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arXiv · 2003.00716

Integrability of point-vortex dynamics via symplectic reduction: a survey

Klas Modin·Milo Viviani
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Abstract

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on 2-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here we give a unified framework for proving integrability results for $N=2$, $3$, or $4$ point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any 2-dimensional manifold; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in 2-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

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Klas Modin, Milo Viviani. 2020-03-02. Integrability of point-vortex dynamics via symplectic reduction: a survey. https://doi.org/10.1007/s40598-020-00162-8

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