Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T15:20:08.167Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcations of relative equilibria in the 4- and 5-body problem

Published online by Cambridge University Press:  10 December 2009

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.

The equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family as m passes through the degenerate value.

The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family as m passes through the degenerate value.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Volume 8: Charles Conley Memorial Issue , December 1988 , pp. 215 - 225
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

Andoyer, M. H.. Sur l'équilibre relatif de n corps. Bull. Astronomique 23 (1906), 50–59.Google Scholar
Breglia, E.. Su alcuni casi particolari del problema di tre corpi. Giorn. Math. 7 (1916), 3, 151–173.Google Scholar
Dziobek, O.. Über einen merkwürdigen Fall des Vielkörperproblem. Astron. Nach. 152 (1900), 32–46.CrossRefGoogle Scholar
Euler, L.. De motu restilineo trium corporum se mutus attrahentium. Novi Comm. Acad. Sci. Imp. Petrop. 11 (1767), 144–151.Google Scholar
Hagihara, Y.. Celestial Mechanics I. MIT Press, Cambridge, MA (1970).Google Scholar
Hoppe, R.. Erweiterung der bekannten Speciallösung des Dreikörper problems. Archiv. Math. Phys. 64 (1879), 218–223.Google Scholar
Lagrange, J. L.. Essai sur le problem des trois corps. Oeuvres 6 (1772), 272–292.Google Scholar
Lehmann-Filhes, R.. Über zwei Fälle des Vielkörpersproblem. Astron. Nach. 127 (1891), 137–144.CrossRefGoogle Scholar
Longley, W. R.. Some particular solutions in the problem of n-bodies. Bull. Amer. Math. Soc. 13 (1906), 324–335.CrossRefGoogle Scholar
Moulton, F. R.. The straight line solutions of the problem of N-bodies. Ann. Math. 12 (1910), 2, 1–17.10.2307/2007159CrossRefGoogle Scholar
Palmore, J. I.. Relative equilibria of the n-body problem. Thesis. University of California, Berkeley, CA (1973).Google Scholar
Wintner, A.. The Analytic Foundations of Celestial Mechanics. Princeton Mathematics Series 5. Princeton University Press, Princeton, NJ (1941).Google Scholar