A Hamiltonian approach for point vortices on non-orientable surfaces II: the Klein bottle

TL;DR

This paper develops a Hamiltonian framework for analyzing the dynamics of one and two point vortices on the Klein bottle, a non-orientable surface, including explicit equations and equilibrium descriptions.

Contribution

It provides explicit Hamiltonian formulations and equations of motion for vortices on the Klein bottle, extending previous work on non-orientable surfaces.

Findings

Explicit Hamiltonian for vortices on the Klein bottle

Descriptions of relative equilibria on the Klein bottle

Analysis of vortex dynamics on non-orientable surfaces

Abstract

This is the second of two companion papers dedicated to the investigation of vortex motion on non-orientable surfaces. The first paper of the pair is predominantly concerned with establishing the Hamiltonian approach to systems of point vortices on non-orientable manifolds and investigating the limits of the intrinsic (restricted to the non-orientable manifold) approach. In addition, point vortex motion on the Mobius band is closely examined. In this paper, we investigate dynamics of one and two point vortices on the Klein bottle through establishing explicit forms of the Hamiltonian, equations of motion on the charts, describing relative equilibria, etc.