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

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.