Collinear Central Configuration in Four-Body Problem |
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| Authors: | Tiancheng Ouyang Zhifu Xie |
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| Institution: | (1) Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA |
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| Abstract: | In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center
of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of
four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible
to choose three positive masses such that the given three positions with the masses form a central configuration. However
for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration.
In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region
in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of
the compact region, it is always possible to choose positive masses to make the configuration central. |
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| Keywords: | central configuration n-body problem inverse problem of central configuration Descartes’ rule |
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