The family of planar linear chains are found as collision-free action minimizers of the spatial N-body problem with equal masses under $D_N$ and $D_N\times {\mathbb{Z}}_2$-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in [32] for the planar N-body problem. In particular, the monotone constraints required in [32] are proven to be unnecessary, as it will be implied by the action minimization property.

For each type of topological constraints, by considering the corresponding action minimization problem in a coordinate frame rotating around the vertical axis at a constant angular velocity ω, we find an entire family of simple choreographies (seen in the rotating frame), as ω changes from 0 to N. Such a family starts from one planar linear chain and ends at another (seen in the original non-rotating frame). The action minimizer is collision-free, when $\omega =0$ or N, but may contain collision for $0<\omega <N$. However it can only contain binary collisions and the corresponding collision solutions are $C^0$ block-regularizable.

These families of solutions can be seen as a generalization of Marchal's $P_{12}$ family for $N=3$ to arbitrary $N\ge 3$. In particular, for certain types of topological constraints, based on results from [3] and [7], we show that when ω belongs to some sub-intervals of $[0,N]$, the corresponding minimizer must be a rotating regular N-gon contained in the horizontal plane.

平面线性链族被发现是质量相等的空间N体问题的无碰撞动作最小化器$D_N$ 和 $D_N\times {\mathbb{Z}}_2$-对称约束和不同类型的拓扑约束。这概括了作者在 [32] 中针对平面N体问题的先前结果。特别是，[32] 中要求的单调约束被证明是不必要的，因为它会被动作最小化属性所暗示。

对于每种类型的拓扑约束，通过考虑以恒定角速度ω绕垂直轴旋转的坐标系中相应的动作最小化问题，我们找到了整个简单编排系列（在旋转框架中可见），因为ω从0至ñ。这样的家庭从一个平面线性链开始，到另一个结束（在原始的非旋转框架中看到）。动作最小化器是无碰撞的，当$\omega =0$或N，但可能包含碰撞$0<\omega <N$. 但是它只能包含二进制碰撞，相应的碰撞解决方案是$C^0$ 块可正则化。

这些解决方案系列可以看作是 Marchal 的一般化 ${磷}_{12}$ 家庭为 $N=3$ 随意 $N\ge 3$. 特别是，对于某些类型的拓扑约束，基于 [3] 和 [7] 的结果，我们表明当ω属于$[0,N]$，相应的最小化器必须是包含在水平平面中的旋转规则N边形。