Generalized point vortex dynamics on CP2

James Montaldi, Amna Shaddad

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)601
Number of pages619
JournalJournal of Geometric Mechanics
Issue number4
Publication statusPublished - Dec 2019


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