Abstract

The paper deals with the study of the model of point vortices proposed by the German scientist Hermann Helmholtz. Necessary and sufficient conditions for the existence of infinitely many nonequivalent stationary configurations are found for a system consisting of two point vortices of intensity \(\Gamma_1\) and an arbitrary number of point vortices of intensity \(\Gamma_2\). A classification of such configurations is carried out. For the first time, a connection is discovered between the negative Diophantine Pell equation and stationary configurations of point vortices on the plane.

中文翻译：

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负佩尔方程与平面上点涡的稳态构型

摘要

本文研究了德国科学家赫尔曼·亥姆霍兹提出的点涡模型。对于由两个强度为\(\Gamma_1\)的点涡旋和任意数量的强度为\(\Gamma_2\) 的点涡旋组成的系统，找到了存在无限多个非等价稳态构型的必要和充分条件。对此类配置进行分类。首次发现负丢番图佩尔方程与平面上点涡的稳态构型之间的联系。