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Classical Perturbation Theory in Celestial Mechanics. The Equations of Planetary Motion

  • Dino Boccaletti
  • Giuseppe Pucacco
Part of the Astronomy and Astrophysics Library book series (AAL)

Abstract

In this chapter, we mainly aim to give an idea of the infancy and the adolescence of perturbation theory. Keeping in mind the general assumption according to which the whole book is built up, we have tried, as far as possible, to account for the developments and the changes which perturbation theory has undergone in the long term. This is almost always the only way to understand a problem thoroughly: that is, to understand why, at a certain instant in history, the problem is stated in a particular way and what are the missing answers and the reason it is not possible to answer certain questions hic et nunc. As the reader can see, in perturbation theory, although with difficulty but ineluctably, the idea that one can, nay must, average over short-period terms has won. This enables one to obtain results valid over the long period, that is, results of “secular” validity. It has been remarked that here, as in other fields, the idea has just been accepted that the end justifies the means (instead of the moral justification, here the mathematical proof is missing); we shall come back to this subject at the end of the whole presentation of the theory. If the introduction of the averaging procedure removes the so-called secular terms, nonetheless it does not remove the other plague of perturbation theory: the appearance of small denominators and related problems of convergence of the perturbative series. We shall see about these in the next chapter. After having explained the classical non-canonical methods and briefly mentioned the more modern averaging method, we also give an example of a problem, previously dealt with following a perturbative approach, which has been recognized to be solvable analytically in closed form.

Keywords

Perturbation Theory Perturbative Expansion Secular Term Artificial Satellite Keplerian Orbit 
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References

