Point vortices on the hyperbolic plane

Abstract

We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on N=2, 3. Contrary to the system on the sphere, relative equilibria with non-compact momentum isotropy subgroup are found, and are used to illustrate the different stability types of relative equilibria.

To appear in J. Mathematical Physics

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• Integrability of point-vortex dynamics via symplectic reduction: a survey
• A Hamiltonian approach for point vortices on non-orientable surfaces