A second order variational principle for the Lorentz force equation: conjugacy and bifurcation

We give the notion of a conjugate instant along a solution of the relativistic Lorentz force equation (LFE). Electromagnetic conjugate instants are defined as zeros of solutions of the linearized LFE with fixed value of the charge-to-mass ratio; equivalently, we show that electromagnetic conjugate points are the critical values of the corresponding electromagnetic exponential map. We prove a second-order variational principle relating every solution of the LFE to a canonical lightlike geodesic in a Kaluza–Klein manifold, whose metric is defined using the value of the charge-to-mass ratio. Electromagnetic conjugate instants correspond to conjugate points along the lightlike geodesic, and therefore they are isolated; based on such correspondence and on a recent result of bifurcation for light rays, we prove a bifurcation result for solutions of the LFE in the exact case.