Topology of the Liouville foliation in the generalized constrained three-vortex problem

G. P. Palshin

Sbornik. Mathematics, Tome 215 (2024) no. 5, pp. 667-702

Résumé

We consider a completely Liouville integrable model of Hamiltonian mechanics with two degrees of freedom. This model describes the motion of two point vortices with a third vortex fixed at the origin. This problem covers, as a particular case, the problems of the motion of hydrodynamic vortices in an unbounded perfect fluid and magnetic vortices in a ferromagnetic medium. We study the topology of the Liouville foliation of this system using the bifurcation diagram of the momentum map. We prove some results on the general form of the bifurcation diagram and discover some properties of critical trajectories in the inverse image of its bifurcation curves. Using these results we show the presence of two important bifurcations of Liouville tori passing through a singular leaf of the form $\mathbb S^1 \times (\mathbb S^1 \,\dot{\cup}\, \mathbb S^1 \,\dot{\cup}\, \mathbb S^1)$. In the first case one Liouville torus bifurcates into three tori when passing through the singular leaf; in the second case two tori bifurcate into two tori. Bibliography: 46 titles.

Détail

Keywords: vortex dynamics, integrable Hamiltonian system, bifurcation diagram, momentum map
Mots-clés : Liouville foliation.