Solar models and McKean's breakdown theorem for the $\mu$CH and $\mu$DP equations

We study the breakdown for $\mu$CH and $\mu$DP equations on the circle, given by $$m_t + u m_{\theta} + \lambda u_{\theta} m = 0,$$ for $m = \mu(u) - u_{\theta\theta}$, where $\mu$ is the mean and $\lambda=2$ or $\lambda=3$ respectively. It is already known that if the initial momentum $m_0$ never changes sign, then smooth solutions exist globally. We prove the converse: if the initial momentum changes sign, then $C^2$ solutions $u$ must break down in finite time. The technique is similar to that of McKean, who proved the same for the Camassa-Holm equation, but we introduce a new perspective involving a change of variables to treat the equation as a family of planar systems with central force for which the conserved angular momentum is precisely the conserved vorticity. We also demonstrate how this perspective can apply to give some insights for other PDEs of continuum mechanics, such as the Okamoto-Sakajo-Wunsch equation (and in particular the De Gregorio equation).