Celestial mechanics

, Volume 27, Issue 1, pp 3–22 | Cite as

Capture into resonance: An extension of the use of adiabatic invariants

  • J. Henrard


The theory of the adiabatic invariant predicts the long term evolution of mechanical systems with slowly varying parameters.

Unfortunately, it is not valid when the system goes across a critical trajectory. This case is important because it can lead to capture into resonance.

Analysing the motion in the vincinity of the critical trajectory, we are able to give general formulae for the probability of capture and to show that in general, the adiabatic invariant is conserved (allowance being made for the geometrical discontinuity in its definition at the critical orbit).


Mechanical System General Formula Term Evolution Critical Orbit Adiabatic Invariant 
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  1. Allan, R. R.: 1969,Astron. J. 74, 497–506.Google Scholar
  2. Borderies, N.: 1980, Thesis, University Paul Sabatier, Toulouse.Google Scholar
  3. Bretagnon, P.: 1974,Astron. Astrophys. 30, 141–154.Google Scholar
  4. Burns, T. J.: 1979,Celest. Mech. 19, 297–313.Google Scholar
  5. Colombo, G.: 1965,Nature,208, 575.Google Scholar
  6. Colombo, G.: 1966,Astron. J. 71, 891.Google Scholar
  7. Councelman, C. C. and Shapiro, I. I.: 1970.Symposia Mathematica 3, 121.Google Scholar
  8. Goldreich, P.: 1965,Monthly Notices Roy. Astron. Soc. 130, 159–181.Google Scholar
  9. Goldreich, P.: 1966,Astron. J. 71, 1.Google Scholar
  10. Goldreich, P. and Peale, S.: 1966,Astron. J. 71, 425.Google Scholar
  11. Greenberg, R.: 1973,Astron. J. 78, 338–346.Google Scholar
  12. Greenberg, R.: 1977,Vistas Astron. 21, 209–239.Google Scholar
  13. Landau, L. and Lifchitz, E.: 1960,Mécanique, Editions, en Langues Etrangères, Moscow.Google Scholar
  14. Message, P. J.: 1966,IAU Symp. 25.Google Scholar
  15. Meyer, K.: 1974,Celest. Mech. 9, 517–522.Google Scholar
  16. Murdock, J. A.: 1978,Celest. Mech. 18, 237–253.Google Scholar
  17. Peale, S. J.: 1969,Astron. J. 74, 483–489.Google Scholar
  18. Peale, S. J.: 1974,Astron. J. 79, 722.Google Scholar
  19. Peale, S. J.: 1976,Ann. Rev. Astron. Astrophys. 14, 215–246.Google Scholar
  20. Sinclair, A. T.: 1972,Monthly Notices Roy. Astron. Soc. 160, 169.Google Scholar
  21. Sinclair, A. T.: 1974,Monthly Notices Roy. Astron. Soc. 166, 165–179.Google Scholar
  22. Urabe, M.: 1954,J. Sc. Hiroshima Univ. 18A, 183–213.Google Scholar
  23. Ward, W. R.: 1975,Astron. J. 80, 64.Google Scholar
  24. Ward, W. R., Burns, J. A., and Toon, O. B.: 1979,J. Geophys. Res. 84, 243–259.Google Scholar
  25. Yoder, C. F.: 1979a,Celest. Mech. 19, 3–29.Google Scholar
  26. Yoder, C. F.: 1979b,Nature,279, 767–770.Google Scholar

Copyright information

© D. Reidel Publishing Co 1982

Authors and Affiliations

  • J. Henrard
    • 1
  1. 1.Department of MathematicsFacultés Universitaires de NamurNamurBelgium

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