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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2018, Volume 23, Issue 5, Pages 613–636 (Mi rcd349)

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

Alexey V. Borisova, Ivan S. Mamaevb, Ivan A. Bizyaeva

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia

Abstract: This paper is concerned with the problem of three vortices on a sphere $S^2$ and the Lobachevsky plane $L^2$. 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

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.2404.2017/4.65-100 Russian Foundation for Basic Research 17-01-00846-a The work of A.V. Borisov (Sections 1 and 2) and I.A. Bizyaev (Section 4) was carried out within the framework of the state assignment to the Ministry of Education and Science of Russia (1.2404.2017/4.6). The work of I. S. Mamaev (Section 3) is carried out at MIPT under project 5-100 for state support for leading universities of the Russian Federation. Also this work is supported by the Russian Foundation for Basic Research (Project No. 17-01-00846-a).

DOI: https://doi.org/10.1134/S1560354718050106

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Bibliographic databases:

Document Type: Article
MSC: 76M23, 37J05
Accepted:04.09.2018
Language: English

Citation: Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability”, Regul. Chaotic Dyn., 23:5 (2018), 613–636

Citation in format AMSBIB
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