Curvature perturbations and stability of a ring of vortices
Stefanella Boatto - Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal de Rio de Janeiro, Av. Athos de Silveira Ramos 149, Cidade Universitária, Caixa Postal 68530, Ilha do Fundão, CEP 21941-909, Rio de Janeiro, RJ, Brazil (email)
Abstract: Vortex modeling has a long history. Descartes (1644) used it as a model for the solar system. J.J. Thomson (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 relative 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 Kähler 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 is positive or negative definite, when evaluated at the equilibrium configuration.
Keywords: Hamiltonian dynamics, relative equilibria, N-vortex problem, stability, differential geometry.
Received: October 2006; Revised: September 2007; Published: June 2008.
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