2017/11/14 by Yu, Guowei
#37N05 #70F07 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.05071
The family of planar linear chains are found as collision-free action minimizers of the spatial N-body problem with equal masses under DN or DN × \zz2-symmetry constraint and different types of topological constraints. This generalizes a previous result by the author in \citeY15c for the planar N-body problem. In particular, the monotone constraints required in \citeY15c 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 C0 block-regularizable. Moreover for certain types of topological constraints, based on results from \citeBT04 and \citeCF09, 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 P12 family of the three body problem to arbitrary N ≥ 3.