  Recent results on the maximization of the charged-particle action I in a
globally hyperbolic spacetime are discussed and generalized. We focus on the
maximization of I over a given causal homotopy class C of curves connecting two
causally related events x_0 <= x_1. Action I is proved to admit a maximum on C,
and also one in the adherence of each timelike homotopy class. Moreover, the
maximum on C is timelike if C contains a timelike curve (and the degree of
differentiability of all the elements is at least C^2).
  In particular, this last result yields a complete Avez-Seifert type solution
to the problem of connectedness through trajectories of charged particles in a
globally hyperbolic spacetime endowed with an exact electromagnetic field:
fixed any charge-to-mass ratio q/m, any two chronologically related events x_0
<< x_1 can be connected by means of a timelike solution of the Lorentz force
equation (LFE) corresponding to q/m. The accuracy of the approach is stressed
by many examples, including an explicit counterexample (valid for all q/m) in
the non-exact case.
  As a relevant previous step, new properties of the causal path space, causal
homotopy classes and cut points on lightlike geodesics are studied.
