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

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 $N \ge 3$.