Exclusion of quadruple collisions in minimizers of the planar equal-mass N-body problem

Abstract

It has been shown by Chen that if all masses are in the same proper vector subspace of Rd(d≥2) on one of the free boundaries, local minimizers connecting the two free boundaries have no collision on that boundary. However, not much is known for other types of boundary configurations.In this paper, we consider minimizers connecting two free boundaries and study possible quadruple collisions under the following four symmetric configurations: isosceles trapezoid configuration, rectangular configuration, double isosceles configuration and diamond configuration. Whenever the boundary configuration of four bodies coincides with one of the four symmetric configurations, we show that these four bodies are free of quadruple collision on that boundary set. New ideas and detailed analysis are introduced in order to construct deformed paths in some of the cases and estimate their action values. As its applications, we show the existence of several sets of periodic orbits in the planar four-body and five-body problems.