Advertisement

The Theory of Adiabatic Invariants

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

Abstract

In this chapter, devoted to the theory of adiabatic invariants and its applications in the field of astronomy, we try to point out how procedures, seemingly different and originating in different fields, in the end derive from the same concepts. What closely relates perturbation theory, which originated from the demand for suitable solutions to the problems of planetary motion, and the theory of adiabatic invariants, which was developed to give a more solid bases to the quantization rules of early quantum mechanics, is recourse to the averaging procedure. In both cases, the average is performed over a periodic motion having a period by far shorter than the time which characterizes the evolution of the physical system one is studying. Through the application of Noether’s theorem, together with the averaging method, one then sees how also the approximate invariance of the quantities, called the adiabatic invariants, is connected with the symmetry properties of the system. It is clear that one could, as is sometimes done, overturn the argument and start from the element recognized as the central one, i.e. the averaging procedure, and apply it to the different classes of problems. But it seemed to us more effective from a didactic point of view to follow the inverse path of progressively identifying the unifying element in the different problems belonging to apparently disparate fields of research. But at this point one cannot omit the fact that what we have called the unifying element is far from being rigorously proven. Just to quote one of the most authoritative “users” of the averaging principle, “We note that this principle is neither a theorem, an axiom, nor a definition, but rather a physical proposition, i.e., a vaguely formulated and, strictly speaking, untrue assertion. Such assertions are often fruitful sources of mathematical theorems.”1 The introduction of the use of methods based on the use of the Lie transform on the one hand has emphasized the link of the adiabatic invariants theory with the traditional perturbation theory, on the other hand has enabled people to resort to automatic computations in actual applications. To conclude, we once more call the attention of the reader to the universality of the paradigm of the perturbed oscillator which, by help of a transformation of variables, can also be applied to the case of a charged particle in a magnetic field.

Keywords

Solar Wind Hamiltonian System Globular Cluster Fundamental Matrix Linear Oscillator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. 1.
    V. I. Arnold: Mathematical Methods of Classical Mechanics, 2nd edn (Springer, 1989 ), p. 292.Google Scholar
  2. 4.
    A. J. Lichtenberg, M. A. Lieberman: Regular and Chaotic Dynamics (Springer, 1992), Sect. 2. 5c.Google Scholar
  3. 5.
    For all the results given without proof, the reader may consult, for instance, F. Brauer, J. A. Nohel: The Qualitative Theory of Ordinary Differential Equations (Dover, 1989 ).Google Scholar
  4. 6.
    This was shown for the first time by O. Haupt. See O. Haupt: ber lineare homogene Differentialgleichungen zweiter Ordnung mit periodischen Koeffizienten, Math. Ann. 79, 278–285 (1919).Google Scholar
  5. 7.
    See, for instance, L. Ruby: Applications of the Mathieu equation. Am. J. Phys. 64, 39–44 (1996).Google Scholar
  6. 8.
    Lord Rayleigh: On maintained vibrations, Phil. Mag. S. 5 15, 229–235 (1883).Google Scholar
  7. 9.
    See, for instance, Lochak and Meunier: op. cit., Sect. 8. 3.Google Scholar
  8. 10.
    We just give the main lines of the demonstration included in the celebrated paper by Arnold: Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Mathematical Surveys 18: 6, 86–191 (1963).Google Scholar
  9. 11.
    T. G. Northrop, E. Teller: Stability of the adiabatic motion of charged particles in the earth’s field, Phys. Rev. 117, 215–225 (1960).MathSciNetADSCrossRefGoogle Scholar
  10. 12.
    T.G. Northrop, J. A. Rome: Extensions of guiding centre motion to higher order, Phys. Fluids 21, 384–389 (1978).ADSCrossRefGoogle Scholar
  11. 13.
    R. G. Littlejohn: A guiding center Hamiltonian: a new approach, J. Math. Phys. 20, 2445–2458 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 14.
    Hamiltonian formulation of guiding centre motion, Phys. Fluids 24, 1730–1749 (1981)MathSciNetCrossRefGoogle Scholar
  13. 15.
    Hamiltonian perturbation theory in noncanonical coordinates, J. Math. Phys. 23, 742–747 (1982).MathSciNetCrossRefGoogle Scholar
  14. See, for instance: R. H. Rand, D. Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra (Springer, 1987).Google Scholar
  15. 17.
    See J. R. Cary, D. F. Escande, J. L. Tennyson: Adiabatic invariant change due to separatrix crossing, Phys. Rev. A 34, 4256–4275 (1986)Google Scholar
  16. 18.
    J. Henrard: The adiabatic invariant in classical mechanics, in Dynamics Reported - Expositions in Dynamical Systems, New Series: Vol. 2 (Springer, 1993) Sect. 6.Google Scholar
  17. 19.
    To avoid making this concise section overwhelmed by notes we defer the citation of original papers to the bibliographical notes.Google Scholar
  18. 20.
    The reader will find an exhaustive survey in J. Hajimetriou: Secular variation of mass and the evolution of binary systems Advances in Astronomy 5, 131–187 (1967) See Sect. II of the paper quoted in Footnote 29.Google Scholar
  19. 22.
    Again, see the paper quoted in Footnote 29.Google Scholar
  20. 23.
    J. E. Littlewood: Adiabatic invariance II: elliptic motion about a slowly varying center of force, Annals of Physics 26, 131–156 (1964).MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 24.
    T. Levi-Civita: Applicazioni Astronomiche degli Invarianti Adiabatici - Proceedings of the International Congress of Mathematicians, Bologna (1928), pp. 17–28.Google Scholar
  22. 25.
    See, for instance, L. Spitzer, Jr.: Dynamical Evolution of Globular Clusters (Princeton University Press, 1987), pp. 110 and ff. See the three papers byGoogle Scholar
  23. 26.
    M. D. Weinberg: Adiabatic invariants in stellar dynamics I, II, III, Astron. J. 108, 1398–1402, 1403–1413, 1414–1420 (1994).Google Scholar
  24. 27.
    J. Binney, A. May: The spheroids of galaxies before and after disc formation, Mon. Not. Roy. Astr. Soc. 218, 743–760 (1986).ADSGoogle Scholar
  25. 38.
    See, for instance, J. Binney, S. Tremaine: Galactic Dynamics (Princeton University Press, 1987), Chap. 4.Google Scholar
  26. 39.
    J. Binney, S. Tremaine, op. cit., p. 181.Google Scholar
  27. 40.
    For the history of the studies and experiments on the Earth’s magnetosphere, see the two papers by D. P. Stern: A brief history of magnetospheric physics, Reviews of Geophysics 27, 103–114 (1989); 34, 1–31 (1996).Google Scholar
  28. 41.
    H. Poincaré: Remarques sur une experience de M. Birkeland, C. R. Acad. Sci. 123, 530–533 (1896).Google Scholar
  29. 42.
    See A. J. Dragt, J. M. Finn: Insolubility of trapped particle motion in a magnetic dipole field, J. Geophys. Res. 81, 2327–2340 (1976).ADSCrossRefGoogle Scholar
  30. 43.
    See D. P. Stern, loc. cit., and also D. P. Stern, N. F. Ness: Planetary magnetospheres, Ann. Rev. Astron. Astrophys. 20, 139–161 (1982).ADSCrossRefGoogle 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

Personalised recommendations