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Strictly convex central configurations of the planar five-body problem


Authors: Kuo-Chang Chen and Jun-Shian Hsiao
Journal: Trans. Amer. Math. Soc. 370 (2018), 1907-1924
MSC (2010): Primary 70F10, 70F15; Secondary 37J45
DOI: https://doi.org/10.1090/tran/7010
Published electronically: July 19, 2017
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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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Additional Information

Kuo-Chang Chen
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
Email: kchen@math.nthu.edu.tw

Jun-Shian Hsiao
Affiliation: Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
Email: d9621804@oz.nthu.edu.tw

DOI: https://doi.org/10.1090/tran/7010
Keywords: $n$-body problem, central configuration, relative equilibrium
Received by editor(s): February 29, 2016
Received by editor(s) in revised form: April 28, 2016, May 10, 2016, and June 27, 2016
Published electronically: July 19, 2017
Additional Notes: This work was supported in part by the Ministry of Science and Technology (Grant NSC 102-2628-M-007-004-MY4) and the National Center for Theoretical Sciences in Taiwan.
Article copyright: © Copyright 2017 American Mathematical Society

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