It has been shown by Chen that if all masses are in the same proper vector subspace of ${\mathbb{R}}^d(d\ge 2)$ on one of the free boundaries, local minimizers connecting the two free boundaries have no collision on that boundary. However, not much is known for other types of boundary configurations.

In this paper, we consider minimizers connecting two free boundaries and study possible quadruple collisions under the following four symmetric configurations: isosceles trapezoid configuration, rectangular configuration, double isosceles configuration and diamond configuration. Whenever the boundary configuration of four bodies coincides with one of the four symmetric configurations, we show that these four bodies are free of quadruple collision on that boundary set. New ideas and detailed analysis are introduced in order to construct deformed paths in some of the cases and estimate their action values. As its applications, we show the existence of several sets of periodic orbits in the planar four-body and five-body problems.

Chen已经证明，如果所有质量都在的相同的适当矢量子空间中 ${\mathbb{[R}}^d（d\ge \mathrm{2个}）$在其中一个自由边界上，连接两个自由边界的局部极小值在该边界上没有冲突。但是，对于其他类型的边界配置知之甚少。

在本文中，我们考虑连接两个自由边界的极小值，并研究以下四种对称配置下的可能的四重碰撞：等腰梯形配置，矩形配置，双等腰配置和菱形配置。只要四个物体的边界构型与四个对称构型之一重合，我们就会证明这四个物体在该边界集上没有四重碰撞。为了在某​​些情况下构造变形路径并估计其作用值，引入了新的思想和详细的分析。作为其应用，我们显示了平面四体和五体问题中存在几组周期轨道。