  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.
