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

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.

26 pages, 4 figures, accepted in Arnold Math. J

References in corpus (4)

• A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
• Lie algebras in vortex dynamics and celestial mechanics - IV
• Lie-Poisson methods for isospectral flows
• A minimal-variable symplectic method for isospectral flows

Cited by in corpus (1)

• Generalised point vortices on a plane