Advertisement

Enrico Fermi pp 85-114 | Cite as

On the adiabatic invariants

  • Tullio Levi-Civita
Chapter

Abstract

Of the canonical systems
$$ \frac{{d{p_i}}}{{dt}} =- \frac{{\partial H}}{{\partial {q_i}}},\;\frac{{d{q_i}}}{{dt}} = \frac{{\partial H}}{{\partial {p_i}}}\;\left( {i = 1,2,...,n} \right) $$
with characteristic function H, independent of t, which contains slowly varying parameters a, two particularly conspicuous types of adiabatic invariants are known:
  1. 1° (Gibbs-Hertz’s theorems)
    The volume V enclosed in phase space by a generic isoenergetic manifold
    $$ H = E\;\left( {E\;constant} \right) $$
    which applies to quasi-ergodic systems; systems that do not allow other uniform integrals apart from H = E (see, e.g., nos. 3–5 of this paper).
     
  2. 2° (Burgers’s Theorem)
    Sommerfeld’s n loop integrals
    $$ {J_i} = \oint {{p_i}\;d{q_i}} \;\left( {i = 1,2,...,n} \right) $$
    which are adiabatic invariants for (Stäckel’s) systems that are integrable by means of separation of the variables and that admit in total n integrals, (quadratic in the p’s).
     

Keywords

Phase Space Canonical System Adiabatic Invariant Uniform Integral Total Differential 
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).
    See, e.g., Les spectres et la theorie de Vatome. Paris, Hermann, 1923.Google Scholar
  2. (2).
    Atombau und Spektrallinien. Braunschweig, Vieweg, 1922; 4a ed., 1924.Google Scholar
  3. (3).
    Vorlesungen ueber Atomdynamik. Berlin, Springer, Bd. I, 1925.Google Scholar
  4. (4).
    The structure of the atom. London, Bell, 1923; 3a ed., 1927.Google Scholar
  5. (5).
    Mecanique analytique et theorie des quanta. Paris, Blanchard, 1926.Google Scholar
  6. (6).
    Atomicity and quanta. Cambridge University Press, 1926.Google Scholar
  7. (7).
    In several articles, particularly in “Zeitschrift fur Physik”, 1924–1927.Google Scholar
  8. (8).
    See, above all for the German authors, the years 1926 and 1927 of the already quoted “Zeitschrift fur Physik”, and, for Dirac’s papers, Vol. 112, 1926, of the “Proc. of the R. S. of London”.Google Scholar
  9. (9).
    Ondes et mouvements. Paris, Gauthier-Villars, 1926.Google Scholar
  10. (10).
    Abhandlungen zur Wellenmechanik. Leipzig, Barth, 1927.Google Scholar
  11. (11).
    See a paper by these three authors Ueber die Grundlagnen der Quantentheorie in “Math. Ann.”, B. 98, pp. 130; and also Von Neumann, Mathematische Begriindung der Quantenmechanik, “Gottinger Nachr.”, 1927, pp. 157. (12) Adiabatic invariants and the theory of quanta, “Phil. Mag.”, vol. XXXIII, 1917, pp. 500’513.Google Scholar
  12. (14).
    Statistical mechanics. Yale University Press, 1902.Google Scholar
  13. (15).
    See, Weber-Gans — Repertorium der Physik. Leipzig, Teubner, 1916, Bd I, N. 270, pp. 535.Google Scholar
  14. (17).
    Cf., e.g., Levi-Civita and Amaldi — Lezioni di meccanica razionale. Bologna, Zanichelli, vol. (II) 1, chapter I, § 6.Google Scholar
  15. (18).
    Ibidem, vol. (II) 2, chapter II, no. 13.Google Scholar
  16. (20).
    See, e.g., op. cit. (18), chapter I, no. 38.Google Scholar
  17. (21).
    Adiabatic invariants of mechanical systems, “Phil. Mag.”, volume XXXIII, 1917, pp. 514’520.Google Scholar
  18. (22).
    See in the latter case Burgers’S dissertation (presented at the University of Leiden; Haarlem, 1918; in Dutch); alternatively Born, loc. cit.(3), pp. 98148.Google Scholar
  19. (24).
    Alcuni teoremi di meccanica analitica importanti per la teoria del quanti, “Nuovo Cimento”, VII, vol. 25, 1923, pp. 271’285.Google Scholar
  20. (25).
    Lie-Engel, Theorie der Transformationsgruppen, B. II. Leipzig, Trubner, 1980, chapter X, pp. 207’209.Google Scholar
  21. (26).
    Intorno ai sistemi di equazioni a derivate parziali del 1° ordine in involuzione. “Rend, del R. 1st. Lombardo”, vol. XXXVI, 1903, pp. 775–790.Google Scholar
  22. (27).
    Cf., e.g., loc. cit. (17), vol. (II) 2, chapter X, no. 29.Google Scholar
  23. (28).
    Ibidem, no. 30.Google Scholar
  24. (29).
    Anyone seeking guidance on the general theory of systems with total differentials can consult our Lezioni di calcolo differenziale assoluto, collected by Prof. E. Persico, Roma, Stock, 1925, chapter II.Google Scholar
  25. (31).
    Loc. cit. (17) vol. (II), chapter X, nos. 44, 45.Google Scholar
  26. (32).
    Ibidem, no. 64.Google Scholar
  27. (33).
    I.e. algebraic complements divided by the value of the determinant.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Tullio Levi-Civita
    • 1
  1. 1.RomeItaly

Personalised recommendations