Zero-canonical-angular-momentum ring oscillation (${\Omega }^2=4{\pi }^{2}k/m+q^2{B}^2/(4m^2)$)

With $J=0$, define ${\omega }_s^2=4{\pi }^{2}k/m$ and ${\omega }_c=qB/m$. Then $r_e={\omega }_s^2{r}_0/({\omega }_s^2+{\omega }_c^2/4)$, the radial frequency is $\Omega =\sqrt{{\omega }_s^2+{\omega }_c^2/4}$, and release at $r_0$ gives $r=r_e+(r_0-r_e)\cos \Omega t$ with $\dot{\varphi }=-{\omega }_c/2$. The physical solution requires $r>0$. If $2r_e<r_0$, the formal oscillation reaches zero and cannot be continued as a negative ring radius without a new collapse model.