Completeness of trajectories of relativistic particles under stationary magnetic fields

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.

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