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For central potential $V(r)=\beta r^k$ and nonzero angular momentum $J$, a circular orbit satisfies $\beta k{r}_0^{k+2}=J^2/m$, so attraction requires $\beta k>0$. Twice differentiating the radial effective potential $V+J^2/(2m{r}^2)$ gives $V_{\mathrm{eff}}^{\prime \prime }(r_0)=(k+2)J^2/(m{r}_0^4)$. Its harmonic approximation therefore has radial angular frequency ${\omega }_r=\sqrt{k+2}|J|/(m{r}_0^2)$, whereas $|{\omega }_{\theta }|=|J|/(m{r}_0^2)$. This proves the ratio and shows stability precisely when $k>-2$. At $k=-2$ the tuned effective potential is flat; at $k<-2$ the radial disturbance grows exponentially. Rationality of this local frequency ratio is only a necessary condition for all nearby orbits to close; Bertrand's theorem makes a stronger finite-amplitude assertion.

Separation of variables in a central potential writes a wavefunction as $R_{nl}(r)Y_l^m(\theta ,\varphi )$. With normalized spherical harmonics the radial factor satisfies ${\int }_0^{\infty }{r}^2|R_{nl}{|}^{2}dr=1$. The reduced radial function is $u=rR$, with normalization $\int |u{|}^{2}dr=1$; the two functions have different behavior at the origin. For a regular Coulomb eigenstate, $R_{nl}\propto r^l$ near the origin.

An energy and orbital angular momentum eigenstate for a real two-dimensional central potential can be chosen with integer $k$ and real radial function $f$. Its radial equation isQuantum degeneracy permits other stationary states, including $f(r)\cos (k\varphi )/\sqrt{\pi }$, which are not single angular exponentials. Separation of variables supplies a basis rather than describing every superposition in a degenerate eigenspace.

For a central potential $V=\beta r^k$ and fixed nonzero angular momentum $L$, the effective potential $L^2/(2m{r}^2)+\beta r^k$ 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 $\Omega =|L|/(m{r}_0^2)$. 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.