论文标题
$ N $ - 涡流问题的柏拉图固体和对称解决方案
Platonic solids and symmetric solutions of the $N$-vortex problem on the sphere
论文作者
论文摘要
假设所有涡流都具有相同的强度，我们认为球体上的$ n $涡旋问题。我们开发了一个理论框架，以分析具有规定对称性的运动方程的解决方案。我们的构建依赖于整个对称组的扭曲亚组对系统的离散降低，该组旋转并置入涡流。我们的方法形式化并扩展了Tokieda（C. R.Acad。Sci。，Paris I 333（2001））和Soulière和Tokieda（J. FluidMech。460（2002））先前概述的想法，并允许我们证明存在几个1-参数周期性的Orbits家族。这些家族要么从平衡中散发出来，要么汇聚为具有特定对称性的碰撞。我们的结果用于显示出柏拉图固体平衡产生的小非线性振荡家族的存在。
We consider the $N$-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the discrete reduction of the system by twisted subgroups of the full symmetry group that rotates and permutes the vortices. Our approach formalises and extends ideas outlined previously by Tokieda (C. R. Acad. Sci., Paris I 333 (2001)) and Soulière and Tokieda (J. Fluid Mech. 460 (2002)) and allows us to prove the existence of several 1-parameter families of periodic orbits. These families either emanate from equilibria or converge to collisions possessing a specific symmetry. Our results are applied to show existence of families of small nonlinear oscillations emanating from the platonic solid equilibria.
