Period behavior near separatrix in Hamiltonian system

Question

Given the periodic potential Hamiltonian $H=\frac{p^2}{2} - \omega_0^2 \cos(q)$ I would like to show that near the separatrix the period has this behavior: $T(E)\sim |\log(\delta E)|$ with $\delta E=|E-\omega_0^2|$.

More generally given an Hamiltonian system of the form $H=\frac{p^2}{2} + V(q)$ with $V''(q^*)\ne 0$ for a non stable fixed point, I would like to show that near the separatrix we get the same kind of law.

I could prove that $p$ is a solution on the separatrix and found an infinite period. Then I tried doing different development of $E$ to first order and second order but didn't get any result. Do you have any idea on how to do that for the first case and then maybe the general case?

Answer 1

The period is calculated by the integral $$ \sqrt{2}\int_0^{2\pi} \frac{d q}{ \sqrt{\omega_0 ^2 \cos (q)+E}} $$ which can be represented by special functions. After applying a replacement $E\to \delta E+\omega^2_0$, you need to expand this integral around separatrix $\delta E=0$, the leading term is $-2\omega_0^{-1}\ln(\delta E)$. Thus the leading term of period around $\delta E=0$ will be $T\sim-2\omega_0^{-1}\ln(\delta E)$.