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Transactions of the American Mathematical Society

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On the distribution of mass
in collinear central configurations

Author: Peter W. Lindstrom
Journal: Trans. Amer. Math. Soc. 350 (1998), 2487-2523
MSC (1991): Primary 70F10
MathSciNet review: 1422613
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Abstract: Moulton's Theorem says that given an ordering of masses, $m_1,m_2, \break \dotsc,m_n$, there exists a unique collinear central configuration with center of mass at the origin and moment of inertia equal to 1. This theorem allows us to ask the questions: What is the distribution of mass in this standardized collinear central configuration? What is the behavior of the distribution as $n\to\infty$? In this paper, we define continuous configurations, prove a continuous version of Moulton's Theorem, and then, in the spirit of limit theorems in probability theory, prove that as $n\to\infty$, under rather general conditions, the discrete mass distributions of the standardized collinear central configurations have distribution functions which converge uniformly to the distribution function of the unique continuous standardized collinear central configuration which we determine.

References [Enhancements On Off] (What's this?)

  • 1. G. Buck, 1991, The collinear central configuration of $n$ equal masses, Celestial Mech. Dynam. Astronom. 51, 305-317. MR 92k:70013
  • 2. L. Euler, 1767, De moto recilineo trium corporum se mutuo attahentium, Novi Comm. Acad. Sci. Imp. Petrop. 11, 144-151.
  • 3. P. W. Lindstrom, 1996, Limiting mass distributions of minimal potential central configurations, Hamiltonian Dynamics and Celestial Mechanics, Contemporary Mathematics, vol. 198, Amer. Math. Soc., Providence, RI, pp. 109-129. MR 97g:70015
  • 4. K. R. Meyer and G. R. Hall, 1992, Introduction to Hamiltonian Dynamical Systems and the $N$-Body Problem, Springer-Verlag. MR 93b:70002
  • 5. R. Moeckel, 1990, On central configurations, Math. Z. 205, 499-517. MR 92b:70012
  • 6. R. Moeckel, Some relative equilibria of $n$ equal masses, $n=4,5,6,7,8$, unpublished.
  • 7. F. R. Moulton, 1910, The straight line solutions of the problem of $N$ bodies, Ann. Math., II. Ser. 12, 1-17.
  • 8. D. G. Saari, 1980, On the role and properties of $n$ body central configurations, Celestial Mech. 21, 9-20. MR 81a:70016

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Additional Information

Peter W. Lindstrom
Affiliation: Department of Mathematics, Saint Anselm College, Manchester, New Hampshire 03102

Received by editor(s): October 1, 1995
Received by editor(s) in revised form: September 20, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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