# Connecting planar linear chains in the spatial N-body problem

### Guowei Yu

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

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## Abstract

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 $N \geq 3$. 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.

## Cite this article

Guowei Yu, Connecting planar linear chains in the spatial N-body problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 4, pp. 1115–1144

DOI 10.1016/J.ANIHPC.2020.10.004