On the existence of maximizing curves for the charged-particle action

The classical Avez-Seifert theorem is generalized to the case of the Lorentz force equation for charged test particles with fixed charge-to-mass ratio. Given two events x(0) and x(1), with x(1) in the chronological future of x(0), and a ratio q/m, it is proved that a timelike connecting solution of the Lorentz force equation exists provided there is no null connecting geodesic and the spacetime is globally hyperbolic. As a result, the theorem answers affirmatively to the existence of timelike connecting solutions for the particular case of Minkowski spacetime. Moreover, it is proved that there is at least one C-1 connecting curve that maximizes the functional I[gamma] = integral(gamma)ds + q/(mc(2))omega over the set of C-1 future-directed non-spacelike connecting curves.