  1. 1.
    In any case, what makes these periodic solutions so precious for us is that they are, so to speak, the only breach through which we can try to penetrate into a place so far deemed inaccessible“. H. Poincaré: Les Méthodes Nouvelles de la Mécanique Céleste (Gauthiers-Villars, Paris, 1892), Vol. 1, p. 82.Google Scholar
  2. 2.
    For C1 Hamiltonians, the point has been established by Pugh and Robinson. See C. Pugh, C. Robinson: The C1 closing lemma, including Hamiltonians. Erg. Theory and Dyn. Syst. 3, 261–313 (1977).Google Scholar
  3. 3.
    For a demonstration of this statement and for a further deepening of the subject, the reader is referred to C. L. Siegel, J. K. Moser: Lectures on Celestial Mechanics (Springer, 1971), Sect. 21, of which our exposition is an incomplete summary.Google Scholar
  4. 4.
    L. A. Pars: A Treatise on Analytical Dynamics (Heinemann, 1964) Sects. 30.7, 30.8.Google Scholar
  5. 5.
    V. Szebehely: Theory of Orbits (Academic Press, 1967) Sect. 8.6.Google Scholar
  6. 6.
    See G. D. Birkhoff: Dynamical Systems, rev. edn. (Am. Math. Society, 1966) pp. 124–128 and the paper “Stabilità e Periodicità nella Dinamica”, Periodico di Matematiche, Ser. 4, 6 262–271 (1926). Our exposition is necessarily concise and so some subtleties have been neglected.Google Scholar
  7. 7.
    J. Horn: Beiträge zur Theorie der kleinen Schwingungen, Z. Math. Phys. 48, 400434 (1903); A. Lyapunov: Problème général de la stabilité des mouvements, Ann. Fac. Sci. de Toulose, 2, 203–474 (1907) - reprinted in Annals of Mathematical Studies Vol. 17 (Princeton University Press, 1949).Google Scholar
  8. 9.
    The example is taken from E. Zehnder: Periodic solutions of Hamiltonian equations, in Lecture Notes in Mathematics 1031, Dynamics and Processes, ed. by Ph. Blanchard, L. Streit (Springer, 1983).Google Scholar
  9. 10.
    A. Weinstein: Normal modes for non-linear Hamiltonian systems, Inv. Math. 20, 47–57 (1973).Google Scholar
  10. 11.
    The reader will find the proof of both theorems in J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Comm. Pure Appl. Math. 29, 727–747 (1976).Google Scholar
  11. 12.
    H. Poincaré: Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Mathematica 7 259–380 (1885).Google Scholar
  12. 13.
    For a proof, see P. Hartman: Ordinary Differential Equations (Wiley, 1964).Google Scholar
  13. 16.
    A. Andronov: Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues, C. R. Acad. Sci. Paris, 189 559–561 (1929).Google Scholar
  14. 17.
    See Sect. 8.5 and, for the subject we are dealing with, J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1983), Sect. 3.4.Google Scholar
  15. 20.
    H. Koçak, F. Bisshopp, Th. Banchoff, D. Laidlaw: Topology and mechanics with computer graphics — linear Hamiltonian systems in four dimensions. Advances in APplied Mathematics 7, 282–308 (1986).Google Scholar
  16. 21.
    See, for instance, H. Koçak: Normal forms and versal deformations of linear Hamiltonian systems, J. Diff. Eq. 51, 359–407 (1984).Google Scholar
  17. 22.
    See, for instance, K. R. Meyer, G. R. Hall: Introduction to the Hamiltonian Dynamical Systems and the N-body Problem II, G.Google Scholar
  18. 23.
    Obviously, we are speaking of a parameter in addition to h. Google Scholar
  19. 24.
    D. Buchanan: Trojan satellites - limiting case, Trans. Roy. Soc. Canada 35, 9–25 (1941).Google Scholar
  20. 25.
    See A. Deprit: Sur les orbites periodiques issues de L4 à la resonance interne 1/4, Astron. Astrophys. 3, 88–93 (1969).Google Scholar
  21. 26.
    K. R. Meyer, D. R. Schmidt: Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh, Celest. Mech. 4, 99–109 (1971).Google Scholar
  22. 28.
    We limit ourselves to quoting a few papers: J. Henrard: Periodic orbits emanating from a resonant equilibrium, Celest. Mech. 1, 437–466 (1970); Lyapunov’s center theorem for resonant equilibrium, J. Diff. Eqs. 14 431–441 (1973); D. Schmidt: Periodic solutions near a resonant equilibrium of a Hamiltonian system,Celest. Mech. 9 81–103 (1974); G. F. Dell’Antonio: Fine tuning of resonances and pcriodic solutions of Hamiltonian systems near equilibrium, Commun. Math. Phys. 120 529–546 (1989).Google Scholar
  23. 29.
    H. Seifert: Periodische Bewegungen mechanischen Systeme, Math. Z. 51 197–216 (1948).Google Scholar
  24. 30.
    P H. Rabinowitz: Periodic solutions of Hamiltonian systems. Commun. Pure and Appl. Math. XXXI,157–184 (1978).Google Scholar
  25. 31.
    P. H. Rabinowitz. Periodic solutions of Hamiltonian systems: A Survey. SIAM J. Math. Anal. 13, 343–352 (1982).Google Scholar
  26. 32.
