Point vortices on a sphere: Stability of relative equilibria

Point vortices on a sphere: Stability of relative equilibria

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 cor...

Relative equilibria of point vortices on the sphere

We prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non-rotating sphere. The proofs use the rotational symmetry group SO(3) and the resulting conservation laws, the time-reversing reflectional symmetries in O(3), and the finite symmetry group of permutations of identical vortices. Results include both global existence theor...

Stability of relative equilibria of three vortices

Dynamics of Perturbed Relative Equilibria of Point Vortices on the Sphere or Plane

The system of point vortices on the sphere is a Hamiltonian system with symmetry SO(3), and there are stable relative equilibria of four point vortices, where three identical point vortices form an equilateral triangle circling a central vortex. These relative equilibria have zero (nongeneric) momentum and form a family that extends to arbitrarily small diameters. Using the energy-momentum meth...

Relative Equilibria of Point Vorti es on the Sphere