Stability of Relative Equilibria in the Planar N-Vortex Problem

TL;DR

This paper investigates the stability of relative equilibria in the planar N-vortex problem, establishing conditions for linear and nonlinear stability, and analyzing specific symmetric configurations to identify stable solutions.

Contribution

It adapts celestial mechanics methods to vortex dynamics, providing new criteria for stability and detailed analysis of symmetric vortex configurations.

Findings

Linear stability corresponds to being a nondegenerate minimum of the Hamiltonian.

Positive vorticity equilibria are nonlinearly stable if linearly stable.

Stable solutions are identified in rhombus and isosceles trapezoid configurations.

Abstract

We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.