On polygonal relative equilibria in the N-vortex problem

TL;DR

This paper investigates the conditions for polygonal relative equilibria in the N-vortex problem, revealing that equal vorticities are necessary for certain configurations and classifying equilibria with two concentric regular polygons.

Contribution

It provides new classifications of polygonal relative equilibria, including conditions for equal vorticities and configurations with two concentric polygons, extending classical results.

Findings

Equal vorticities are required for regular polygons with more than three vertices.

All equilibria with two concentric regular n-gons and same vorticity are characterized.

Completes classical studies by analyzing cases with varying vorticity distributions.

Abstract

Helmholtz's equations provide the motion of a system of N vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we are interested in relative equilibria formed of concentric regular polygons of vortices. We show that in the case of one regular polygon (and a possible vortex at the center) with more than three vertices (two if there is a vortex at the center), a relative equilibrium requires equal vorticities (on the polygon). We also determine all the relative equilibria with two concentric regular n-gons and the same vorticity on each n-gon. This result completes the classical studies for two regular n-gons when all the vortices have the same vorticity or when the total vorticity vanishes.