Repulsive relativistic Coulomb Hamiltonian phase system

Let $m,c,\kappa >0$, let $x,p\in {\mathbb{R}}^3$, and set

$D:=({\mathbb{R}}^3\setminus \{0\})\times {\mathbb{R}}^3,r:=\mathrm{\mid }x\mathrm{\mid },\mathcal{E}(p):=\sqrt{m^2{c}^4+c^2\mathrm{\mid }p{\mathrm{\mid }}^2}\mathrm{.}$

The repulsive relativistic Coulomb Hamiltonian is

$H(x,p):=\mathcal{E}(p)+\frac{\kappa }{r},$

and its phase system is the autonomous ODE on the open set $D$

$\dot{x}={\mathrm{\nabla }}_{p}H=\frac{c^{2}p}{\mathcal{E}(p)},\dot{p}=-{\mathrm{\nabla }}_{x}H=\frac{\kappa x}{r^3}\mathrm{.}$

Here the system parameter $t$ is time in seconds and dot denotes $d\mathrm{/}dt$; $m$ is a positive mass parameter (kilograms in the SI interpretation), $c$ a positive speed parameter (metres per second), and $\kappa$ a positive energy-times-length parameter (joule-metres). Define the radial momentum $p_r:=x\cdot p\mathrm{/}r$ and angular-momentum vector $J:=x\times p$. A trajectory is a $C^1$ map $z=(x,p):I\to D$ on a real interval $I$ with nonempty interior, satisfying this ODE in the interval-solution sense of First-order systems, initial value problems, and solutions on intervals. A trajectory $\tilde{z}:\tilde{I}\to D$ extends $z$ if $I\subseteq \tilde{I}$ and $\tilde{z}{\mathrm{\mid }}_I=z$; it is a proper extension if $I⊊\tilde{I}$. A trajectory is maximal if it has no proper extension as a solution of this same system in $D$. This definition alone asserts neither existence nor uniqueness of maximal trajectories.

This is only a mathematical phase system. Its physical interpretation as a charged particle in an externally prescribed Coulomb field requires separate model assumptions.