## 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 CP2 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 coecients of the symplectic forms on the copies of

CP2. 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.

of the point vortex system on surfaces to a Hamiltonian dynamical

system consisting of two or three points on complex projective space CP2 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 coecients of the symplectic forms on the copies of

CP2. 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.

Original language | English |
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Pages (from-to) | 601 |

Number of pages | 619 |

Journal | Journal of Geometric Mechanics |

Volume | 11 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2019 |