I have a very specific problem that I'm not even sure how to approach.
I have a function $G(t,s,W,W')$, such that $G(s,t,W,W') = -G(t,s,W,W')$, where $t$ and $s$ are real variables while $W$ is a real function and $W'$ is its derivative. I then need to find conditions on $W$ such that $G$ has the form $$G(t,s,W,W') = f(t,W,W')g(s,W,W') - f(s,W,W')g(t,W,W')$$ for some $f$ and $g$ (which are not know). This is very similar to finding conditions on where the function $G$ would be separable, but here instead of separable it is of this anti-symmetric form.
As simple example, take $G (t,s,W,W') = W(t)^2W(s) - W(s)^2 W(t)$. This function is already of the form we desire explicitly for $f(t,W,W') = W(t)^2$ and $g(t,W,W') = W(t)$.
The way I'm trying to approach this is finding some property of $ f(t,W,W')g(s,W,W') - f(s,W,W')g(t,W,W')$, for example some functional derivatives, that lead to a condition where only functions of this form would be zero, which then would lead to an equation form $W$ that hopefully would be solvable.
Is this a well know problem?