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.