The topology of reduced phase spaces of the motion of vortices on a sphere

Abstract

The motion of point vortices in a perfect incompressible fluid on a two-dimensional sphere can be represented by a Hamiltonian flow which is invariant under the action of the special orthogonal group SO(3). Following the reduction method of Marsden and Weinstein this motion can be described by Hamiltonian flows on reduced phase spaces. Some topological invariants of these reduced phase spaces are calculated. For suitable vorticities the results obtained can be used to give lower bounds on numbers of relative equilibria of the original system.