Relative Dynamics of Vortices in Confined Bose--Einstein Condensates

We consider the relative dynamics -- the dynamics modulo rotational symmetry in this particular context -- of $N$ vortices in confined Bose--Einstein Condensates (BEC) using a finite-dimensional vortex approximation to the two-dimensional Gross--Pitaevskii equation. We give a Hamiltonian formulation of the relative dynamics by showing that it is an instance of the Lie--Poisson equation on the dual of a certain Lie algebra. Just as in our accompanying work on vortex dynamics with the Euclidean symmetry, the relative dynamics possesses a Casimir invariant and evolves in an invariant set, yielding an Energy--Casimir-type stability condition. We consider three examples of relative equilibria -- those solutions that are undergoing rigid rotations about the origin -- with $N=2, 3, 4$, and investigate their stability using the stability condition.