2018/08/31 by Tomoki Ohsawa · 1 citation
Mathematics · Physics and Astronomy · #math-ph #math.MP #math.SG #msc:37J15 #msc:53D20 #msc:70H05 #msc:70H06 #msc:76B47 #nlin.SI #physics.flu-dyn
paper · pdf · doi:10.1088/1361-6544/ab28aa
published as Nonlinearity, Volume 32, Number 10, pp. 3820-3842, 2019 · 21 pages, 4 figures, in press at Nonlinearity
arxiv created 2019/08/28 · arxiv updated 2019/09/11
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.