Separation of variables for Hamilton-Jacobi equation (HJE) in elliptic (prolate spheroidal) coordinates I'm reading Landau & Lifshitz's Mechanics, and try to understand the result of equation 48.22. I agree with the result of 48.20 and 48.21, but when it comes to 48.22, I calculate it as follows:
because the Hamilton-Jacobi equation (HJE) is
$$H + \frac{\partial S}{\partial t} = 0$$
replace H with 48.20 and 48.1, then multiply $2m\sigma^2(\xi^2 - \eta^2)$ on both side, one got
$$(\xi^2 - 1)(\frac{\partial S_0}{\partial\xi})^2 + (1 - \eta^2)(\frac{\partial S_0}{\partial\eta})^2 + (\frac{1}{\xi^2 - 1} + \frac{1}{1 - \eta^2})p_\varphi^2 + 2m\sigma^2[a(\xi) + b(\eta)] - 2m\sigma^2(\xi^2 - \eta^2)E = 0$$
separation of the variables gives
$$S = -Et + p_\varphi\varphi + \int\sqrt{\frac{2m\sigma^2E\xi^2 + \beta- 2m\sigma^2a(\xi)}{\xi^2 -1} -\frac{p_\varphi^2}{(\xi^2 - 1)^2}} ~ d\xi + \int\sqrt{\frac{-2m\sigma^2E\eta^2 - \beta - 2m\sigma^2b(\eta)}{1 - \eta^2} -\frac{p_\varphi^2}{(1 - \eta^2)^2}} ~ d\eta$$
however, according to the textbook, the result should be
$$S = -Et + p_\varphi\varphi + \int\sqrt{2m\sigma^2E + \frac{\beta- 2m\sigma^2a(\xi)}{\xi^2 -1} -\frac{p_\varphi^2}{(\xi^2 - 1)^2}} ~ d\xi + \int\sqrt{2m\sigma^2E - \frac{\beta + 2m\sigma^2b(\eta)}{1 - \eta^2} -\frac{p_\varphi^2}{(1 - \eta^2)^2}} ~ d\eta$$
Am I wrong? or the textbook is wrong?
 A: The textbook is correct, though your result seems to be correct too. The text-book is just taking
\begin{align}
2m\sigma^2(\xi^2 - \eta^2)E = &+ (\xi^2 - 1)[(\frac{\partial S_0}{\partial\xi})^2 + \frac{p_\varphi^2}{(\xi^2 - 1)^2} + \frac{2m\sigma^2 a(\xi)}{\xi^2 - 1} ] \\
&+ (1 - \eta^2)[(\frac{\partial S_0}{\partial\eta})^2  + \frac{p_\varphi^2}{(1 - \eta^2)^2} + \frac{2m\sigma^2  b(\eta)}{1 - \eta^2}] 
\end{align}
and re-writing the left-hand side as
$$2m\sigma^2(\xi^2 - \eta^2)E  = 2m\sigma^2(\xi^2 - 1)E + 2m\sigma^2(1 - \eta^2)E$$
then setting
$$\beta = (\xi^2 - 1)[(\frac{\partial S_0}{\partial\xi})^2 + \frac{p_\varphi^2}{(\xi^2 - 1)^2} + \frac{2m\sigma^2 a(\xi)}{\xi^2 - 1} ] - 2m\sigma^2(\xi^2 - 1)E $$
and
$$2m\sigma^2(1 - \eta^2) E = (1 - \eta^2)[(\frac{\partial S_0}{\partial\eta})^2  + \frac{p_\varphi^2}{(1 - \eta^2)^2} + \frac{2m\sigma^2  b(\eta)}{1 - \eta^2}] + \beta $$
so that e.g. the first equation reduces as
$$(\frac{\partial S_0}{\partial\xi})^2  = 2m\sigma^2 E + \frac{\beta - 2m\sigma^2 a(\xi)}{(\xi^2 - 1)}  - \frac{p_\varphi^2}{(\xi^2 - 1)^2}   $$
and similarly for the second.
Without re-writing the left-hand side at the first step, one gets your expression which looks like a more complicated complicated integral but the end result should be equivalent, though it does not look like one can directly re-write the final integrals in the same way because of the intermediate solution steps e.g. taking square roots.
