Generalized point vortex dynamics on $CP ^2$

TL;DR

This paper extends point vortex dynamics to complex projective space CP^2, analyzing the system's symmetries, reduced spaces, and relative equilibria for two and three vortices, revealing integrability properties.

Contribution

It introduces a Hamiltonian system of generalized vortices on CP^2, analyzes its symplectic reduction, and characterizes the reduced spaces and dynamics for two and three vortices.

Findings

Reduced space for three vortices is generically a 2-sphere.

For two vortices, the reduced space is a point, leading to collective Hamiltonian motion.

The system is completely integrable in the non-abelian sense.

Abstract

This is the second of two companion papers. We describe a generalization of the point vortex system on surfaces to a Hamiltonian dynamical system consisting of two or three points on complex projective space CP^2 interacting via a Hamiltonian function depending only on the distance between the points. The system has symmetry group SU(3). The first paper describes all possible momentum values for such systems, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of such interacting generalized vortices. The different types of polytope depend on the values of the `vortex strengths', which are manifested as coefficients of the symplectic forms on the copies of CP^2. We show that the reduced space for this Hamiltonian action for 3 vortices is generically a 2-sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other non-trivial reduced spaces are topological spheres with isolated singular points. For 2 generalized vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian, whereas for 3 the reduced spaces are of dimension at most 2. In both cases the system will be completely integrable in the non-abelian sense.