Shape dynamics of point vortices

Applied and Computational Mathematics Seminar
Monday, April 1, 2019 - 1:55pm for 1 hour (actually 50 minutes)
Skiles 005
Prof. Tomoki Ohsawa – UT Dallas –
Molei Tao
We present a Hamiltonian formulation of the dynamics of the ``shape'' of N point vortices on the plane and the sphere: For example, if N=3, it is the dynamics of the shape of the triangle formed by three point vortices, regardless of the position and orientation of the triangle on the plane/sphere.For the planar case, reducing the basic equations of point vortex dynamics by the special Euclidean group SE(2) yields a Lie-Poisson equation for relative configurations of the vortices. Particularly, we show that the shape dynamics is periodic in certain cases. We extend the approach to the spherical case by first lifting the dynamics from the two-sphere to C^2 and then performing reductions by symmetries.