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</=6 also the nonlinear orbital stability takes place. The case n=7 was doubtful-one can find in the literature statements about both stability and instability with incomplete or erroneous proofs. In this paper we prove that for n=7 the nonlinear stability still takes place. Thus the full answer to Kelvin's question is that the regular vortex n-gon is stable at n</=7, while at 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.