Stability of relative equilibria of point vortices on a sphere and symplectic integrators

This paper analyzes the dynamics of N point vortices moving on a sphere fro m the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are provided for the (integ rable) case N = 3. Stability of relative equilibria is analyzed by the ener gy-momentum method. Explicit criteria for stability of different configurat ions with generic and non-generic momenta are obtained. In each case, a gro up of transformations is specified, such that motion in the original (unred uced) phase space is stable module this group. Finally, we outline the cons truction of a symplectic-momentum integrator for vortex dynamics on a spher e.