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A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

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Abstract

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzyński (Int. J. Theor. Phys. 29(11):1277–1284, [1990]) helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzyński helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.

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References

  1. Abraham, R., Marsden, J.: Foundations of Mechanics. Cummings, New York (1978)

    MATH  Google Scholar 

  2. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer, New York (1998)

    MATH  Google Scholar 

  3. Holm, D., Kupershmidt, B.: Poisson structures of superfluids. Phys. Lett. A 91, 425–430 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  4. Holm, D., Marsden, J., Ratiu, T., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123(1/2), 1–116 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  5. Moffat, H.K.: The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35(1), 117–129 (1969)

    Article  ADS  Google Scholar 

  6. Owczarek, R.: Topological defects in superfluid Helium. Int. J. Theor. Phys. 30(12), 1605–1612 (1991)

    Article  MathSciNet  Google Scholar 

  7. Owczarek, R.: Frames and fermionic excitations of vortices in superfluid helium. J. Phys. Condens. Matter 5, 8793–8798 (1993)

    Article  ADS  Google Scholar 

  8. Peradzyński, Z.: Helicity theorem and vertex lines in superfluid 4He. Int. J. Theor. Phys. 29(11), 1277–1284 (1990)

    Article  Google Scholar 

  9. Prykarpatsky, A., Mykytiuk, I.: Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects. Kluwer Academic, Dordrecht (1998)

    MATH  Google Scholar 

  10. Prykarpatsky, A., Zagrodziński, J.: Dynamical aspects of Josephson type media. Ann. Inst. Henri Poincaré Phys. Theor. 70(5), 497–524 (1999)

    MATH  Google Scholar 

  11. Schwarz, K.W.: Phys. Rev. B 38, 2398–2417 (1988)

    Article  ADS  Google Scholar 

  12. Troshkin, O.V.: Nontraditional Methods in Mathematical Hydrodynamics. Transl. Math. Monogr., vol. 114. Am. Math. Soc., Providence (1995)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dept. of Applied Mathematics, The AGH University of Science and Technology, Kraków, 30-059, Poland

    Anatoliy K. Prykarpatsky & Jolanta Golenia

  2. The IAPMM of the National Academy of Sciences, Lviv, Ukraine

    Anatoliy K. Prykarpatsky

  3. The Abdus Salam International Center for Theoretical Physics, Trieste, Italy

    Anatoliy K. Prykarpatsky & Nikolai N. Bogoliubov Jr.

  4. V.A. Steklov Mathematical Institute of RAN, Moscow, Russia

    Nikolai N. Bogoliubov Jr.

Authors
  1. Anatoliy K. Prykarpatsky
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  2. Nikolai N. Bogoliubov Jr.
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  3. Jolanta Golenia
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Correspondence to Anatoliy K. Prykarpatsky.

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Prykarpatsky, A.K., Bogoliubov, N.N. & Golenia, J. A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications. Int J Theor Phys 47, 1919–1928 (2008). https://doi.org/10.1007/s10773-007-9636-3

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  • DOI: https://doi.org/10.1007/s10773-007-9636-3

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