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

TL;DR

This paper demonstrates how symplectic reduction of $N$ point vortex dynamics on the plane leads to a Lie--Poisson structure for relative configurations, revealing new Casimirs and periodic shape dynamics.

Contribution

It establishes a connection between symplectic reduction and Lie--Poisson structures for vortex dynamics, introducing new Casimirs and analyzing shape dynamics.

Findings

Reduced space with non-zero angular impulse is a coadjoint orbit.

Identifies a family of Casimirs including new ones.

Shows some shape dynamics are periodic using Casimirs.

Abstract

We show that the symplectic reduction of the dynamics of $N$ point vortices on the plane by the special Euclidean group $\mathsf{SE}(2)$ 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 $\mathsf{SE}(2)$-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.