The stability of stationary rotation of a regular vortex polygon

This paper is devoted to the Lord Kelvin's (1878) problem on stability of the stationary rotation of the system of n equal vortices located in the vertices of a regular n-gon. During the last decades this problem again became actual in connection with the investigation of point vortices in liquid helium and electron columns in plasma physics. This regime is described by the explicit solution of the Kirchhoff equations. The corresponding eigenvalue problem for the linearization matrix can be also decided explicitly. This was used in the works of Thomson (1883) and Havelock (1931) to obtain exhaustive results on the linear stability. Kurakin (1994) proved that for n/=8 it is unstable. We also present the general theory of stationary motions of a dynamical system with symmetry group. The definitions of stability and instability are necessary to modify in the specific case of stationary regimes. We do not assume that the system is conservative. Thus, the results can be applied not only to various stationary regimes of an ideal fluid flows but, for instance, also to motions of viscous fluids. (c) 2002 American Institute of Physics.