Three-Point Vortex Dynamics as a Lie-Poisson System

This paper studies the reduced dynamics of the three-vortex problem from the point of view of Lie-Poisson reduction on the dual of the Lie algebra of $U(2)$. The algebraic study leading to this point of view has been given by Borisov and Lebedev 1998 (see also Bolsinov, Borisov and Mamaev 1999). The main contribution of this paper is to properly describe the dynamics as a Lie-Poisson reduced system on $(\mathfrak{u}(2) ^\ast, \{\;,\,\}_{\text{LP}} )$, giving a systematic construction of a one-parameter family of covectors $\{ \sigma _1, \sigma _2, \sigma _3 \}$ closely related to Pauli spin matrices, and to bring light to the relation between Lie-Poisson reduction and symplectic reduction using Jacobi-Bertrand-Haretu coordinates.