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

DOI:
https://doi.org/10.1090/S0002-9947-98-01964-3

MathSciNet review:
1422613

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Moulton's Theorem says that given an ordering of masses, , 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 ? 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 , 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.

**1.**G. Buck, 1991,*The collinear central configuration of 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 -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 MR**92b:70012****6.**R. Moeckel,*Some relative equilibria of equal masses*, , unpublished.**7.**F. R. Moulton, 1910,*The straight line solutions of the problem of bodies*, Ann. Math., II. Ser. 12, 1-17.**8.**D. G. Saari, 1980,*On the role and properties of body central configurations*, Celestial Mech.**21**, 9-20. MR**81a:70016**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
70F10

Retrieve articles in all journals with MSC (1991): 70F10

Additional Information

**Peter W. Lindstrom**

Affiliation:
Department of Mathematics, Saint Anselm College, Manchester, New Hampshire 03102

DOI:
https://doi.org/10.1090/S0002-9947-98-01964-3

Received by editor(s):
October 1, 1995

Received by editor(s) in revised form:
September 20, 1996

Article copyright:
© Copyright 1998
American Mathematical Society