Three-Point Vortex Dynamics as a Lie-Poisson System

2016/09/19 by Antonio Hernández-Garduño, Hernández-Garduño, Antonio
Chemistry · Mathematics · Physics and Astronomy · #53D20 #70F99 #76B47 #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #math-ph #math.MP #msc:53D20 #msc:70F99 #msc:76B47

paper · pdf · doi:10.48550/arxiv.1609.05851

22 pages, 1 figure

openalex publication_date 2016/09/19 · arxiv created 2019/01/25 · arxiv updated 2019/01/29 · openalex created_date 2019/02/21 · openalex updated_date 2026/07/28

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 (\mathfraku(2) ^∗, \ , \_LP ) , giving a systematic construction of a one-parameter family of covectors \ σ₁, σ₂, σ₃ \ 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.

Related

• Lie algebras in vortex dynamics and celestial mechanics - IV
• Poisson structures of some heavenly type dynamical systems
• Dynamics and statics of vortices on a plane and a sphere - I
• On the dynamics of a Hamilton-Poisson system
• The Global Phase Space of the Three-Vortex Interaction System