Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2004.08437 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910516298907648 |
|---|---|
| author | Hoyer-Leitzel, Alanna Le, Sophie Phuong |
| author_facet | Hoyer-Leitzel, Alanna Le, Sophie Phuong |
| contents | We investigate the symmetry of point vortices with one dominant vortex and four vortices with infinitesimal circulations in the (1+4)-vortex problem, a subcase of the five-vortex problem. The four infinitesimal vortices inscribe quadrilaterals in the unit circle with the dominant vortex at the origin. We consider symmetric configurations which have one degree of spacial freedom, namely the (1+N)-gon, kites, rectangles, and trapezoids with three equal sides. We show there is only one possible rectangular configuration (up to rotation and ordering of the vortices) and one possible trapezoid with three equal sides (up to rotation and ordering), while there are parametrically defined families of kites. Additionally we consider the (1+4)-gon and show that the infinitesimal vortices must have equal circulations on opposite corners of the square. The proofs are heavily dependent on techniques from algebraic geometry and require the use of a computer to calculate Grobner bases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_08437 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Symmetric relative equilibria with one dominant and four infinitesimal point vortices Hoyer-Leitzel, Alanna Le, Sophie Phuong Dynamical Systems Algebraic Geometry We investigate the symmetry of point vortices with one dominant vortex and four vortices with infinitesimal circulations in the (1+4)-vortex problem, a subcase of the five-vortex problem. The four infinitesimal vortices inscribe quadrilaterals in the unit circle with the dominant vortex at the origin. We consider symmetric configurations which have one degree of spacial freedom, namely the (1+N)-gon, kites, rectangles, and trapezoids with three equal sides. We show there is only one possible rectangular configuration (up to rotation and ordering of the vortices) and one possible trapezoid with three equal sides (up to rotation and ordering), while there are parametrically defined families of kites. Additionally we consider the (1+4)-gon and show that the infinitesimal vortices must have equal circulations on opposite corners of the square. The proofs are heavily dependent on techniques from algebraic geometry and require the use of a computer to calculate Grobner bases. |
| title | Symmetric relative equilibria with one dominant and four infinitesimal point vortices |
| topic | Dynamical Systems Algebraic Geometry |
| url | https://arxiv.org/abs/2004.08437 |