Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T19:48:37.949Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative equilibria of the four-body problem

Published online by Cambridge University Press:  19 September 2008

Richard Moeckel
Richard Moeckel
Affiliation:
School of Mathematics, University of Minnesota-Twin Cities, Minneapolis, MN 55455, USA
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.

By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 3 , September 1985 , pp. 417 - 435
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Andoyer, M. H.. Sur l'equilibre relatif de n corps. Bull. Astron. 25 (1906), 50–59.CrossRefGoogle Scholar
[2]Artzy, R.. Linear Geometry. Addison-Wesley, 1978, p. 42.Google Scholar
[3]Dziobek, O.. Über einen merkwürdigen Fall der Vielkörperproblems. Astron. Nach. 152 (1900), 33–46.CrossRefGoogle Scholar
[4]Euler, L.. De motu restilineo trium corporum se mutuo attrahentium. Novi Comm. Acad. Sci. Imp. Petrop. 11 (1767), 144–151.Google Scholar
[5]Greub, W.. Multilinear Algebra. Springer Verlag: New York 1978, pp. 200–204.CrossRefGoogle Scholar
[6]Hoppe, . Enveiterung der bekannten Speziallosungen der Dreikorperproblem. Archiv der Math, und Phys. 64, 218–223.Google Scholar
[7]Lagrange, J. L.. Ouvres, vol. 6, 272–292. Paris 1873.Google Scholar
[8]Lehmann-Filhes, R.. Über zwei fölle der vielkörpersproblem. Astron. Nach. 127 (1891).CrossRefGoogle Scholar
[9]Levi-Civita, T.. Sur la regularisation du problem des trois corps. Acta Math. 42 (1920), 99–144.CrossRefGoogle Scholar
[10]Longley, W. R.. Some particular solutions of the problem of n bodies. Bull. Amer. Math. Soc. 13 (1907), 324–335.CrossRefGoogle Scholar
[11]MacMillan, W. D. & Bartky, W.. Permanent configurations in the problem of four bodies. Trans. Amer. Math. Soc. 34 (1932), 838–875.CrossRefGoogle Scholar
[12]Milnor, J.. On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15 (1964), 275–280.CrossRefGoogle Scholar
[13]Moulton, F. R.. The straight line solutions of the problem of N bodies. Ann. of Math. (2) 12 (1910), 1–17.CrossRefGoogle Scholar
[14]Mumford, D.. Algebraic Geometry I, Complex Projective Varieties. Springer-Verlag: Berlin, 1976, pp. 42–43.Google Scholar
[15]Pacella, F.. Central configurations of the AT-body problem via the equivariant Morse theory. To appear in Archive for Rational Mechanics and Analysis.Google Scholar
[16]Palmore, J. I.. Classifying relative equilibria, I. Bull. Amer. Math. Soc. 79 (1973), 904–908;CrossRefGoogle Scholar
II, Bull. Amer. Math. Soc. 81 (1975)' 489–491;CrossRefGoogle Scholar
III, Letters in Math. Physics 1 (1975), 71–73.CrossRefGoogle Scholar
[17]Palmore, J. I.. Measure of degenerate relative equilibria, I. Annals of Math. 104 (1976), 421–429.CrossRefGoogle Scholar
[18]Perko, L. M. & Walter, E. L.. Regular polygon solutions- of the JV-body problem. Preprint, 1984.CrossRefGoogle Scholar
[19]Robbin, J.. Relative equilibria in mechanical systems. In Dynamical Systems. Academic Press, New York, 1973.Google Scholar
[20]Saari, D.. On the role and properties of central configurations. Celestial Mechanics 21, No. 1 (1980), 9–20.CrossRefGoogle Scholar
[21]Siegel, C. L. & Moser, J. K.. Lectures on Celestial Mechanics. Springer Verlag: New York, 1971.CrossRefGoogle Scholar
[22]Simo, C.. Relative equilibrium solutions in the four body problem. Celestial Mechanics 18 (1978), 165–184.CrossRefGoogle Scholar
[23]Shub, M.. Diagonals and relative equilibria. In Manifolds-Amsterdam 1970, Lecture Notes in Math., vol. 197, Springer: Berlin (1971), 199–201.CrossRefGoogle Scholar
[24]Smale, S.. Problems on the nature of relative equilibria in celestial mechanics. In Manifolds - Amsterdam 1970, Lecture Notes in Math., Vol. 197, Springer: Berlin (1971).Google Scholar
[25]Smale, S.. Topology and mechanics, I. Inventiones Math. 10 (1970) 305–331;CrossRefGoogle Scholar
II, Inventiones Math. 11 (1970), 45–64.CrossRefGoogle Scholar
[26]Thorn, R.. Sur l'homologie des variétés algebrique réeles. In Differential and Combinational Topology. Princeton Univ. Press (1965), pp 255–265.Google Scholar
[27]Whitney, H.. Elementary structure of real algebraic varieties. Annals of Math. 66 (3) (1957), 545–556.CrossRefGoogle Scholar
[28]Williams, W. L.. Permanent configurations in the problem of five bodies. Trans. Amer. Math. Soc. 44 (1938), 563–579.CrossRefGoogle Scholar
[29]Wintner, A.. The Analytical Foundations of Celestial Mechanics. Princeton Math, series, vol. 5, Princeton Univ. Press, 1941.Google Scholar