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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Number of central configurations and singular surfaces in the mass space in the collinear four-body problem


Authors: Tiancheng Ouyang and Zhifu Xie
Journal: Trans. Amer. Math. Soc. 364 (2012), 2909-2932
MSC (2010): Primary 37N05, 70F10, 70F15, 37N30, 70H05, 70F17
Published electronically: February 10, 2012
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Abstract: For a given $ m=(m_1,m_2,\cdots , m_n)\in (\mathbf {R}^+)^n$, let $ p$ and $ q\in (\mathbf {R}^d)^n$ be two central configurations for $ m$. Then we call $ p$ and $ q$ geometrically equivalent and write $ p\sim q$ if they differ by a rotation followed by a scalar multiplication as well as by a permutation of bodies. Denote by $ L(n,m)$ the set of geometric equivalence classes of $ n$-body collinear central configurations for any given mass vector $ m$. There are other different understandings of equivalence of central configurations in the collinear $ n$-body problem. Under the usual definition of equivalence of central configurations in history, permutations of the bodies are not allowed, and we call them permutation equivalence. In this case Euler found three collinear central configurations and Moulton generalized to $ n!/2$ central configurations for any given mass $ m$ in the collinear $ n$-body problem under permutation equivalence. In particular, the number of central configurations becomes from 12 under permutation equivalence to 1 under geometric equivalence for four equal masses in the collinear four-body problem. The main result in this paper is the discovery of the explicit parametric expressions of the union $ H_4$ of the singular surfaces in the mass space $ m=(m_1,m_2,m_3,m_4)\in $ $ (\mathbf {R}^+)^4$, which decrease the number of collinear central configurations under geometric equivalence. We prove that the number of central configurations $ ^\char93 L(4,m)=4!/2-1=11$ if $ m_1, m_2, m_3$ and $ m_4$ are mutually distinct and $ m\in H_4$.


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Additional Information

Tiancheng Ouyang
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602

Zhifu Xie
Affiliation: Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806
Email: zxie@vsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05426-2
PII: S 0002-9947(2012)05426-2
Keywords: Central configurations, super central configurations, $N$-body problem, geometric equivalence, permutation equivalence, mass equivalence, singular surfaces, relative equilibrium, Descartes’ rule.
Received by editor(s): November 7, 2009
Received by editor(s) in revised form: January 13, 2010, April 4, 2010, and May 8, 2010
Published electronically: February 10, 2012
Additional Notes: The second author was partially supported by RIG Grant (code 2137) from Virginia State University 2008-2009.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.