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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.

In a high-orbital eccentricity interior $k:1$ mean-motion resonance, the resonant argument $\varphi =k{\lambda }_p-\lambda -(k-1)\varpi$ places successive rotating-frame apocentres at $\pi -[\varphi +(2j+1)\pi ]/k$. Avoiding astronomical conjunction at these slow outer excursions favors $\varphi =0$ for $k=2$ and $\varphi =\pi$ for $k=3$. This geometrical phase protection suggests stable resonant-argument libration centres; actual dynamical stability requires the resonant perturbation problem.

A resonant argument is a slow angular combination whose libration diagnoses resonance. For an exterior $j:(j-1)$ eccentric resonance it can be $\varphi =j\lambda -(j-1){\lambda }_p-\varpi$.

A resonant argument is an integer combination of orbital angles whose slow variation can diagnose mean-motion resonance. Its libration, rather than circulation through every angle, characterizes trapping in the resonance.

The nominal semi-major axis of an exterior $(p+q):p$ mean-motion resonance follows from $(p+q)n_r=p{n}_p$ and Kepler's third law. Apsidal precession displaces the actual libration center because stationarity of the resonant argument requires $(p+q)n-p{n}_p-q\dot{\varpi }=0$.