= Resonant pendulum energy
{title2=$E=\dot\phi^2/2+\omega_0^2(1+\cos\phi)$}
For $\ddot\phi=\omega_0^2\sin\phi$ with constant $\omega_0$, multiplication by $\dot\phi$ gives the conserved energy $E=\dot\phi^2/2+\omega_0^2(1+\cos\phi)$. The stable center is $\pi$; the <separatrix> has $E=2\omega_0^2$. Below it the <resonant argument> librates and above it the angle circulates. This is an invariant of the constant-coefficient <pendulum approximation of a mean-motion resonance>, not of arbitrary coupled <orbital element> evolution.
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