On the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces

TL;DR

This paper calculates the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces, including the two-sphere and higher genus surfaces, with applications to magnetic particle dynamics.

Contribution

It provides explicit capacity calculations for twisted tangent bundles over closed surfaces, including an SO(3)-equivariant compactification for the sphere.

Findings

Hofer-Zehnder capacity for magnetic fields on S^2 and higher genus surfaces

Explicit compactification of twisted tangent bundle on S^2

Applications to charged particle dynamics in magnetic fields

Abstract

We determine the Hofer-Zehnder capacity for twisted tangent bundles over closed surfaces for (i) arbitrary constant magnetic fields on the two-sphere and (ii) strong constant magnetic fields for higher genus surfaces. On $S^2$ we further give an explicit $\text{SO}(3)$-equivariant compactification of the twisted tangent bundle to $S^2\times S^2$ with split symplectic form. The former is the phase space of a charged particle moving on the two-sphere in a constant magnetic field, the latter is the configuration space of two massless coupled angular momenta.