Point Vortices on a Sphere: Stability of Relative Equilibria
Pekarsky, S. and J. E. Marsden
J. of Math. Phys., 39, 5894-5907
Abstract:
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
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. Poison
reduction corresponding to this symmetry is performed; the quotient
space is constructed and its Poisson bracket structure and symplectic
leaves are found explicitly. 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
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 non-great circle equilateral
triangle and when the vortices line up on a great circle.