Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey
Journal article, 2021

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-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 two-dimensional manifold with a symmetry group; 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 two-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.

Integrable systems

Point-vortex dynamics

Symplectic reduction

Euler equations


Klas Modin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Milo Viviani

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Arnold Mathematical Journal

21996792 (ISSN) 21996806 (eISSN)

Vol. 7 3 357-385

Geometric numerical methods for computational anatomy

Swedish Research Council (VR) (2017-05040), 2018-01-01 -- 2021-12-31.

Subject Categories

Computational Mathematics


Mathematical Analysis



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