On the distribution of mass in collinear central configurations
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- by Peter W. Lindstrom
- Trans. Amer. Math. Soc. 350 (1998), 2487-2523
- DOI: https://doi.org/10.1090/S0002-9947-98-01964-3
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Abstract:
Moulton’s Theorem says that given an ordering of masses, $m_1,m_2, \dotsc ,m_n$, 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
- Gregory Buck, The collinear central configuration of $n$ equal masses, Celestial Mech. Dynam. Astronom. 51 (1991), no. 4, 305–317. MR 1134436, DOI 10.1007/BF00052924
- L. Euler, 1767, De moto recilineo trium corporum se mutuo attahentium, Novi Comm. Acad. Sci. Imp. Petrop. 11, 144–151.
- Peter W. Lindstrom, Limiting mass distributions of minimal potential central configurations, Hamiltonian dynamics and celestial mechanics (Seattle, WA, 1995) Contemp. Math., vol. 198, Amer. Math. Soc., Providence, RI, 1996, pp. 109–129. MR 1409156, DOI 10.1090/conm/198/02487
- Kenneth R. Meyer and Glen R. Hall, Introduction to Hamiltonian dynamical systems and the $N$-body problem, Applied Mathematical Sciences, vol. 90, Springer-Verlag, New York, 1992. MR 1140006, DOI 10.1007/978-1-4757-4073-8
- Richard Moeckel, On central configurations, Math. Z. 205 (1990), no. 4, 499–517. MR 1082871, DOI 10.1007/BF02571259
- R. Moeckel, Some relative equilibria of $n$ equal masses, $n=4,5,6,7,8$, unpublished.
- F. R. Moulton, 1910, The straight line solutions of the problem of $N$ bodies, Ann. Math., II. Ser. 12, 1–17.
- Donald G. Saari, On the role and the properties of $n$-body central configurations, Celestial Mech. 21 (1980), no. 1, 9–20. MR 564603, DOI 10.1007/BF01230241
Bibliographic 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
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 2487-2523
- MSC (1991): Primary 70F10
- DOI: https://doi.org/10.1090/S0002-9947-98-01964-3
- MathSciNet review: 1422613