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We show that the symplectic reduction of the dynamics of $N$ point vorticeson the plane by the special Euclidean group $\mathsf{SE}(2)$ yields aLie--Poisson equation for relative configurations of the vortices.Specifically, we combine symplectic reduction by stages with a dual pairassociated with the reduction by rotations to show that the$\mathsf{SE}(2)$-reduced space with non-zero angular impulse is a coadjointorbit. This result complements some existing works by establishing arelationship between the symplectic/Hamiltonian structures of the original andreduced dynamics. We also find a family of Casimirs associated with theLie--Poisson structure including some apparently new ones. We demonstratethrough examples that one may exploit these Casimirs to show that some shapedynamics are periodic.

Symplectic Reduction and the Lie--Poisson Shape Dynamics of $N$ Point Vortices on the Plane