Sophus Lie and Harmony in Mathematical Physics, on the 150th Anniversary of His Birth

  • Nail H. Ibragimov

Abstract

“The extraordinary significance of Lie’s work for the general development of geometry can not be overstated; I am convinced that in years to come it will grow still greater”—so wrote Felix Klein [13] in his nomination of the results of Sophus Lie on the group-theoretic foundations of geometry to receive the N. I. Lobachevskii prize. This prize was established by the Physical-Mathematical Society of the Imperial University of Kazan in 1895 and was to recognize works on geometry, especially non-Euclidean geometry, chosen by leading specialists. The first three prizes awarded were to the following:
  • 1897: S. Lie (Nominator: F. Klein)

  • 1900: W. Killing (Nominator: F. Engel)

  • 1904: D. Hilbert (Nominator: H. Poincaré).

Keywords

Manifold Huygens 

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References

  1. 1.
    W. F. Ames, non-linear Partial Differential Equations in Engineering, Vols. I and II, New York: Academic Press (1965, 1972).Google Scholar
  2. 2.
    R. L. Anderson and N. H. Ibragimov, Lie-Bäcklund Transformations in Applications, Philadelphia: SLAM (1979).MATHCrossRefGoogle Scholar
  3. 3.
    Yu. Berest, Construction of fundamental solutions for Huygens equations as invariant solutions, Dokl Akad. Nauk SSR, 317(4), 786–789 (1991).MathSciNetGoogle Scholar
  4. 4.
    L. Bianchi, Lezioni sulla teoria dei groupi continui finiti di trasformazioni, Pisa: Spoerri (1918).Google Scholar
  5. 5.
    G. Birkhoff, Hydrodynamics, Princeton, NJ: Princeton University Press (1950, 1960).MATHGoogle Scholar
  6. 6.
    G. W. Bluman and S. Kumei, Symmetries and Differential Equations, New York: Springer-Verlag (1989).MATHCrossRefGoogle Scholar
  7. 7.
    T. Hawkins, Jacobi and the birth of Lie’s theory of groups, Arch. History Exact Sciences 42(3), 187–278 (1991).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    E. Hille, Functional Analysis and Semi-groups, New York: Amer. Math. Soc. (1948), preface.MATHGoogle Scholar
  9. 9.
    N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Dordrecht: D. Reidel (1985).MATHCrossRefGoogle Scholar
  10. 10.
    N. H. Ibragimov, Primer on the Group Analysis, Moscow: Znanie (1989).Google Scholar
  11. 11.
    N. H. Ibragimov, Essays in the Group Analysis of Ordinary Differential Equations, Moscow: Znanie (1991).Google Scholar
  12. 12.
    N. H. Ibragimov, Group analysis of ordinary differential equations and the invariance principle in mathematical physics, Uspekhi Mat. Nauk, 47(1992), 89–156.Google Scholar
  13. 13.
    F. Klein, Theorie der Transformationsgruppen B. III, Pervoe prisuzhdenie premii N. I. Lobachevskogo, 22 okt. 1897goda, Kazan: Tipo-litografiya Imperatorskago Universiteta (1898), pp. 10–28.Google Scholar
  14. 14.
    P. S. Laplace, Mécanique céleste, T. I. livre 2, Chap. III (1799).Google Scholar
  15. 15.
    S. Lie, Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen, Arch. for Math. VI (1881).Google Scholar
  16. 16.
    S. Lie, Klassifikation und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten, Arch. Math. VIII, 187–453 (1883).Google Scholar
  17. 17.
    S. Lie, Theorie der Transformationsgruppen, Bd. 1 (Bearbeitet unter Mitwirkung von F. Enget), Leipzig: B. G. Teubner (1888).Google Scholar
  18. 18.
    S. Lie, Die infinitesimalen Berührungstransformationen der Mechanik, Leipz. Ber. (1889).Google Scholar
  19. 19.
    S. Lie, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen (Bearbeitet und herausgegeben von Dr. G. Scheffers), Leipzig: B. G. Teubner (1891).Google Scholar
  20. 20.
    S. Lie, Zur allgemeinen Theorie der partiellen Differentialgleichunen beliebiger Ordnung, Leipz. Ber. I, 53–128 (1895).Google Scholar
  21. 21.
    S. Lie, Gesammelte Abhandlungen, Bd. 1–6, Leipzig-Oslo.Google Scholar
  22. 22.
    M. Noether, Sophus Lie, Math. Annalen 53, 1–41 (1900).MATHCrossRefGoogle Scholar
  23. 23.
    P. J. Olver, Applications of Lie Groups to Differential Equations, New York: Springer-Verlag (1986).MATHCrossRefGoogle Scholar
  24. 24.
    L. V. Ovsiannikov, Group properties of differential equations, Novosibirsk: USSR Academy of Science, Siberian Branch (1962).Google Scholar
  25. 25.
    L. V. Ovsiannikov, Group Analysis of Differential Equations, Boston: Academic Press (1982).MATHGoogle Scholar
  26. 26.
    A. Z. Petrov, Einstein Spaces, Oxford: Pergamon Press (1969).MATHGoogle Scholar
  27. 27.
    E. M. Polischuk, Sophus Lie, Leningrad: Nauka (1983).Google Scholar
  28. 28.
    V. V. Pukhnachev, Invariant solutions of Navier-Stokes equations describing free-boundary motions, Dokl. Akad. Nauk SSSR 202(2), 302–305 (1972).Google Scholar
  29. 29.
    W. Purkert, Zum Verhältnis von Sophus Lie und Friedrich Engel, Wiss. Zeitschr. Ernst-Moritz-Arndt-Universität Greifswald, Math.-Naturwiss. Reihe XXXIII, Heft 1–2,29–34,(1984).MathSciNetGoogle Scholar
  30. 30.
    G. F. B. Riemann, Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abh. K. Ges. Wiss. Göttingen 8 (1860).Google Scholar
  31. 31.
    L. I. Sedov, Similarity and Dimensional Methods in Mechanics, 4th ed., New York Academic Press (1959).MATHGoogle Scholar
  32. 32.
    H. Stephani, Differential Equations: Their Solution Using Symmetries, Cambridge: Cambridge University Press (1989).MATHGoogle Scholar

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© Springer Science+Business Media New York 2001

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  • Nail H. Ibragimov

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