Kurakin L. G., Yudovich V. I.
Department of Mechanics and Mathematics, Rostov University, ul. Zorge 5, 344090, Rostov-on-Don, Russia.
Chaos. 2002 Sep;12(3):574-595. doi: 10.1063/1.1482175.
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.
本文致力于解决开尔文勋爵（1878 年）提出的关于位于正(n)边形顶点的(n)个等涡旋系统定常旋转稳定性的问题。在过去几十年里，由于对液氦中的点涡旋以及等离子体物理中的电子柱的研究，这个问题再次变得重要起来。这种情况由基尔霍夫方程的显式解来描述。线性化矩阵的相应特征值问题也可以明确求解。汤姆森（1883 年）和哈夫洛克（1931 年）在其著作中利用这一点得到了关于线性稳定性的详尽结果。库拉金（1994 年）证明，对于(n\leq6)，非线性轨道稳定性也成立。(n = 7)的情况尚不确定——在文献中可以找到关于稳定性和不稳定性的表述，但证明不完整或有误。在本文中，我们证明对于(n = 7)，非线性稳定性仍然成立。因此，对开尔文问题的完整答案是，正涡旋(n)边形在(n\leq7)时是稳定的，而在(n\geq8)时是不稳定的。我们还给出了具有对称群的动力系统定常运动的一般理论。在定常状态的特定情况下，稳定性和不稳定性的定义需要修改。我们不假定系统是保守的。因此，这些结果不仅可以应用于理想流体流动的各种定常状态，例如，也可以应用于粘性流体的运动。（c）2002 美国物理研究所。
