= Circular orbit in a quadratic central potential
{title2=$r_0^4=h^2/(2km^2)$}
For $V(r)=kmr^2$, fixed nonzero <angular momentum> $h$ gives the <effective potential> $V_{\rm eff}=h^2/(2mr^2)+kmr^2$. A positive-radius circular orbit exists when $k>0$, at the displayed radius, and $V_{\rm eff}''(r_0)=8km>0$ 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 $k\le0$, 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.
Back to article page
