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Maslov index and Morse theory for the relativistic Lorentz force equation. (English) Zbl 1060.58010
The Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold are investigated. The Maslov index of a solution of the Lorentz force equation is defined and its properties are outlined. Moreover, for a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field which preserves the flow of the Killing vector field, it is introduced a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. The value of this Morse index is shown to be equal to the Maslov index, and the Morse relations for the solutions of the Lorentz force equation in a static spacetime are proved.

MSC:
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
53D12 Lagrangian submanifolds; Maslov index
83C10 Equations of motion in general relativity and gravitational theory
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