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.

No abstract

Helical (Beltrami) flow with nonuniform coefficient is considered for the case of compressible fluid and a class of exact solutions is proposed. A paradox of helical flow is discussed and the compressibility is considered as a possible resolution of the paradox. Examples with different symmetries are given for the compressible helical flow and, in particular, the generalization of the ABC (Arnold-Beltrami-Childress) flow for the compressible case is proposed. It was shown in [31 that for incompressible fluid and nondegenerate level surfaces of the Bernoulli function qo (nondegenerate Bernoulli surfaces) the field v is integrable, which means that the Bernoulli surfaces are invariant tori and cylinders, and that motion along the surfaces is quasiperiodic.

No abstract

No abstract