A Bifurcation Result for Semi-riemannian Trajectories of the Lorentz Force Equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semiRiemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the magnetic field. The flow of the Jacobi equation along each solution preserves the so-called magnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We study magnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see [13]). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see [10]), we are able to prove that each non degenerate and non-null magnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.