Algebraization of 2-D Ideal Fluid Hydrodynamical Systems and Their Finite-Mode Approximations

  • V. Yu. Zeitlin
Conference paper


Starting from the description of ideal 2-D hydrodynamics in the framework of the Lie algebra of area-preserving diffeomorphisms sdiff two observations are made: 1st — this construction may be generalized by means of the central extension of the latter algebra, giving equations equivalent to variants of the vorticity equation on the β-plane; 2nd — a regular way to obtain finite-mode analogs of 2-D hydrodynamical equations preserving the essential algebraic features of these latter exists due to the relation between sdiff and su(N), N— ≻ ∞, algebras recently pointed out in literature.


Central Extension Vorticity Equation Fundamental Cell Integrable Dynamical System Generalize Euler Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V.V. Trofimov, A.T. Fomenko Russian Math. Surveys, 39 (1984) 3MathSciNetGoogle Scholar
  2. 2.
    V. Arnold Ann. Inst. Fourier, 16 (1966) 319CrossRefGoogle Scholar
  3. 3.
    A.M. Obukhov Sov. Phys. Doklady, 184 (1969) 309Google Scholar
  4. 4.
    J. Hoppe Phys. Lett., 215B (1988) 706CrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Pedlosky “Geophysical Fluid Dynamics”, Springer, 1978Google Scholar
  6. 6.
    E.B. Gledzer, F.V. Dolzhansky, A.M. Obukhov “The hydrodynamical-type systems and their applications”, Nauka, 1981 (in Russian, in English the essential ideas of this book are presented e.g. in A.M. Obukhov, F.V. Dolzhansky Geoph. Fluid Dyn., 6 (1975) 195 )CrossRefGoogle Scholar
  7. 7.
    D.B. Fairlie, P. Fletcher, C.K. Zachos Phys. Lett., 218B (1988) 203MathSciNetGoogle Scholar
  8. 8.
    D.B. Fairlie, C.K. Zachos Phys. Lett., 224B (1989) 101MATHMathSciNetGoogle Scholar
  9. 9.
    J. Patera, H. Zassenhaus Journ. Math. Phys., 29 (1989) 665CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    E.G. Floratos, J. Iliopoulos, Phys. Lett., 217B (1989) 285 G. TictopoulosGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1991

Authors and Affiliations

  • V. Yu. Zeitlin
    • 1
  1. 1.Institute of atmospheric physicsMoscowUSSR

Personalised recommendations