How do we know the solutions to the Hamilton Jacobi equation are seperable? So I usually start my questions here with "I'm sure this is a stupid question", but I am even more convinced than usual that it's true this time. We are trying to find a solution $S$ of the Hamilton Jacobi equation. It is often said in passing that if a certain generalized coordinate ''$q_k$'' and its derivative $ \partial S/\partial q_k$ appear together as a single function
$\psi \left(q_k, \frac{\partial S}{\partial q_k} \right)$
in the Hamiltonian
$H = H(q_1,q_2,\ldots, q_{k-1}, q_{k+1},\ldots, q_N; p_1,p_2,\ldots, p_{k-1}, p_{k+1},\ldots, p_N; \psi; t). $
then he function ''S'' can be partitioned into two functions, one that depends only on ''q_k'' and another that depends only on the remaining generalized coordinate.
$S = S_k(q_k) + S_\text{rem}(q_1,\ldots, q_{k-1}, q_{k+1}, \ldots, q_N, t). $
It is said so nonchalantly everywhere, I assume that this fact is entirely obvious to literally everyone but me, but unfortunately it's not obvious to me. Is clear, once we have the above condition, that ANY solution of the Hamilton Jacobi is seperable? Can anybody clear this up? Thank you :)
 A: You shouldn't feel stupid about this question: the Hamilton-Jacobi formalism is a tricky one, and this leads several authors to expose the subject in a somewhat confusing manner...
A solution $S$ to the Hamilton-Jacobi equation (associated to a Hamiltonian function $H$) indirectly parametrizes some (special) subset of solutions to the Hamilton equations for some time interval. In particular, given $S$, there are always solutions to the Hamiltonian equations which are not described by $S$. This interpretation of $S$ as a "choice of subset of solutions to the Hamilton equations" suggests that it might be possible to choose a complicated subset which is not described by a separable function $S$ even when $H$ itself is separable.
To find an example of such, one can happily work backwards: consider $S(q_1,q_2,t) = q_1q_2$, which is clearly not separable (in those coordinates). We want to find $H$ such that this $S$ describes some subset of solutions to the Hamilton equations for $H$. Assuming that such a $H$ exists, the Hamilton-Jacobi formalism here implies $p_1 = \partial S/\partial q_1 = q_2$, $p_2 = \partial S/\partial q_2 = q_1$ (for the solutions that $S$ describes) and $\partial S/\partial t = 0$. The imposition of the Hamilton-Jacobi equation $$H(q_1, \partial S/\partial q_1, q_2, \partial S/\partial q_2, t) + \partial S/\partial t = 0$$ constrains the possible $H$; it is easy to see that the choice $H := p_1 q_1 - p_2 q_2$ solves this constraint, and that it is a separable Hamiltonian function.
For a somewhat physically more relevant counterexample, consider two free particles, $$H = \frac{p_1^2}{2} + \frac{p_2^2}{2} ,$$
and consider the nonseparable function
$$ S(q_1, q_2, t) = \frac{(q_1 + q_2)^2}{4t} \, .$$
That being said, one usually tries to solve the Hamilton-Jacobi equation precisely in order to find some set of solutions to the Hamilton equations; to find some "simple" set of solutions (corresponding to a "simple" $S$) might give a much better insight about the physical system than just any set. With that in mind, to look for separable functions $S$ when $H$ is separable might be the relevant thing to do.
