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Annals of Global Analysis and Geometry

, Volume 41, Issue 1, pp 1–24 | Cite as

Dual pairs in fluid dynamics

  • François Gay-Balmaz
  • Cornelia VizmanEmail author
Original Paper

Abstract

This article is a rigorous study of the dual pair structure of the ideal fluid (Phys D 7:305–323, 1983) and the dual pair structure for the n-dimensional Camassa–Holm (EPDiff) equation (The breadth of symplectic and poisson geometry: Festshrift in honor of Alan Weinstein, 2004), including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads naturally to central extensions of diffeomorphism groups such as the group of quantomorphisms and the Ismagilov central extension.

Keywords

Dual pair Momentum map Euler equation n-Camassa–Holm equation Central extension Quantomorphisms 

Mathematics Subject Classification (2000)

53D17 53D20 37K65 58D05 58D10 

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References

  1. 1.
    Arnold V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier Grenoble 16, 319–361 (1966)CrossRefGoogle Scholar
  2. 2.
    Banyaga A.: The Structure of Classical Diffeomorphism Groups. Kluwer Academic Publishers, Dordrecht (1997)zbMATHGoogle Scholar
  3. 3.
    Camassa R., Holm D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cushman R., Rod D.: Reduction of the semisimple 1:1 resonance. Phys. D 6, 105–112 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gay-Balmaz, F., Tronci, C., Vizman C.: Geodesic flows on the automorphism group of principal bundles. arXiv preprint (2010)Google Scholar
  6. 6.
    Golubitsky M., Stewart I.: Generic bifurcation of Hamiltonian systems with symmetry. Phys. D 24, 391–405 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Greub W., Halperin S., Vanstone R.: Connections, Curvature and Cohomology, Vol. I. Pure and Applied Mathematics 47. Academic Press, New York, London (1972)Google Scholar
  8. 8.
    Haller S., Vizman C.: Non-linear Grassmannians as coadjoint orbits. Math. Ann. 329, 771–785 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hirsch M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, New York (1976)Google Scholar
  10. 10.
    Holm, D.D., Marsden, J.E.: Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation. In: Marsden, J.E., Ratiu, T.S. (eds.) The Breadth of Symplectic and Poisson Geometry: Festshrift in Honor of Alan Weinstein. Progress in Mathematics, vol. 232, pp. 203–235. Birkhäuser, Boston (2004)Google Scholar
  11. 11.
    Holm D.D., Tronci C.: Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions. Proc. R. Soc. A 465(2102), 457–476 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Holm D.D., Tronci C.: The geodesic Vlasov equation and its integrable moment closures. J. Geom. Mech. 2, 181–208 (2009)MathSciNetGoogle Scholar
  13. 13.
    Ismagilov R.S.: Representations of Infinite-Dimensional Groups. Translations of Mathematical Monographs 152. American Mathematical Society, Providence, RI (1996)Google Scholar
  14. 14.
    Iwai T.: On reduction of two degree of freedom Hamiltonian systems by an S 1 action, and SO 0(1, 2) as a dynamical group. J. Math. Phys. 26, 885–893 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kriegl A., Michor P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs 53. American Mathematical Society, Providence, RI (1997)Google Scholar
  16. 16.
    Kostant B.: Quantization and Unitary Representations, Lectures in Modern Analysis and Applications III. Lecture Notes in Mathematics, vol. 170, pp. 87–208. Springer, Berlin (1970)Google Scholar
  17. 17.
    Lichnerowicz A.: Algèbre de Lie des automorphismes infinitésimaux d’une structure unimodulaire. Ann. Inst. Fourier 24, 219–266 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Marsden J.E.: Generic bifurcation of Hamiltonian systems with symmetry, appendix to Golubitsky and Stewart. Phys. D 24, 391–405 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Marsden J.E., Weinstein A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Phys. D 7, 305–323 (1983)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marsden J.E., Ratiu T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, New York (1999)zbMATHGoogle Scholar
  21. 21.
    Ortega, J.P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction. Progress in Mathematics (Boston, Mass.) 222. Birkhäuser, Boston (2004)Google Scholar
  22. 22.
    Roger C.: Extensions centrales d’algèbres et de groupes de Lie de dimension infinie, algèbre de Virasoro et généralisations. Rep. Math. Phys. 35, 225–266 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Souriau J.M.: Structure des systemes dynamiques. Dunod, Paris (1970)zbMATHGoogle Scholar
  24. 24.
    Vizman, C.: Natural differential forms on manifolds of functions. Arch. Math. (2011)Google Scholar
  25. 25.
    Weinstein A.: The local structure of Poisson manifolds. J. Diff. Geom. 18, 523–557 (1983)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Control and Dynamical SystemsCalifornia Institute of Technology 107-81PasadenaUSA
  2. 2.LMD, Ecole Normale Supérieure/CNRSParisFrance
  3. 3.Department of MathematicsWest University of TimişoaraTimişoaraRomania

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