On the Stability of Stationary Rotation of a Point Vortices System in a Circular Domain
The stability problem of a system of point vortices is considered. The charges are located in the vertices of a regular N-gon (  ) in a circular area. The interaction potential between the vortices is inversely proportional to the distance between them. The quadratic terms of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically. The conditions of Routh stability and exponential instability are obtained. The range of parameters that require additional non-linear analysis is indicated. All the resonances arising in this problem up to fourth order has been listed and investigated numerically. Instability is found numerically in two of them. The results of the theoretical analysis are confirmed by numerical calculation of the trajectories of point vortices.