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Fractional Nambu Mechanics

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Abstract

The fractional generalization of Nambu mechanics is constructed by using the differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained and one example is investigated in details.

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References

  1. Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A. Math. Gen. 39, 10375–10384 (2006)

    Article  MATH  ADS  Google Scholar 

  3. Baleanu, D.: Killing-Yano tensors and Nambu tensors. Nuovo Cim. B 114(9), 1065–1072 (1999)

    ADS  MathSciNet  Google Scholar 

  4. Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72, 119–121 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)

    MATH  Google Scholar 

  6. Cayley, A.: Collected Mathematical Papers, vol. 3, pp. 156–204. Cambridge University Press, Cambridge (1890)

    Google Scholar 

  7. Fecko, M.: On a geometric formulation of the Nambu dynamics. J. Math. Phys. 33, 926–929 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)

    Google Scholar 

  9. Kilbas, A.A., Srivastava, H.H., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  10. Klimek, K.: Fractional sequential mechanics-models with symmetric fractional derivative. Czechoslov. J. Phys. E 51, 1348–1354 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Klimek, K.: Lagrangean and Hamiltonian fractional sequential mechanics. Czechoslov. J. Phys. E 52, 1247–1253 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135–3145 (2000)

    Article  ADS  Google Scholar 

  13. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)

    Google Scholar 

  14. Miller, K.S., Ross, B.: An Introduction to the Fractional Integrals and Derivatives-Theory and Application. Wiley, New York (1993)

    Google Scholar 

  15. Mukund, N., Sudarshan, E.C.G.: Relation between Nambu and Hamiltonian mechanics. Phys. Rev. D 13(10), 2846–2851 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  16. Muslih, S., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. J. Math. Anal. Appl. 304, 599–606 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Naber, M.: Time fractional Schrodinger equation. J. Math. Phys. 45, 3339–3352 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Nambu, Y.: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  19. Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Status Solidi B 133, 425–430 (1986)

    Article  Google Scholar 

  20. Ogawa, T., Sagae, T.: Int. J. Theor. Phys. 39(12), 2875–2890 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic, New York (1974)

    MATH  Google Scholar 

  22. Pandit, S.A., Gangal, A.D.: On generalized Nambu mechanics. J. Phys. A: Math. Gen. 31, 2899–2912 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)

    MATH  Google Scholar 

  24. Rabei, E.M., Nawafleh, K.I., Hiijawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  26. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  27. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon & Breach, New York (1993)

    MATH  Google Scholar 

  28. Takhtajan, L.: On foundation of generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Tarasov, V.E.: Continuous medium model for fractal media. Phys. Lett. A 336, 167–174 (2005)

    Article  MATH  ADS  Google Scholar 

  30. Tarasov, V.E.: Fractional variation for dynamical systems: Hamilton and Lagrange approaches. J. Phys. 39(26), 8409–8425 (2006)

    MATH  ADS  MathSciNet  Google Scholar 

  31. Tarasov, V.E.: Fractional statistical mechanics. Chaos 16, 033108–033115 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  32. Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371, 461–580 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

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Correspondence to Dumitru Baleanu.

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On leave of absence from Institute of Space Sciences, P.O. Box, MG-23, 76900, Magurele-Bucharest, Romania.

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Baleanu, D., Golmankhaneh, A.K. & Golmankhaneh, A.K. Fractional Nambu Mechanics. Int J Theor Phys 48, 1044–1052 (2009). https://doi.org/10.1007/s10773-008-9877-9

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  • DOI: https://doi.org/10.1007/s10773-008-9877-9

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