The family of planar linear chains are found as collision-free action minimizers of the spatial N-body problem with equal masses under and -symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in  for the planar N-body problem. In particular, the monotone constraints required in  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 or N, but may contain collision for . However it can only contain binary collisions and the corresponding collision solutions are block-regularizable.
These families of solutions can be seen as a generalization of Marchal's family for to arbitrary . In particular, for certain types of topological constraints, based on results from  and , we show that when ω belongs to some sub-intervals of , 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. 1115–1144DOI 10.1016/J.ANIHPC.2020.10.004