Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey

  title={Integrability of Point-Vortex Dynamics via Symplectic Reduction: A Survey},
  author={Klas Modin and Milo Viviani},
  journal={Arnold Mathematical Journal},
  pages={357 - 385}
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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage… 

Canonical scale separation in two-dimensional incompressible hydrodynamics

It is shown that Euler’s equations posses an intrinsic, canonical splitting of the vorticity function, which accounts for the “broken line” in the power law for the energy spectrum, observed in both experiments and numerical simulations.

Platonic Solids and Symmetric Solutions of the N-vortex Problem on the Sphere

We consider the N-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics

On Maximally Mixed Equilibria of Two-Dimensional Perfect Fluids

The vorticity of a two-dimensional perfect (incompressible and inviscid) fluid is transported by its area preserving flow. Given an initial vorticity distribution ω0\documentclass[12pt]{minimal}

Vortex Motion of the Euler and Lake Equations

  • Cheng Yang
  • Physics, Mathematics
    Journal of Nonlinear Science
  • 2021
It is proved that the 2-vortex system in the half-plane is nonintegrable for $N>2$ and the skew-mean-curvature flow in R^n, n with certain symmetry can be regarded as point vortex motion of the 2D lake equations.



Coadjoint Orbits for A

A complete description of the coadjoint orbits for A + n−1 , the nilpo-tent Lie algebra of n × n strictly upper triangular matrices, has not yet been obtained, though there has been steady progress

Co-adjoint Orbits

The motion of three point vortices on a sphere

We consider the incompressible and inviscid flow on a sphere. The vorticity distributes as a point vortex. The governing equation for point vortices on a sphere is given by Bogomolov [3]. In the

Integrable four vortex motion

It follows from the Poisson brackets between constants of motion that the motion of four vortices of zero net vorticity is integrable if the total momentum vanishes. The phase space motion of this

Théorie des tourbillons

Motion of three point vortices in a periodic parallelogram

The motion of three interacting point vortices with zero net circulation in a periodic parallelogram defines an integrable dynamical system. A method for solving this system is presented. The

Integrable four-vortex motion on sphere with zero moment of vorticity

We consider the motion of four vortex points on sphere, which defines a Hamiltonian dynamical system. When the moment of vorticity vector, which is a conserved quantity, is zero at the initial

Point vortices on the hyperbolic plane

We investigate the dynamical system of point vortices on the hyperboloid. This system has non-compact symmetry SL(2, R) and a coadjoint equivariant momentum map. The relative equilibrium conditions

A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics

An efficient numerical method is developed for long-time simulations that preserve the geometric features of the exact flow, in particular conservation of Casimirs, and shows that there is a correlation between a first integral of $\unicode[STIX]{x1D6FE}$ (the ratio of total angular momentum and the square root of enstrophy) and the long- time behaviour.

A minimal-coordinate symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the