JournalsaihpcVol. 38, No. 4pp. 1115–1144

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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The family of planar linear chains are found as collision-free action minimizers of the spatial N-body problem with equal masses under DND_{N} and DN×Z2D_{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 ω=0\omega = 0 or N, but may contain collision for 0<ω<N0 < \omega < N. However it can only contain binary collisions and the corresponding collision solutions are C0C^{0} block-regularizable.

These families of solutions can be seen as a generalization of Marchal's P12P_{12} family for N=3N = 3 to arbitrary N3N \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][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