Magnetic Geodesics via the Heat Flow

TL;DR

This paper uses the heat flow method to prove the existence of magnetic geodesics on Riemannian manifolds, under certain conditions related to the magnetic field and initial data.

Contribution

It introduces a heat flow approach to establish existence results for magnetic geodesics, including conditions for convergence and examples illustrating the theory.

Findings

Heat flow converges to magnetic geodesics under boundedness assumptions.

Existence is guaranteed if the magnetic field has a global potential.

Small initial curves lead to convergence in the absence of a potential.

Abstract

Magnetic geodesics describe the trajectory of a particle in a Riemannian manifold under the influence of an external magnetic field. In this article, we use the heat flow method to derive existence results for such curves. We first establish subconvergence of this flow to a magnetic geodesic under certain boundedness assumptions. It is then shown that these conditions are satisfied provided that either the magnetic field admits a global potential or the initial curve is sufficiently small. Finally, we discuss different examples which illustrate our results.