Celestial Mechanics and Dynamical Astronomy

, Volume 123, Issue 3, pp 351–361 | Cite as

Generic uniqueness of the minimal Moulton central configuration

Original Article
First Online:
Received:
Revised:
Accepted:

Abstract

We prove that, for generic (open and dense) values of the masses, the Newtonian potential function of the collinear N-body problem has Open image in new window critical values when restricted to a fixed inertia level. In particular, we prove that for generic values of the masses, there is only one global minimal Moulton configuration.

Keywords

N-body problem Central configuration Genericity Homothetic motions 

Mathematics Subject Classification

70F10 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The authors would like to thank the anonymous referees and Professor Alain Albouy for their suggestions and comments. Following the suggestions, we have included several improvements in the manuscript.

References

  1. Albouy, A., Fu, Y.: Euler configurations and quasi-polynomial systems. Regul. Chaotic Dyn. 12, 39–55 (2007)MATHMathSciNetCrossRefADSGoogle Scholar
  2. Albouy, A., Kaloshin, V.: Finiteness of central configurations of five bodies in the plane. Ann. Math. (2) 176, 535–588 (2012)MATHMathSciNetCrossRefGoogle Scholar
  3. Euler, L.: De motu rectilineo trium corporum se mutuo attrahentium. Novi commentarii academiae scientiarum Petropolitanae 11, 144–151 (1765). (read at St Petersburg in december 1763. Also in Opera Omnia S. 2, vol. 25, 281–289)Google Scholar
  4. Feingold, D., Varga, R.: Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem. Pac. J. Math. 12, 1241–1250 (1962)MATHMathSciNetCrossRefGoogle Scholar
  5. Krantz, S., Parks, H.: The Implicit Function Theorem. Birkhäuser, Boston (2002)MATHGoogle Scholar
  6. Maderna, E., Venturelli, A.: Globally minimizing parabolic motions in the Newtonian N-body problem. Arch. Ration. Mech. Anal. 194(1), 283–313 (2009)MATHMathSciNetCrossRefGoogle Scholar
  7. Moulton, F.: The straight line solutions of the problem of n bodies. Ann. Math. (2) 12(1), 1–17 (1910)MATHMathSciNetCrossRefGoogle Scholar
  8. Percino, B., Sánchez-Morgado, H.: Busemann functions for the \(N\)-body problem. Arch. Ration. Mech. Anal. 213(3), 981–991 (2014)MATHMathSciNetCrossRefGoogle Scholar
  9. Smale, S.: Topology and mechanics. II (the planar N-body problem). Invent. Math. 11, 45–64 (1970)MATHMathSciNetCrossRefADSGoogle Scholar
  10. Yoccoz, J.C.: Configurations centrales dans le problème des quatre corps dans le plan. In: Conference at Palaiseau, April 21 (1986)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.CIMATGuanajuatoMexico
  2. 2.Centro de MateméticaUniversidad de la RepúblicaMontevideoUruguay

Personalised recommendations