A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

Abstract

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic 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 electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.

A Morse Complex for Lorentzian Geodesics

“…As in the Riemannian case, this relative Morse index plays an important role in the local bifurcation theory of geodesics, see for instance [PPT04] and [PP05].…”

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

“…One of the first results of the paper is that the electromagnetic exponential map is a local diffeomorphism around non-electromagnetic conjugate points (propositions 2.4 and 2.5). We must observe here that our notion of the electromagnetic Jacobi field does not coincide with that given in [5] or [13]; in those papers, electromagnetic Jacobi fields were defined as solutions of the linear second-order equation satisfied by vector fields in the kernel of the second variation of the electromagnetic action functional. With such a definition, which makes sense only in the exact case (recall that the electromagnetic action functional can be defined only in the exact case), Jacobi fields would correspond to variational vector fields associated to variations by solutions with non-constant charge-to-mass ratio.…”

“…Piccione et al . [13] have studied the phenomenon of bifurcation of solutions at a given conjugate instant. Their result, however, is not physically interesting due to the fact that each solution of the bifurcating branch has a different value of the charge-to-mass ratio, and such values are not even constant up to first-order infinitesimals.…”

Bifurcation results for critical points of families of functionals

“…[FPR00]), bifurcation of families of geodesics in semi-Riemannian manifolds (cf. [MPP07]) and for the study of conjugate points in [PPT03] and [PP05]. Recently the first author has considered the more general situation of C 2 functions f : X×H → R, where X is a compact, orientable smooth manifold of dimension at least 2 and 0 ∈ H is again a critical point of all functionals f λ : H → R, λ ∈ X.…”