Kuo-Chang Chen

In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian $N$-body problem. Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds for action values. Explicit formulae will be proved by iterative arguments. We demonstrate some applications to actionminimizing solutions for three- and four-body problems.

In this paper we investigate rhomboidal central configurations for the planar fivebody problem. These are configurations whose convex hulls are rhombuses, and one interior mass lies on a diagonal of the rhombus. Consider partition of the configuration space by orderings of mutual distances, our main theorem provides criteria for central configurations in every region of this partition. For some regions we obtain existence and uniqueness of central configurations, for some we obtain nonexistence, for others our criteria are necessary conditions on masses. Our proofs are elementary and completely analytic.