Generalized point vortex dynamics on CP^2

Montaldi, James and Shaddad, Amna (2018) Generalized point vortex dynamics on CP^2. [MIMS Preprint] (Submitted)

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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 simple Hamiltonian function. The system has symmetry group SU(3). The first paper describes all possible momentum polytopes for this system, and here we apply methods of symplectic reduction and geometric mechanics to analyze the possible relative equilibria of 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 spaces 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. For 2 vortices, the reduced spaces are just points, and the motion is governed by a collective Hamiltonian.

Item Type: MIMS Preprint
Uncontrolled Keywords: Hamiltonian systems, momentum map, symplectic geometry
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
MSC 2010, the AMS's Mathematics Subject Classification > 53 Differential geometry
Depositing User: Dr James Montaldi
Date Deposited: 27 Sep 2018 15:52
Last Modified: 27 Sep 2018 15:55
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2661

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