Connecting planar linear chains in the spatial N-body problem

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 $N\ge 3$. 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.