Symplectic reduction and the Lie-Poisson shape dynamics of N point vortices on the plane

We show that the symplectic reduction of the dynamics of N point vortices on the plane by the special Euclidean group SE(2) yields a Lie-Poisson equation for relative configurations of the vortices. Specifically, we combine symplectic reduction by stages with a dual pair associated with the reduction by rotations to show that the SE(2)-reduced space with non-zero angular impulse is a coadjoint orbit. This result complements some existing works by establishing a relationship between the symplectic/Hamiltonian structures of the original and reduced dynamics. We also find a family of Casimirs associated with the Lie-Poisson structure including some apparently new ones. We demonstrate through examples that one may exploit these Casimirs to show that some shape dynamics are periodic.