Circular orbit in a quadratic central potential ($r_0^4=h^2/(2k{m}^2)$)

For $V(r)=km{r}^2$, fixed nonzero angular momentum $h$ gives the effective potential $V_{\mathrm{eff}}=h^2/(2m{r}^2)+km{r}^2$. A positive-radius circular orbit exists when $k>0$, at the displayed radius, and $V_{\mathrm{eff}}^{\prime \prime }(r_0)=8km>0$ proves radial stability by the effective potential stability criterion. Its angular speed has magnitude $\sqrt{2k}$ and its small radial oscillation frequency is $\sqrt{8k}$. If $k\le 0$, no such orbit with nonzero angular momentum exists; for $h=0,k>0$, the sole minimum is at the origin, not a positive-radius circular orbit.