Celestial Mechanics and Dynamical Astronomy

, Volume 56, Issue 4, pp 523–540 | Cite as

Adiabatic invariants for the nonconservative Kepler's problem

  • Djordje S. Djukic


This paper considers adiabatic invariants for the classical Kepler problem with resisting forces. The analysis is based on the theory of integrating factors and theory of adiabatic invariants in the Krylov-Bogoliubov-Mitropolski variables. The adiabatic invariants are series with respect to a small parameter. Also, for every particular case of nonconservative forces, it is shown that, with a complete set of adiabatic invariants, an approximate solution of the problem can be obtained. Four problems are analyzed in detail where approximate solutions are compared with numerical.

Key words

Kepler problem integral of motion adiabatic invariant 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Burgers, J.M.: 1917,Ann. Physik 52, 195–203.Google Scholar
  2. Andronov, A.A., Leontovich, M.A. and Meholelstam, L. I.: 1928, L.I.J.R.F.H.O.60, 413–457.Google Scholar
  3. Kulsrud, R.M.: 1957,Phys. Rev. 106, 205.Google Scholar
  4. Chandrasekhar, S.: 1960,Plasma Physics, Chap. 3, Chicago, University of Chicago Press.Google Scholar
  5. Kasuga, T.: 1961,Proc. Japan Acad. 37, 366–382.Google Scholar
  6. Kruskal, M.: 1961,Adiabatic Invariants, Princeton University Press, Princeton.Google Scholar
  7. Arnold, V.I.: 1962,D.A.N. USSR 142, 758–761.Google Scholar
  8. Arnold, V.I.: 1963, V. Mat. Nauk.18, 91–192.Google Scholar
  9. Wasow, W.: 1971,Nat. Am. Soc. 18, No. 1.Google Scholar
  10. Lewis, H.R.: 1968,J. Math. Physics 9, 1976–1986.Google Scholar
  11. Stern, D.P.: 1970,J. Math. Physics 11, 2771–2775.Google Scholar
  12. Stern, D.P.: 1971,J. Math. Physics 12, 2231–2242.Google Scholar
  13. Symon, K.R.: 1970,J. Math. Physics 11, 1320–1330.Google Scholar
  14. Djukic, Dj.S.: 1981,Int. J. Non-Linear Mechanics,16, 489–498.Google Scholar
  15. Bakai, A.S. and Stepanovsky, Y.P.: 1981,Adiabatic Invariants, Naukova Dumka, Kiev.Google Scholar
  16. Landau, L.D. and Lifshic, E.M.: 1958,Mechanics, G.I.F.M.L., Moscow.Google Scholar
  17. Djukic, Dj.S. and Sutela, T.: 1984,Int. J. Non-Linear Mechanics 19, 331–339.Google Scholar
  18. Gravitation and Topology, problems: 1966, S.S.I. Mir, Moscow.Google Scholar
  19. Gradstein, I.S. and Rizik, I.M.: 1971,Tables of Integrals, Sums, Series and Derivatives, I. Nauka, Moscow.Google Scholar
  20. Nayfeh, A.H.: 1973,Perturbation Methods, John Wiley, New York.Google Scholar
  21. Moisev, N.N.: 1981,Asymptotic Methods of Non-Linear Mechanics, Nauka, Moscow.Google Scholar
  22. Mittelman, D. and Jezewski, D.: 1982,Celes. Mech. 28, 401–413.Google Scholar
  23. Jezewski, D.J. and Mittelman, D.: 1983,Int. J. Non-Linear Mechanics 18, 119–124.Google Scholar
  24. Vujanovic, D.B. and Jones, E.S.: 1989,Variational Methods in Nonconservative Phenomena, Academic Press, Inc., New York.Google Scholar
  25. Danby, J.M.A.: 1962,Fundamentals of Celestial Mechanics, MacMillian Co., New York.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Djordje S. Djukic
    • 1
  1. 1.Fakultet tehnickih naukaUniversity of Novi SadNovi SadSerbia

Personalised recommendations