Successive Elimination of Harmonics: A way to Explore the Resonant Structure of a Hamiltonian System

  • A. Morbidelli
Part of the NATO ASI Series book series (NSSB, volume 331)


This paper deals with the problem of exploring Hamiltonian systems with semi-numerical methods. I come back to an original idea of Delaunay, consisting in successively eliminating all the perturbation harmonics up to a given order. Here the elimination is performed via the introduction of suitable Arnold action-angle variables, along the lines of the well known theorem of Arnold (1963b), with numerical evaluation of the action integrals. More precisely, given the Fourier expansion of the Hamiltonian, I first ignore all the harmonics in the perturbation except a single one; this is an inte-grable system and the introduction of Arnold action-angle variables allows the complete elimination of the considered harmonic without remainder terms. Next, I compute the Fourier expansion of the till now ignored terms in the new variables. This algorithm can be implemented on computer and iterated so that one attains a global description of the dynamics. By the way, this allows to point out all the power of the method of Delaunay, which seems to have been almost forgotten.


Fourier Expansion Invariant Torus Main Resonance Asteroid Belt Successive Elimination 
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. Arnold, V.I., (1963a): “Small denominators and problems of stability of motion in classical and celestial mechanics.”, Russ. Math. Surveys, 18, 85–191.CrossRefADSGoogle Scholar
  2. Arnold, V., I., (1963b): “On a theorem of Liouville concerning integrable problems of dynamics”, Sib. mathem. zh., 4, 2.Google Scholar
  3. Bendjoya, Ph., (1993): “A classification of 6479 asteroids into families by means of wavelets clastering method.”, Astron. Astrophys., in press.Google Scholar
  4. Born, M., (1967): “The Mechanics of the Atom”, Frederick Ungar Publ. Co.Google Scholar
  5. Chirikov, B., V., (1979), Phys. Reports., 52, 265.CrossRefMathSciNetADSGoogle Scholar
  6. Delaunay, C., (1867): “Théorie du mouvement de la Lune”, Mem. Ac. Se, Paris, 29.Google Scholar
  7. Escande, D., F., and Doveil, F., (1981), J. Stat. Phys., 26, 257.CrossRefMathSciNetADSGoogle Scholar
  8. Escande, D., F., and Doveil, F., (1981), Phys. Lett., 83A, 307.CrossRefMathSciNetADSGoogle Scholar
  9. Farinella, P., Gonczi, R., Froeschlé, Ch. and Froeschlé C., (1993): “The injection of asteroid fragments into resonances”, Icarus, 101, 174–187.CrossRefADSGoogle Scholar
  10. Ferraz-Mello, S., (1989): “A semi-numerical expansion of the averaged disturbing function for some very-high-eccentricity orbits.”, Celest. Mech., 45, 65–68.CrossRefADSGoogle Scholar
  11. Henrard, J., (1990): “A semi-numerical perturbation method for separable Hamiltonian systems.”, Celest. Mech., 49, 43–67.CrossRefzbMATHMathSciNetADSGoogle Scholar
  12. Moons, M., and Henrard, J., (1993): “Surfaces of sections in the Miranda-Umbriel 1/3 inclination problem”, submitted to Celest. Mech..Google Scholar
  13. Lichtenberg, A.J., and Lieberman, M.A., (1983): “Regular and stochastic motion.”, Springer Verlag ed.Google Scholar
  14. Morbidelli, A., (1993a): “On the successive eliminations of perturbation harmonics.”, Celest. Mech., 55, 101–130.CrossRefzbMATHMathSciNetADSGoogle Scholar
  15. Morbidelli, A., and Giorgilli, A., (1992): “A quantitative perturbation theory by sucessive elimination of harmonics.”, Celest. Mech., 55, 131–159.CrossRefMathSciNetADSGoogle Scholar
  16. Morbidelli, A., (1993b): “Asteroid resonant proper elements”, Icarus, in press.Google Scholar
  17. Morbidelli, A., and Moons, M., (1993): “Secular resonances in mean motion commensurabilities: the 2/1 and 3/2 cases, Icarus, 102, 316–332.CrossRefADSGoogle Scholar
  18. Morbidelli, A., Scholl, H., and Froeschlé, Ch., (1993): “The location of secular resonances close to the 2/1 commensurability”, Astron. Astrophys., in press.Google Scholar
  19. Nekhoroshev, N., N., (1977): “Exponential estimates of the stability time of near-integrable Hamiltonian systems.”, Russ. Math. Surveys, 32, 1–65.CrossRefzbMATHADSGoogle Scholar
  20. Nekhoroshev, N., N., (1979): “Exponential estimates of the stability time of near-integrable Hamiltonian systems, 2.”, Trudy Sem. Petrovs., 5, 5–50.zbMATHMathSciNetGoogle Scholar
  21. Poincaré, H., Les methodes nouvelles de la mechanique celeste, Gauthier-Villars, Paris.Google Scholar
  22. Tisserand, M. F., (1882): Ann. Obs. Paris, 16, El.Google Scholar

Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • A. Morbidelli
    • 1
  1. 1.CERGA departmentObservatory of NiceNice Cedex 4France

Personalised recommendations