论文标题
对称相对平衡，具有一个显性和四个无限点涡流
Symmetric relative equilibria with one dominant and four infinitesimal point vortices
论文作者
论文摘要
我们研究了点涡流的对称性，其中一个显性涡流和四个具有无限循环的涡流（1+4） - 涡流问题，这是五涡式问题的子案例。四个无穷小的涡旋在单位圆中刻有四边形，与原点的主要涡流。我们考虑具有一定程度的空间自由的对称配置，即（1+n）-gon，风筝，矩形和具有三个相等侧面的梯形。我们表明，只有一种可能的矩形配置（直至涡流的旋转和排序）和一个可能具有三个相等侧的梯形（直至旋转和排序），而有参数定义的风筝家族。另外，我们考虑（1+4）-gon，并表明无穷小的涡旋必须在正方形的相对角上具有相等的循环。这些证明在很大程度上取决于代数几何形状的技术，并且需要使用计算机来计算Grobner碱基。
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.
