论文标题
$ n $ point涡流的形状动力学
Shape Dynamics of $N$ Point Vortices on the Sphere
论文作者
论文摘要
我们给出了相对运动或$ n $ point涡流的相对运动动力学的几何说明，从而利用了系统的$ \ mathsf {so}（3）$ - 系统的对称性。主要的想法是绕过$ \ mathsf {so}（3）$的技术难度 - 首先将动态从$ \ Mathbb {s}^{2} $提升为$ \ Mathbb {c}^{2} $。然后，我们使用双对进行$ \ mathsf {u}（2）$ - 缩减二重对，以获得形状动力学的lie-poisson动力学。这种谎言 - 贫民的结构可以帮助我们找到形状动力学的卡西米尔家族。我们进一步通过$ \ mathbb {t}^{n-1} $ - 对称性来减少系统，以获得与Borisov和Pavlov先前工作的形状变量更少的形状动力学的泊松结构。作为形状动力学的应用，我们证明四面体相对平衡是稳定的，当它们的所有循环都具有相同的符号，从而将一些现有结果推广到四面体相对涡流的相对平衡上。
We give a geometric account of the relative motion or the shape dynamics of $N$ point vortices on the sphere exploiting the $\mathsf{SO}(3)$-symmetry of the system. The main idea is to bypass the technical difficulty of the $\mathsf{SO}(3)$-reduction by first lifting the dynamics from $\mathbb{S}^{2}$ to $\mathbb{C}^{2}$. We then perform the $\mathsf{U}(2)$-reduction using a dual pair to obtain a Lie--Poisson dynamics for the shape dynamics. This Lie--Poisson structure helps us find a family of Casimirs for the shape dynamics. We further reduce the system by $\mathbb{T}^{N-1}$-symmetry to obtain a Poisson structure for the shape dynamics involving fewer shape variables than those of the previous work by Borisov and Pavlov. As an application of the shape dynamics, we prove that the tetrahedron relative equilibria are stable when all of their circulations have the same sign, generalizing some existing results on tetrahedron relative equilibria of identical vortices.
