Three-Point Vortex Dynamics as a Lie-Poisson System

TL;DR

This paper reformulates the three-vortex problem as a Lie-Poisson system on the dual of u(2), linking algebraic structures with vortex dynamics and clarifying the relation to symplectic reduction.

Contribution

It provides a systematic Lie-Poisson reduction framework for three-vortex dynamics, introducing a family of covectors related to Pauli matrices and connecting to symplectic reduction methods.

Findings

Describes three-vortex dynamics as a Lie-Poisson system on u(2)*

Constructs a family of covectors related to Pauli spin matrices

Links Lie-Poisson reduction with symplectic reduction using Jacobi-Bertrand-Haretu coordinates

Abstract

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.