Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-09T13:36:48.665Z Has data issue: false hasContentIssue false
  • English
  • Français

The zero angular momentum, three-body problem: All but one solution has syzygies

Published online by Cambridge University Press:  01 December 2007

RICHARD MONTGOMERY
RICHARD MONTGOMERY*
Affiliation:
Mathematics Department, UC Santa Cruz, Santa Cruz, CA, 95064, USA (email: rmont@math.ucsc.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A syzygy in the three-body problem is a collinear instant. We prove that, with the exception of Lagrange’s solution, every solution to the zero angular momentum, Newtonian three-body problem suffers syzygies. The proof works for all mass ratios.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 6 , December 2007 , pp. 1933 - 1946
Copyright
Copyright © Cambridge University Press 2007

References

[1]Fujiwara, T., Fukuda, H., Kameyama, A., Ozaki, H. and Yamada, M.. Synchronised similar triangles. J. Phys. A: Math. Gen. 37 (2004), 10571–10584.CrossRefGoogle Scholar
[2]Lagrange, J.-L.. Essai sur le Probléme des Trois Corps (œuvres, 6), tome IX. Prix de l’Académie Royale des Sciences de Paris, 1772, p. 292.Google Scholar
[3]Levi-Civita, T.. Sur la régularisation du problème des trois corps. Acta Math. 42 (1921), 99–144.CrossRefGoogle Scholar
[4]Marchal, C.. The Three-Body Problem. Elsevier, Amsterdam, 1990.Google Scholar
[5]Chenciner, A. and Montgomery, R.. A remarkable periodic solution of the three-body problem in the case of equal masses. Ann. of Math. 152 (2000), 881–901.CrossRefGoogle Scholar
[6]Moeckel, R.. Chaotic dynamics near triple collision. Arch. Ration. Mech. 107(1) (1989), 37–69.CrossRefGoogle Scholar
[7]Montgomery, R.. Infinitely many syzygies. Arch. Ration. Mech. Anal. 164 (2002), 311–340.CrossRefGoogle Scholar
[8]Montgomery, R.. Fitting hyperbolic pants to a three-body problem. Ergod. Th. & Dynam. Sys. 25(3) (2005), 921–947.CrossRefGoogle Scholar
[9]Montgomery, R.. The N-body problem, the braid group, and action-minimizing periodic orbits. Nonlinearity 11(2) (1998), 363–376.CrossRefGoogle Scholar
[10]Sitnikov, K. A.. The existence of oscillatory motions in the three-body problem. Sov. Phys. Dokl. 5 (1961), 647–650.Google Scholar
[11]Saari, D.. Collisions, Rings, and Other Newtonian N-Body Problems (CBMS Regional Conference Series in Mathematics, 104). American Mathematical Society, Providence, RI, 2005.CrossRefGoogle Scholar
[12]Sundman, K.. Memoire sur le probleme des trois corps. Acta Math. 36 (1912), 105–179.CrossRefGoogle Scholar