Advertisement

The Group and Hamiltonian Descriptions of Hydrodynamical Systems

  • Boris KhesinEmail author
Chapter
  • 1.8k Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1973)

Abstract

We survey applications of the Hamiltonian approach and group theory to ideal fluid dynamics and integrable systems. In particular, we review the derivations of the Landau-Lifschitz and Korteweg-de Vries equations as Euler equations on certain infinite-dimensional groups.

Keywords

Euler Equation Symplectic Structure Coadjoint Orbit Virasoro Algebra Hydrodynamical System 
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.

References

  1. 1.
    Arnold, V. I. (1966) Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361.CrossRefGoogle Scholar
  2. 2.
    Arnold, V. I. (1973) The asymptotic Hopf invariant and its applications. Proc. Summer School in Diff. Equations at Dilizhan, Erevan (in Russian); English transl.: Sel. Math. Sov. 5 (1986), 327–345.Google Scholar
  3. 3.
    Arnold, V. I. & Khesin, B. A. (1998) Topological methods in hydrodynamics.Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, pp. xv+374.Google Scholar
  4. 4.
    Calini, A. (2000) Recent developments in integrable curve dynamics. In Geom. Approaches to Diff. Equations; Australian Math. Soc. Lect. Notes Ser., 15, Cambridge University Press, 56–99.Google Scholar
  5. 5.
    Ebin, D. & Marsden, J. (1970) Groups of diffeomorphisms and the notion of an incompressible fluid. Ann. of Math. (2) 92, 102–163.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hasimoto, H. (1972) A soliton on a vortex filament, J. Fluid Mechanics 51, 477–485.ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Khesin, B. & Misiolek, G. (2003) Euler equations on homogeneous spaces and Virasoro orbits. Advances in Math. 176, 116–144.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Khesin, B. & Wendt, R. (2007) The geometry of infinite-dimensional groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag, to appear.Google Scholar
  9. 9.
    Marsden, J. E., Weinstein, A. (1983) Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D 7, 305–323.MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Misiolek, G. (1998) A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24:3, 203–208; Classical solutions of the periodic Camassa-Holm equation. Geom. Funct. Anal. 12:5 (2002), 1080–1104.MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Moffatt, H. K. (1969) The degree of knottedness of tangled vortex lines, J. Fluid. Mech. 106, 117–129ADSCrossRefGoogle Scholar
  12. 12.
    Ovsienko, V. Yu. & Khesin, B. A. (1987) Korteweg-de Vries super-equation as an Euler equation. Funct. Anal. Appl. 21:4, 329–331.zbMATHGoogle Scholar
  13. 13.
    Ricca, R. L. (1996) The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res. 18:5, 245–268.MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Turski, L. A. (1981) Hydrodynamical description of the continuous Heisenberg chain, Canad. J. Phys. 59:4, 511–514.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.University of TorontoCanada

Personalised recommendations