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
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.
Poisson geometry, point vortices, reduction, quadratic Poisson bracket, spaces of constant curvature, symplectic leaf, collinear configurations
|Ministry of Education and Science of the Russian Federation
|Russian Foundation for Basic Research
|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).
MSC: 76M23, 37J05
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
\by Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev
\paper Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability
\jour Regul. Chaotic Dyn.
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