Classification of Stationary Configurations of Four Identical Point Vortices

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.