Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2003.00716 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929220555374592 |
|---|---|
| author | Modin, Klas Viviani, Milo |
| author_facet | Modin, Klas Viviani, Milo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_00716 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Integrability of point-vortex dynamics via symplectic reduction: a survey Modin, Klas Viviani, Milo Mathematical Physics Dynamical Systems Symplectic Geometry 37J15, 53D20, 70H06, 35Q31, 76B47 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. |
| title | Integrability of point-vortex dynamics via symplectic reduction: a survey |
| topic | Mathematical Physics Dynamical Systems Symplectic Geometry 37J15, 53D20, 70H06, 35Q31, 76B47 |
| url | https://arxiv.org/abs/2003.00716 |