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

Stability of Relative Equilibria in the Planar n-Vortex Problem

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...

On polygonal relative equilibria in the N-vortex problem

Helmholtz’s equations provide the motion of a system of N vortices which describes a planar incompressible fluid. A relative equilibrium is a particular solution of these equations for which the distances between the particles 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 ...

Relative Equilibria of the (1+N)-Vortex Problem

We examine existence and stability of relative equilibria of the nvortex problem specialized to the case where N vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stabi...

Finiteness of Relative Equilibria in the Planar Generalized N-body Problem with Fixed Subconfigurations

We prove that a fixed configuration of N − 1 masses in the plane can be extended to a central configuration of N masses by adding a specified additional mass only in finitely many ways. This holds for a family of potential functions including the Newtonian gravitational case and the classical planar point vortex model.

Central configurations and relative equilibria in the n–body problem