Strictly convex central configurations of the planar five-body problem

Chen, Kuo-Chang; Hsiao, Jun-Shian

doi:10.1090/tran/7010

Strictly convex central configurations of the planar five-body problem
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Trans. Amer. Math. Soc. 370 (2018), 1907-1924 Request permission

Abstract:

In this paper we investigate strictly convex central configurations of the planar five-body problem, and prove some necessary conditions for such configurations. In particular, given such a central configuration with multiplier $\lambda$ and total mass $M$, we show that all exterior edges are less than $r_0=(M/\lambda )^{1/3}$, at most two interior edges are less than or equal to $r_0$, and its subsystem with four masses cannot be a central configuration. We also obtain some other necessary conditions for strictly convex central configurations with five bodies, and show numerical examples of strictly convex central configurations with five bodies that have either one or two interior edges less than or equal to $r_0$. Our work develops some formulae in a classic work by W. L. Williams in 1938 and we rectify some unsupported assumptions there.

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