Montaldi, James and Shaddad, Amna (2019) Generalized point vortex dynamics on CP^2. J. Geometric Mechanics. (In Press)
This is the latest version of this item.
Text
CP2Vortices2.pdf  Accepted Version Download (737kB) 
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 2sphere, and proceed to describe the reduced dynamics under simple hypotheses on the type of Hamiltonian interaction. The other nontrivial 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 nonabelian sense.
Item Type:  Article 

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:  17 Jun 2019 20:52 
Last Modified:  17 Jun 2019 20:52 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2719 
Available Versions of this Item

Generalized point vortex dynamics on CP^2. (deposited 27 Sep 2018 15:52)
 Generalized point vortex dynamics on CP^2. (deposited 17 Jun 2019 20:52) [Currently Displayed]
Actions (login required)
View Item 