Completeness of trajectories of relativistic particles under stationary magnetic fields

TL;DR

This paper investigates the conditions under which the trajectories of relativistic particles, governed by a second order differential equation involving electromagnetic and conservative forces on a Lorentzian manifold, can be extended or continued.

Contribution

It provides the first analysis on the extendability of solutions to this class of differential equations under natural assumptions.

Findings

Identifies conditions ensuring solution extendability

Establishes criteria for trajectory completeness

Lays groundwork for further studies on relativistic particle dynamics

Abstract

The second order differential equation $\frac{D\dot\gamma}{dt}(t) = F_{\gamma(t)}(\dot\gamma(t)) - \nabla V(\gamma(t))$ on a Lorentzian manifold describes, in particular, the dynamics of particles under the action of a electromagnetic field $F$ and a conservative force $-\nabla V$. We provide a first study on the extendability of its solutions, by imposing some natural assumptions.