Three vortices in spaces of constant curvature: reduction, Poisson geometry, and stability. (English) Zbl 1411.76116
Summary: 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.
MSC:
| 76M23 | Vortex methods applied to problems in fluid mechanics |
| 37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
Keywords:
Poisson geometry; point vortices; reduction; quadratic Poisson bracket; spaces of constant curvature; symplectic leaf; collinear configurationsReferences:
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