How to introduce action-angle variables in the following integrable 2 d.o.f. Hamiltonian system?
$$H(q,p,x,y) = \frac{y^{2}}{2} - \frac{x^{2}}{2}\left(p^{2} + \omega^{2}q^{2}\right) + \frac{x^{4}}{4}$$
So $(q,p)$, $(x,y)$ are conjugated variables, and the first integrals are $F_{0}=H, F_{1} = p^{2} + \omega^{2}q^{2}$.
Observe, for fixed $(q,p)$, the Hamiltonian has a saddle point in the $(x,y)$ plane, so one has to change $(q,p,x,y) \mapsto (\varphi, I, X, Y)$ where action-angle variables are $(\varphi, I)$ and $(X,Y)$ are a new canonically conjugate pair, i.e. we have $H(q,p,x,y) = H(I, X, Y)$. So I need to find a generating function $S$.
The question is, how to find this $S$?
Note that if I take $F_{1}(q,p)$ as a "Hamiltonian" on its own, action-angle variables $(q,p) \mapsto (\varphi, I)$ are introduced through a generating function of the form $\tilde{S}(q, \varphi) = \frac{\omega q^{2} \cot 2\pi\varphi}{2}$.
Can I use this function $\tilde{S}$ as an "inspiration" and modify it to take $S(q, \varphi, x, Y) = \tilde{S} + xY$ ?
Or in such a setup, it is not as trivial?