New Periodic Orbits for the n-Body Problem

[+] Author and Article Information
Cristopher Moore

Computer Science Department, and Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 87501 and the Santa Fe Institute

Michael Nauenberg

Physics Department, University of California, Santa Cruz, Santa Cruz, CA

J. Comput. Nonlinear Dynam 1(4), 307-311 (Mar 15, 2006) (5 pages) doi:10.1115/1.2338323 History: Received November 08, 2005; Revised March 15, 2006

Since the discovery of the figure-eight orbit for the three-body problem [Moore, C., 1993, Phys. Rev. Lett., 70, pp. 3675–3679] a large number of periodic orbits of the n-body problem with equal masses and beautiful symmetries have been discovered. However, most of those that have appeared in the literature are either planar or are obtained from perturbations of planar orbits. Here we exhibit a number of new three-dimensional periodic n-body orbits with equal masses and cubic symmetry, including some whose moment of inertia tensor is a scalar. We found these orbits numerically, by minimizing the action as a function of the trajectories’ Fourier coefficients. We also give numerical evidence that a planar three-body orbit first found in [Hénon, M., 1976, Celest. Mech., 13, pp. 267–285], rediscovered by [Moore, 1993], and found to exist for different masses by [Nauenberg, M., 2001, Phys. Lett., 292, pp. 93–99], is dynamically stable.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

The figure-eight orbit for three equal masses. The points show the masses at five equal time intervals.

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Figure 2

Left, a cuboctahedron, and its four hexagonal “great circles;” right, the 12-mass orbit with cubic symmetry

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Figure 3

(a) The criss-cross orbit for three equal masses shown at 16 equal time intervals. Starting with the masses aligned horizontally, at the second time interval they lie at the vertices of an isosceles triangle. (b) The corresponding criss-cross orbit with three masses in the ratio 1:2:3.

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Figure 4

(a) The equal mass criss-cross orbit during 40 periods, for an initial small deviation δx=0.001 of the initial position of one of the masses. (b) The same orbit with a larger perturbation δx=0.005.

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Figure 5

Perturbing the criss-cross along the z direction leads to this beautiful quasiperiodic orbit, indicating that the criss-cross is dynamically stable in three dimensions. The vertical component of motion has been exaggerated by a factor of 50 for purposes of illustration.



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