Point vortices on a sphere: Stability of relative equilibria

Abstract

In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy–momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.