Pendulum approximation of a mean-motion resonance ($\ddot{\varphi }=K\sin \varphi$)

Keeping one resonant harmonic and treating orbital eccentricity as fixed reduces mean-motion resonance dynamics to a physical pendulum equation. For the inner first-order term $\mathcal{R}=(G{M}_2/a_2)f(\alpha )e\cos \varphi$, leading Lagrange planetary equations give $\ddot{\varphi }\simeq 3j^2{\mu }_2{n}^2\alpha fe\sin \varphi$. Around its stable phase, the small-amplitude libration frequency is $jn\sqrt{3{\mu }_2\alpha |f|e}$. Evolution of eccentricity, other harmonics, and approach to a separatrix change this approximation.

For $\ddot{\varphi }={\omega }_0^2\sin \varphi$ with constant ${\omega }_0$, multiplication by $\dot{\varphi }$ gives the conserved energy $E={\dot{\varphi }}^2/2+{\omega }_0^2(1+\cos \varphi )$. 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.