Vortices on closed surfaces

TL;DR

This paper develops an intrinsic geometric framework for analyzing point vortices on general closed surfaces, extending known results beyond spheres and revolution surfaces, and explores integrability and quantization aspects.

Contribution

It introduces a geometric formulation for vortex dynamics on arbitrary closed surfaces and investigates integrability and quantization possibilities.

Findings

Proof of Kimura's conjecture on dipole geodesic motion

Extension of vortex pair systems to triaxial ellipsoids

Discussion on integrability and quantization of vortex systems

Abstract

We consider $N$ point vortices $s_j$ of strengths $\kappa_j$ moving on a closed (compact, boundaryless, orientable) surface $S$ with riemannian metric $g$. As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature. The aim of this note is to present an intrinsic geometric formulation for the general case. We give a simple proof of Kimura's conjecture that a dipole describes geodesic motion. Searching for integrable vortex pairs systems on Liouville surfaces is in order. The vortex pair system on a triaxial ellipsoid extends Jacobi's geodesics. Is it Arnold-Liouville integrable? Not in our wildest dreams is another possibility: that quantizing a vortex system could relate with a million dollars worth question, but we took courage - nerve is more like it - to also present it.