Classification of Stationary Configurations of Four Identical Point Vortices

Woojin Kim

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arXiv · 2609.15090

Classification of Stationary Configurations of Four Identical Point Vortices

Woojin Kim

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Abstract

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.

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Woojin Kim. 2026-09-19. Classification of Stationary Configurations of Four Identical Point Vortices. https://arxiv.org/abs/2609.15090

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