Bifurcation of Relative Equilibria for Vortices and General Homogeneous Potentials

Abstract

We examine the relative equilibria of the four vortex problem where three vortices have equal strength, that is $$ \varGamma _1 = \varGamma _2 = \varGamma _3 = 1 $$ and $$ \varGamma _4 = m $$, where m is a parameter. We study the problem in the case of the classical logarithmic vortex Hamiltonian and in the case the Hamiltonian is a homogeneous function of degree $$ \alpha $$. First we study the bifurcations emanating from the equilateral triangle configuration with the fourth vortex in the barycenter. Then, in the vortex case, we show that all the convex configurations contain a line of symmetry. We also give a precise count of the number of concave asymmetric configurations. Our techniques involve a combination of analysis and computational algebraic geometry.