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Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 613–636 | Cite as

Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability

  • Alexey V. Borisov
  • Ivan S. Mamaev
  • Ivan A. Bizyaev
Article
  • 10 Downloads

Abstract

This paper is concerned with the problem of three vortices on a sphere S2 and the Lobachevsky plane L2. After reduction, the problem reduces in both cases to investigating a Hamiltonian system with a degenerate quadratic Poisson bracket, which makes it possible to study it using the methods of Poisson geometry. This paper presents a topological classification of types of symplectic leaves depending on the values of Casimir functions and system parameters.

Keywords

Poisson geometry point vortices reduction quadratic Poisson bracket spaces of constant curvature symplectic leaf collinear configurations 

MSC2010 numbers

76M23 37J05 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 2
  • Ivan A. Bizyaev
    • 1
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia

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