Integrability of point-vortex dynamics via symplectic reduction: a survey. (English) Zbl 1482.37056

The authors consider point-vertex dynamics and their integrability. This dynamics is described via idealized non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. The aim is to provide a unified treatment for proving integrability results for 2-, 3-, or 4-point-vertices. Part of their goal is to show how the symplectic reduction can provide a broader approach for proving integrability results, especially for point-vertex dynamics.
Euler equations on an orientable Riemannian manifold that govern an incompressible inviscid fluid have the form \(\dot{\mathbf{v}} + \nabla _\mathbf{v} \mathbf{v} = - \nabla p\) with div \(\mathbf{v} = 0\), where \(\mathbf{v}\) is a vector field on a manifold \(M\) incorporating the motion of the fluid’s particles, \(p\) is the pressure function and \(\nabla _\mathbf{v}\) is the covariant derivative along \(\mathbf{v}\). H. Helmholtz [J. Reine Angew. Math. 55, 25–55 (1858; ERAM 055.1448cj)] showed that the two-dimensional Euler equations have special solutions with a finite number of point-vertices. These solutions are not smooth and are characterized by the vorticity \mbox{curl \(\mathbf{v} = \sum_{i=1} ^ {n} \Gamma _i \delta_{\mathbf{r}_i}\)} where non-zero \(\Gamma_i\) is the strength of the vortex \(i\), \(\mathbf{r}_i\) is its position, and \(\delta_{\mathbf{r}_i}\) is a delta function.
The authors describe point-vertex equations and their Hamiltonian structures on the sphere, the plane, the hyperbolic plane, and the flat torus. Each of these cases has a different symmetry group, and the symplectic reduction is treated separately in each case to establish integrability.
The authors also briefly review nonintegrability results. They conclude with some observations about how their results pertain to long term predictions for the Euler equations. An appendix offers a kind of visual portrait of point-vortex solutions.


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
53D20 Momentum maps; symplectic reduction
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
76B47 Vortex flows for incompressible inviscid fluids


ERAM 055.1448cj
Full Text: DOI arXiv


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