Curvature perturbations and stability of a ring of vortices
The author considers point-vortex systems, which can be regarded as ``discrete'' solutions of Euler equations. Their dynamics is described by a Hamiltonian system of equations. The paper focuses on the vortex dynamics on simply connected surfaces of constant curvature \(K\), i.e. planes, spheres and hyperbolic surfaces. It is known that polygonal configurations of \(N\) point vortices are relative equilibria of the system. The author studies the stability of such polygonal configurations, and, more specifically, examines how stability depends upon the number of vortices \(N\) and on the curvature \(K\) of the surface. The problem is formulated in a unified geometric way, and is simplified with the help of fact that the surfaces of interest can be viewed as Kähler manifolds. Nonlinear stability is then studied by making use of Dirichlet criterion. Stability ranges are intervals for which the Hessian of the Hamiltonian is positive or negative definite, when evaluated at the equilibrium configuration.
scientific article; zbMATH DE number 5359477