 Solution

The initial angular velocity cancels the flux term in the canonical angular momentum:Define ${\omega }_c=qB/m$ and ${\omega }_s^2=4{\pi }^{2}k/m$. The radial equation becomes $\ddot{r}+{\Omega }^{2}r={\omega }_s^2{r}_0$, soThese identify the paper's $\omega$ with $\Omega$. For nonzero radius, $J=0$ also fixesA radius is nonnegative. The formal sinusoidal solution remains strictly positive for a full cycle only if $2r_e>r_0$, or ${\omega }_s^2>{\omega }_c^2/4$. At equality it touches zero; at stronger magnetic coupling it first reaches zero and its negative continuation is not a physical ring radius. No collision or re-expansion rule at collapse is supplied, so the circular-ring solution is then valid only up to the first zero. This domain qualification leaves the requested differential-equation solution intact.