Morse Theory and Relative Equilibria in the Planar $n$-Vortex Problem

TL;DR

This paper applies Morse theory to analyze the stability and properties of relative equilibria in the planar n-vortex problem, revealing new insights into their Morse indices, instability, and collision behavior.

Contribution

It introduces Morse theoretical methods to study relative equilibria in the planar n-vortex problem, establishing relationships between Morse index and eigenvalues, and proving instability and collision avoidance results.

Findings

Morse index equals the number of real eigenvalue pairs for positive circulations.

Some families of relative equilibria in the four-vortex problem are proven unstable.

Relative equilibria with positive circulations cannot accumulate on collision sets.

Abstract

Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative equilibria cannot accumulate on the collision set.