= Pendulum approximation of a mean-motion resonance
{title2=$\ddot\phi=K\sin\phi$}
= Resonant pendulum approximation
{synonym}
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 $\mathcal R=(GM_2/a_2)f(\alpha)e\cos\phi$, leading <Lagrange planetary equations> give $\ddot\phi\simeq3j^2\mu_2n^2\alpha f e\sin\phi$. 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.
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