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arXiv · 1808.01769

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

Tomoki Ohsawa
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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.

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Tomoki Ohsawa. 2018-08-06. Symplectic Reduction and the Lie--Poisson Shape Dynamics of $N$ Point Vortices on the Plane. https://doi.org/10.1088/1361-6544/ab28aa

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