Communications in Mathematical Physics

, Volume 90, Issue 2, pp 235–250 | Cite as

Canonical maps between semidirect products with applications to elasticity and superfluids

  • Boris A. Kupershmidt
  • Tudor Ratiu


It is shown that two canonical maps arising in the Poisson bracket formulations of elasticity and superfluids are particular instances of general canonical maps between duals of semidirect product Lie algebras.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. 1.
    Abraham, R., Marsden, J.: Foundations of mechanics. Reading, MA: Addison-Wesley 1978Google Scholar
  2. 2.
    Eringen, A. C.: Mechanics of continua. Huntington, NY: Robert Krieger Publishing Co. 1980Google Scholar
  3. 3.
    Guillemin, V., Sternberg, S.: The moment map and collective motion. Ann. Phys.127, 220–253 (1980)Google Scholar
  4. 4.
    Hochschild, G.: The structure of Lie groups. San Francisco: Holden-Day 1965Google Scholar
  5. 5.
    Holm, D., Kupershmidt, B.: Poisson brackets and Clebsch representations for magnetohydrodyamics, multifluid plasmas, and elasticity. Physica D (in press)Google Scholar
  6. 6.
    Holm, D., Kupershmidt, B.: Poisson structures of superfluids. Phys. Lett.91A, 425–430 (1982)Google Scholar
  7. 7.
    Leggett, A. J.: A theoretical description of the new phases of liquid He. Rev. Mod. Phys.47, 331 (1975)Google Scholar
  8. 8.
    Marsden, T., Ratiu, T., Weinstein, A.: Semidirect products and reduction in mechanics. Trans. Am. Math. Soc. (in press)Google Scholar
  9. 9.
    Weinstein, A.: Lectures on sympletic manifolds. CBMS Conf. Series No. 27, Providence, RI: Am. Math. Soc. 1977Google Scholar
  10. 10.
    Weinstein, A.: The local structure of Poisson manifolds. Diff. Geom. (in press)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Boris A. Kupershmidt
    • 1
  • Tudor Ratiu
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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