The Gauss-Landau-Hall problem on Riemannian surfaces

TL;DR

This paper introduces the Gauss-Landau-Hall magnetic field on Riemannian surfaces, linking it to boson dynamics, and explores properties of flowlines, stability, and specific surface characterizations.

Contribution

It formulates the Landau-Hall problem as a variational issue on Riemannian surfaces and analyzes stability and geometric properties of flowlines, including uniform magnetic fields.

Findings

Flowlines as critical points of an action functional.

Global stability analysis of flowlines.

Characterization of revolution surfaces with normal flowlines.

Abstract

We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.