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

Abstract

We show that the symplectic reduction of the dynamics of N point vortices on the plane by the special Euclidean group 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 -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.