= Radial oscillation of an elastic-string orbit
{title2=$\Omega^2=(k/m)(4-3L/r_0)$}
For a taut elastic string with potential $k(r-L)^2/2$, a circular orbit has $\omega_0^2=k(r_0-L)/(mr_0)$. A radial impulse preserves angular momentum, so the curvature of the fixed-angular-momentum effective potential gives $\Omega^2=k/m+3\omega_0^2$. The ratio is three at $r_0=6L/5$ and approaches two at $r_0\gg L$. Rational linearized frequency ratios describe leading small-amplitude closed sketches without proving exact closure at finite amplitude.
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