= Small radial oscillations in a power-law potential
{title2=$\omega_r/\Omega=\sqrt{k+2}\quad(V=\beta r^k)$}
For a <central potential> $V=\beta r^k$ and fixed nonzero <angular momentum> $L$, the <effective potential> $L^2/(2mr^2)+\beta r^k$ has a nondegenerate stable <circular orbit> when $\beta k>0$ and $k>-2$. Its radius obeys $r_0^{k+2}=L^2/(m\beta k)$. The radial <small oscillation> frequency satisfies $\omega_r^2=(k+2)\Omega^2$, where $\Omega=|L|/(mr_0^2)$. Rational frequency ratios close the linearized near-circular sketch, but generally not the exact finite-amplitude orbit; <Bertrand's theorem> distinguishes the exceptional harmonic and Kepler potentials.
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