Stability of Relative Equilibria of Point Vortices on a Sphere and Symplectic Integrators
Marsden, J. E., S. Pekarsky, and S. Shkoller
Il Nuovo Cimento, 22, 6
This paper analyzes 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 provided for the (integrable) case
N = 3. Stability of relative equilibria is analyzed by the energy-momentum method.
Explicit criteria for stability of different configurations with generic and non-generic
momenta are obtained. In each case, a group of transformations is specied, such
that motion in the original (unreduced) phase space is stable modulo this group.
Finally, we outline the construction of a symplectic-momentum integrator for vortex
dynamics on a sphere.