Three-Point Vortex Dynamics as a Lie-Poisson System

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 [2, 3] (see also [4]). The main contribution of this paper is to properly describe the dynamics as a Lie-Poisson reduced system on (u(2) * , { , } LP ), giving a systematic construction of a one-parameter family of covectors {σ 1 , σ 2 , σ 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.

Symplectic reduction and the Lie–Poisson shape dynamics of N point vortices on the plane

“…, N − 1} such that n 1 + n 2 = N − 1, depending on the signs of the circulations {Γ j } N j=1 . More recently, Hernández-Garduño [12] (see also Hernández-Garduño and Shashikanth [13]) showed that the reduced dynamics of three point vortices may be written as a Lie-Poisson equation on u(2) * with the standard Lie-Poisson bracket by constructing a set of covectors satisfying the Pauli commutation relations.…”

“…as in (12). Notice that z is in C N−2 as opposed to C N−1 here; compare with (12). Then z provides a set of coordinates for the reduced space I −1 (0)/C.…”

“…Suppose that the total circulation is non-zero, i.e. Γ := N j=1 Γ j = 0, and let q(0) ∈ C N be the initial condition for (4), z 0 ∈ Z be the corresponding element defined by (12), and µ 0 := J(z 0 ). If R(z 0 ) = 0 (i.e.…”

Reconstruction phases in the planar three- and four-vortex problems

“…This is shown using the Poisson structure of the three-vortex problem expressed in terms of symmetry invariants. A further account of this point of view is found in [17].…”

“…It follows that, in coordinates (r, s) ∈ C 2 , the matrix representation of the Hermitian metric restricted to P 0 is A B . Moreover, from (17),…”