Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey
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Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey Klas Modin1
· Milo Viviani1
Received: 8 May 2020 / Revised: 14 September 2020 / Accepted: 28 September 2020 © The Author(s) 2020
Abstract Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few pointvortices 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 twodimensional 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. Keywords Point-vortex dynamics · Integrable systems · Euler equations · Symplectic reduction Mathematics Subject Classification 37J15 · 53D20 · 70H06 · 35Q31 · 76B47
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Point-Vortex Equations and Their Conservation Laws 2.1 The Sphere . . . . . . . . . . . . . . . . . . . . 2.2 The Plane . . . . . . . . . . . . . . . . . . . . . 2.3 The Hyperbolic Plane . . . . . . . . . . . . . . . 2.4 The Flat Torus . . . . . . . . . . . . . . . . . . . 3 Integrability Results . . . . . . . . . . . . . . . . . .
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Klas Modin [email protected] Milo Viviani [email protected]
1
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Gothenburg, Sweden
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K. Modin, M. Viviani 3.1 The Sphere . . . . . . . . . . . . . . . . . . . . . 3.2 The Plane . . . . . . . . . . . . . . . . . . . . . . 3.3 The Hyperbolic Plane . . . . . . . . . . . . . . . . 3.4 The Flat Torus . . . . . . . . . . . . . . . . . . . . 4 Symplectic Reduction Theory . . . . . . . . . . . . . . 5 Proofs by Symplectic Reduction . . . . . . . . . . . . . 5.1 The Sphere . . . . . . . . . . . . . . . . . . . . . 5.2 The Plane . . . . . . . . . . . . . . . . . . . . . . 5.3 The Hyperbolic Plane . . . . . . . . . . . . . . . . 5.4 The Flat Torus . . . . . . . . . . . . . . . . . . . . 6 Non-integrability Results . . . . . . . . . . . . . . . . 7 Outlook: Long-Time Predictions for 2D Euler Equations Appendix A: Gallery of point-vortex solutions . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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