    See the article where Berger himself surveys his work: M. S. Berger: Global aspects of periodic solutions of non-linear conservative systems, Lecture Notes in Physics 252 Local and Global Methods of Non-linear Dynamics, ed. by A. W. Sàenz, W. W. Zachary, R. Cawley (Springer, 1984).Google Scholar
  27. 33.
    C. G. Gray, G. Karl, V. A. Novikov: The four variational principles of mechanics, Annals of Physics 251, 1–25 (1996). The main parts of the work, with the addition of a further example of application, can also be found in C. G. Gray, G. Karl, V. A. Novikov: Direct use of variational principles as an approximation technique in classical mechanics, Am. J. Phys. 64, 1177–1184 (1996).Google Scholar
  28. 35.
    R. G. Helleman: Variational solutions of non-integrable systems, in Topics in Nonlinear Dynamics, Vol. 46 (American Institute of Physics, 1978) ed. by S. Jorna pp. 264–285; R. G. Heileman, T. Bountis: Periodic solutions of arbitrary period, variational methods, in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics, Vol. 93, ed. by G. Casati, J. Ford, (Springer, 1979), pp. 353–375. Our exposition will also take advantage of the exposition of A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics 2nd ed. (Springer, 1992), Sect. 2.6b.Google Scholar
  29. 36.
    See, for instance, B. A. Dubrovin, A. T. Fomenko, S. P. Novikov: Modern Geometry - Methods and Applications. Part I, 2nd edn (Springer, 1992) Sect. 8.Google Scholar
  30. 37.
    See the original papers quoted in Footnote 35 for technical details.Google Scholar
  31. 38.
    R. C. Churchill, G. Pecelli, D. L. Rod: A survey of the Hénon-Heiles Hamiltonian with applications to related examples, in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems,Lecture Notes in Physics, Vol. 93, ed. by G. Casati, J. Ford, (Springer, 1979), pp. 76–133.Google Scholar
  32. 40.
    E. W. Brown: Resonance in the solar system, Bull. Am. Math. Soc. 34, 265–289 (1928).Google Scholar
  33. 41.
    F. J. Dyson: Missed opportunities, Bull. Am. Math. Soc. 78, 635–652 (1972). Both papers are the text of a Josiah Willard Gibbs Lecture given under the auspices of the American Mathematical Society.Google Scholar
  34. 42.
    A. M. Molchanov: The resonant structure of the solar system—the law of planetary distances. Icarus, 8, 203–215 (1968).Google Scholar
  35. 43.
    G. E. Backus: Critique of “The resonant structure of the solar system” by A. M. Molchanov, Icarus 11, 88–92 (1969). M. Hénon: A comment on “The resonant structure of the solar system” by A. M. Molchanov, Icarus 11, 93–94 (1969).Google Scholar
  36. 44.
    A. E. Roy, M. W. Ovenden: I. On the occurrence of commensurable mean motions in the Solar System, Mon. Not. R. Astr. Soc. 114, 232–241 (1954); II. The mirror problem, Mon. Not. R. Astr. Soc. 115, 297–309 (1955).Google Scholar
  37. 47.
    The curious thing is that until 1965 astronomers had always corroborated the observations made by Schiaparelli (1889) which attributed to Mercury a synchronous state. Only the radar observations due to Pettengill and Dice (and later on confirmed) revealed that Mercury was in a 3: 2 resonance. The story is reviewed in P. Goldreich, S. J. Peale: The Dynamics of planetary rotations. Ann. Rev. of Astronomy and Astrophysics 14, 287–320 (1969).Google Scholar
  38. 48.
    For a justification of the model, see A. Celletti: Analysis of resonances in the spin-orbit problem in celestial mechanics: the synchronous resonance (Part I), Journal of Appl. Math. Phys. (ZAMP) 41, 174–204 (1990), a paper which we shall partly follow.Google Scholar
  39. 49.
    See J. M. A. Danby, Fundamentals of Celestial Mechanics 2nd Revised & Enlarged Edition (Willmann-Bell, Richmond, 1988) Sects. 5.2, 13.2.Google Scholar
  40. 50.
    See P. Goldreich, S. J. Peale: Spin—orbit coupling in the solar system, Astron. J. 71, 425–438 (1966).Google Scholar
  41. 51.
    J. Wisdom, S. J. Peale, F. Mignard: The chaotic rotation of Hyperion, Icarus 58, 137–152 (1984).Google Scholar
  42. 52.
    J. J. Klavetter: Rotation of Hyperion, I. Observations - Astron. J. 97, 570–579 (1989); Rotation of Hyperion, II. Dynamics, Astron. J. 98, 1855–1874 (1989). A nice relation of Klavetter’s observations can be found in the fascinating book by Ivars Peterson: Newton’s Clock (Freeman, 1993).Google Scholar
  43. 53.
    See, for instance, A. Morbidelli, M. Moons: Secular resonances in mean motion commensurabilities: the 2/1 and 3/2 cases, Icarus 102, 1–17 (1993).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Dino Boccaletti
    • 1
  • Giuseppe Pucacco
    • 2
  1. 1.Dipartimento di Matematica “Guido Castelnuovo”Università degli Studi di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di FisicaUniversità degli Studi di Roma “Tor Vergata”RomaItaly

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