One smooth free relativistic Hamilton-Jacobi datum loses graph regularity in finite time

Let $\mathcal{P}=T^{\ast }\mathbb{R}$ have position coordinate $q\in \mathbb{R}$ in metres, momentum coordinate $p\in \mathbb{R}$ in kg m/s, and symplectic form $\omega =dq\wedge dp$. For fixed mass $m>0$ in kg and speed $c>0$ in m/s, take the free Hamiltonian $H(p)=E(p)=\sqrt{m^2{c}^4+c^2{p}^2}$, in joules, on $\mathcal{P}$; time $t$ is measured in seconds. The proposed assertion that every smooth initial action $S_0:\mathbb{R}\to \mathbb{R}$ produces a characteristic family extending as a smooth single-valued graph $p=S_q(t,q)$ with finite $S_{qq}$ through every finite future time $t\ge 0$ is false.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the cotangent-shell supplier. With the phase space and Hamiltonian from the Statement, let $a\in \mathbb{R}$ be the initial position in metres and $t$ seconds. Choose the smooth initial action $S_0(a)=-\kappa a^2\mathrm{/}2$, $\kappa >0$ in kg/s. Its characteristics have $p(a)=-\kappa a$ and

$q(t,a)=a+t\frac{c^{2}p(a)}{E(p(a))}\mathrm{.}$

For this characteristic family, call $(t,a)$ a caustic point when the projection $(t,a)\mapsto (t,q(t,a))$ is not locally invertible there.

The projection derivative is ${\mathrm{\partial }}_{a}q=1-t\kappa m^2{c}^6\mathrm{/}E(p{)}^3$. It stays positive for $0\le t<m\mathrm{/}\kappa$ and first vanishes at $(t,a)=(m\mathrm{/}\kappa ,0)$, where ${\mathrm{\partial }}_{a}p=-\kappa \mathrm{\ne }0$. Therefore this characteristic family cannot extend there as a smooth single-valued graph $p=S_q(t,q)$ with finite $S_{qq}$. The initial data and Hamiltonian are smooth; the failure is the projection singularity, not an initial-data defect.