Magnetic geodesics via the heat flow

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. In the former case, we can in particular conclude that there exists a magnetic geodesic in each homotopy class of curves. For non-exact fields, the behavior of the flow depends on the exact choice of the initial curve in relation to the magnetic field. We finally discuss different examples to illustrate these results.

Keywords

magnetic geodesics, gradient flow, convergence

2010 Mathematics Subject Classification

58E20

Full Text (PDF format)

Received 24 November 2015

Accepted 24 June 2016

Published 6 March 2018