  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.
