Hill-Type Stability and Hierarchical Stability of the General Three-Body Problem

  • Yan-Chao Ge
Part of the NATO ASI Series book series (NSSB, volume 272)


Hierarchical stability (HS hereafter) was defined by Walker and Roy (1983) in connection with the Jacobian coordinate system. A dynamical N-body system is held to be HS if, during an interval of time substantially longer than the periods of revolution of the bodies in the system, the following conditions hold:
  1. HS-(A).

    none of the bodies escapes to infinity from the system;

  2. HS-(B).

    no dramatic changes occur in any orbit’s size, shape or orientation to the invariable plane of the system;

  3. HS-(C).

    ρi < ρj for any i < j; where ρi = |ρi| (i=2,..., N), ρi being the Jacobian vectors which connect the barycentre of the first (i-1) masses and the ith mass.



Jacobian Vector Synodic Period Invariable Plane Negative Total Energy Outer Mass 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Easton, R. (1971): J. Diff. Eqs., 10:371.ADSMATHCrossRefMathSciNetGoogle Scholar
  2. Ge, Y.C. (1990): “Ph.D. Thesis”, The University of Glasgow.Google Scholar
  3. Golubev, V.G. (1967): Soviet Phys. Doklady. 12 (No. 6):529.ADSMATHGoogle Scholar
  4. Golubev, V.G. (1968): Soviet Phys. Doklady. 13 (No. 5):373.ADSMATHMathSciNetGoogle Scholar
  5. Marchal, C. and Bozis, G. (1982): Celest. Mech. 26:311.ADSMATHCrossRefMathSciNetGoogle Scholar
  6. Marchal, C. and Sarri, D.G. (1975): Celest. Mech. 12:115.ADSMATHCrossRefGoogle Scholar
  7. Marchal, C., Yoshida, J. and Sun, Y.S. (1984):Celest. Mech. 33:193.ADSMATHCrossRefMathSciNetGoogle Scholar
  8. Poincare, H. (1892): “Les Methods Nouvelles de la Mechanique Celeste”, Gauthier Villars.Google Scholar
  9. Roy, A. E. and Ovenden, M. W. (1955):M.N.R.A.S., 115:296.ADSMATHGoogle Scholar
  10. Saari, D. G. (1976):Celest. Mech. 14:11.ADSMATHCrossRefMathSciNetGoogle Scholar
  11. Saari, D.G. (1984): Celest. Mech.. 33:299.ADSMATHCrossRefMathSciNetGoogle Scholar
  12. Saari, D.G. (1987): Celest. Mech.. 40:197.ADSMATHCrossRefMathSciNetGoogle Scholar
  13. Smale, S. (1970): Invent, Math., 10:305.ADSMATHCrossRefMathSciNetGoogle Scholar
  14. Smale, S. (1970):Invent, Math., 11:45.ADSMATHCrossRefMathSciNetGoogle Scholar
  15. Szebehely, V. G. and Zare K. (1976):Astro. AstroPhys. 58:145.ADSGoogle Scholar
  16. Valsecchi G. B. Carusi, A. and Roy, A. E. (1984): Celest. Mech., 32:217.ADSMATHCrossRefGoogle Scholar
  17. Walker, I.W., Emslie, A. G. and Roy A. E. (1980):Celest. Mech. 22:371.ADSMATHCrossRefMathSciNetGoogle Scholar
  18. Walker, I.W. and Roy, A. E. (1983):Celest. Mech. 29:117.ADSMATHCrossRefMathSciNetGoogle Scholar
  19. Walker, I.W. (1983):Celest. Mech. 29:215.ADSMATHCrossRefGoogle Scholar
  20. Zare, K. (1976):Celest. Mech. 14:73.ADSMATHCrossRefMathSciNetGoogle Scholar
  21. Zare, K. (1977):Celest. Mech. 16:35.ADSMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Yan-Chao Ge
    • 1
  1. 1.Department of Physics and AstronomyThe University of GlasgowGlasgowScotland, UK

Personalised recommendations