四个相同点涡的静止构型分类
Classification of Stationary Configurations of Four Identical Point Vortices
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中文总结 AI 辅助
本文分类了平面中四个相同点涡的所有静止构型，发现恰好三种相对平衡构型，其中正方形为全局最小且唯一稳定，其余为鞍点。
中文摘要 AI 辅助
我们找到并分类了平面中四个相同点涡系统的所有静止构型。在系统的自然对称性下，恰好存在三种构型，它们都是相对平衡：一个正方形、一个共线构型，以及一个带中心涡的等边三角形。其中，正方形构型是哈密顿量的全局最小值，而另外两种构型是鞍点。特别地，正方形是静止构型中唯一稳定的构型。
英文摘要
We find and classify every stationary configuration of the system of four identical point vortices in the plane. Up to the natural symmetries of the system, there are exactly three configurations, which are relative equilibria: a square, a collinear configuration, and an equilateral triangle with a central vortex. Among them, the square configuration is the global minimum of the Hamiltonian, whereas the other two configurations are saddle points. In particular, the square is the unique stable configuration among the stationary configurations. Using the special algebraic structure of four points, we introduce a change of variables based on the Hadamard matrix of order four, which transforms the original four-vortex problem into an equivalent three-point problem.
