, and it's safer not to rely on this convention when there are plenty of other glyphs (like $\xi$) that can be used instead.