Classification of Stationary Configurations of Four Identical Point Vortices
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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BibTeX
@misc{kim2026classification,
title = {{Classification of Stationary Configurations of Four Identical Point Vortices}},
author = {Woojin Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15090},
eprint = {2609.15090},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15090},
}
RIS
TY - UNPB
TI - Classification of Stationary Configurations of Four Identical Point Vortices
AU - Woojin Kim
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.15090
UR - https://arxiv.org/abs/2609.15090
AB - 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.
ER -