Nonlinear Stability of Relative Equilibria in Planar $N$-Vortex Problem

Abstract: We prove a sufficient condition for nonlinear stability of relative equilibria in the planar $N$-vortex problem. This result builds on our previous work on the Hamiltonian formulation of its relative dynamics as a Lie--Poisson system. The relative dynamics recasts the stability of relative equilibria of the $N$-vortex problem as that of the corresponding fixed points in the relative dynamics. We analyze the stability of such fixed points by exploiting the Hamiltonian formulation as well as invariants and constraints that naturally arise in the relative dynamics. The stability condition is essentially an Energy--Casimir method, except that we also incorporate the constraints in an effective manner. We apply the method to two types of relative equilibria: (i) three identical vortices at the vertices of an equilateral triangle along with another one at its center and (ii) four identical vortices at the vertices of a square with another one its center, where the circulation of the vortex at the center is arbitrary in both cases. We show that they are stable/unstable depending on the circulation of the vortex at the center.