Shape Dynamics of $N$ Point Vortices on the Sphere

TL;DR

This paper develops a geometric framework for understanding the shape dynamics of N point vortices on the sphere, utilizing symmetry reductions and Lie--Poisson structures to analyze stability and derive new results.

Contribution

It introduces a novel reduction approach for shape dynamics of vortices on the sphere, providing explicit Casimirs and stability results for tetrahedral configurations.

Findings

Established a Lie--Poisson structure for shape dynamics.

Derived Casimirs for the reduced system.

Proved stability of tetrahedron relative equilibria with same-signed circulations.

Abstract

We give a geometric account of the relative motion or the shape dynamics of $N$ point vortices on the sphere exploiting the $\mathsf{SO}(3)$-symmetry of the system. The main idea is to bypass the technical difficulty of the $\mathsf{SO}(3)$-reduction by first lifting the dynamics from $\mathbb{S}^{2}$ to $\mathbb{C}^{2}$. We then perform the $\mathsf{U}(2)$-reduction using a dual pair to obtain a Lie--Poisson dynamics for the shape dynamics. This Lie--Poisson structure helps us find a family of Casimirs for the shape dynamics. We further reduce the system by $\mathbb{T}^{N-1}$-symmetry to obtain a Poisson structure for the shape dynamics involving fewer shape variables than those of the previous work by Borisov and Pavlov. As an application of the shape dynamics, we prove that the tetrahedron relative equilibria are stable when all of their circulations have the same sign, generalizing some existing results on tetrahedron relative equilibria of identical vortices.