A Hamiltonian approach for point vortices on non-orientable surfaces

TL;DR

This paper develops a modified Hamiltonian framework to analyze point vortex dynamics on non-orientable surfaces like the Mobius band and Klein bottle, extending classical vortex theory to these complex geometries.

Contribution

It introduces a new Hamiltonian approach tailored for non-orientable surfaces, enabling the study of vortex motion in these challenging geometries.

Findings

Derived explicit Hamiltonian equations for vortices on Mobius band and Klein bottle.

Characterized relative equilibria for vortex configurations on these surfaces.

Analyzed vortex motion for one and two vortices in the new framework.

Abstract

We investigate the motion of point vortices on the Mobius band and Klein bottle. Since these are non-orientable surfaces, the standard Hamiltonian approach does not apply. We therefore begin by establishing a modified Hamiltonian approach which works for arbitrary non-orientable surfaces, through describing the phase space, the Hamiltonian and the local equations of motion. We use a combination of twisted functions and oriented double covers to adapt some of the known notions of vortex dynamics to non-orientable surfaces. For both of the surfaces of interest, we write Hamiltonian-type equations of vortex motion explicitly and follow that by the description of relative equilibria and an investigation of the motion of one and two vortices.