Point vortices on a sphere: Stability of relative equilibria

Abstract

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 ͑integrable͒ case of three vortices. The system under consideration is SO͑3͒ invariant; the associated momentum map generated by this SO͑3͒ symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original ͑unreduced͒ phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.

Relative Equilibria and Conserved Quantities in Symmetric Hamiltonian Systems

“…There have been two recent studies of the system of 3 point vortices, by Kidambi and Newton [30] (who also treat the 2 vortex case in an appendix) and by Pekarsky and Marsden [63]. The former describes not only the relative equilibria, but also self-similar collapse, where triple collision occurs in finite time with the three vortices retaining their same shape up to similarity.…”

“…It is shown in [63] that the configuration where the four vortices lie at the vertices of a regular tetrahedron is always a relative equilibrium. It is also easy to show (from the equations of motion) that a square lying in a great circle is always an re, independently of the values of the vorticities.…”

“…The computations for the configurations of vortices lying on great circles are longer, and I will not go into them further here; details can be found in the original papers [30,63]. Remarks 6.2 (i) The case σ 2 (λ) = 0 remains to be understood.…”

“…Bogomolov [21]. A study of the dynamics of 3 point vortices has been carried out by Kidambi and Newton [30] and by Pekarsky and Marsden [63]. The case of N identical point vortices has been treated in [37].…”

“…The following results are mostly taken from [30] and [63], and some are discussed below. Denote λ 1 λ 2 + λ 2 λ 3 + λ 3 λ 1 by σ 2 (λ).…”