- 06-311 Stefanella Boatto
- Curvature Perturbations and Stability of a Ring of Vortices
Nov 1, 06
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Abstract. Vortex modeling has a long history. Descartes (1644) used it as a model for the solar system.
J.J. Thomsom (1883) used it as a model for the atom. We consider point-vortex systems,
which can be regarded as "discrete" solutions of the Euler equation. Their dynamics is described by a Hamiltonian system of equations. In particular we are interested in vortex
dynamics on simply connected surfaces of constant curvature K - i.e. a plane, spheres and
hyperbolic surfaces. It is known that polygonal configurations of N point-vortices are
realtive equilibria of the system. We study the stability of such polygonal configurations, and,
more specifically, how stability depends upon the number of vortices N and the curvature K
of the surface. To address such a question we have to formulate the problem in a unified
geometrical way. The fact that the surfaces of interest can be viewed as K\"alher manifolds
greatly simplify our task. Nonlinear stability is then studied by making use of the Dirichlet
Criterion. Stability ranges are the K-intervals for which the Hessian of the Hamiltonian - evaluated at the equilibrium configuration - is positive or negative definite.