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

Yu, Guowei

Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 1115-1144.

The family of planar linear chains are found as collision-free action minimizers of the spatial N-body problem with equal masses under DN and DN×Z2-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 or N, but may contain collision for 0<ω<N. However it can only contain binary collisions and the corresponding collision solutions are C0 block-regularizable.

These families of solutions can be seen as a generalization of Marchal's P12 family for N=3 to arbitrary N≥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.

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