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

TL;DR

This paper extends the understanding of spatial N-body problem by identifying collision-free action minimizers for linear chains under symmetry constraints, revealing families of choreographies and conditions for collision or regular polygon configurations.

Contribution

It generalizes previous planar results to spatial cases, removes monotone constraints, and shows the existence of continuous families of choreographies connecting linear chains.

Findings

Action minimizers are collision-free at specific rotation speeds.

Families of choreographies connect different linear chains.

Certain rotation speeds lead to regular N-gon configurations.

Abstract

The family of planar linear chains are found as collision-free action minimizers of the spatial $N$-body problem with equal masses under $D_N$ or $D_N \times \zz_2$-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in \cite{Y15c} for the planar $N$-body problem. In particular, the monotone constraints required in \cite{Y15c} 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 $\om$, we find an entire family of simple choreographies (seen in the rotating frame), as $\om$ 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 $\om=0$ or $N$, but may contain collision for $0 < \om < N$. However all possible collisions must be binary and each collision solution is $C^0$ block-regularizable. Moreover for certain types of topological constraints, based on results from \cite{BT04} and \cite{CF09}, we show that when $\om$ belongs to some sub-intervals of $[0, N]$, the corresponding minimizer must be a rotating regular $N$-gon contained in the horizontal plane. As a result, this generalizes Marchal's $P_{12}$ family of the three body problem to arbitrary $N \ge 3$.