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Fractional almost Kähler–Lagrange geometry

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Abstract

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kähler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a “prime” Lagrange (in particular, Finsler) generating function. We emphasize the importance of such constructions for deformation quantization of fractional Lagrange geometries and applications in modern physics.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. Nishimoto, K.: Fractional Calculus: Integrations and Differentiations of Arbitrary Order. University of New Haven Press, New Haven (1989)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  5. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  6. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

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

    Google Scholar 

  8. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)

    Google Scholar 

  9. Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4(2), 153–192 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Momani, S.: A numerical scheme for the solution of multi-order fractional differential equations. Appl. Math. Comput. 182, 761–786 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Scalas, E., Gorenflo, R., Mainardi, F.: Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E 69, 011107-1 (2004)

    Article  MathSciNet  Google Scholar 

  12. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publisher, Connecticut (2006)

    Google Scholar 

  13. Lorenzo, C.F., Hartley, T.T.: Fractional trigonometry and the spiral functions. Nonlinear Dyn. 38, 23–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Silva, M.F., Tenreiro Machado, J.A., Lopes, A.M.: Modelling and simulation of artificial locomotion systems. Robotica 23, 595–606 (2005)

    Article  Google Scholar 

  16. Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13, 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  18. Tricaud, C., Chen, Y.Q.: Solving fractional order optimal control problems in RIOTS 95-a general-purpose optimal control problem solver. In: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and Its Applications, Ankara, Turkey, 5–7 November 2008

  19. Ortigueira, M.D., Tenreiro Machado, J.A.: Fractional signal processing and applications. Signal Process. 83, 2285 (2003)

    Article  Google Scholar 

  20. Baleanu, D.: Fractional Hamilton formalism with Caputo’s derivative. arXiv:math-ph/0612025

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Tarasov, V.E.: Fractional variations for dynamical systems: Hamilton and Lagrange approaches. J. Phys. A 39, 8409–8425 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tarasov, V.E., Zaslavsky, G.M.: Dynamics with low-level fractionality. Physica A 368, 399–415 (2006)

    Article  MathSciNet  Google Scholar 

  24. Baleanu, D., Trujillo, J.J.: On exact solutions of a class of fractional Euler–Lagrange equations. Nonlinear Dyn. 52, 331–335 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Laskin, N.: Fractional Schrodinger equation. Phys. Rev. E 66, 056108 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Vacaru, S.: Fractional nonholonomic Ricci flows. arXiv:1004.0625

  29. Vacaru, S.: Fractional dynamics from Einstein gravity, general solutions, and black holes. arXiv:1004.0628

  30. Baleanu, D., Vacaru, S.: Fedosov quantization of fractional Lagrange spaces. Int. J. Theor. Phys. 50 (2011, accepted). arXiv:1006.5538

  31. Vacaru, S.: Deformation quantization of almost Kahler models and Lagrange–Finsler spaces. J. Math. Phys. 48, 123509 (2007)

    Article  MathSciNet  Google Scholar 

  32. Vacaru, S.: Deformation quantization of nonholonomic almost Kahler models and Einstein gravity. Phys. Lett. A 372, 2949–2955 (2008)

    Article  MathSciNet  Google Scholar 

  33. Vacaru, S.: Generalized Lagrange transforms: Finsler geometry methods and deformation quantization of gravity. An. St. Univ. Al. I. Cuza (S.N.), Mat. LIII, 327–342 (2007)

    MathSciNet  Google Scholar 

  34. Vacaru, S.: Einstein gravity as a nonholonomic almost Kahler geometry, Lagrange–Finsler variables, and deformation quantization. J. Geom. Phys. 60, 1289–1305 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fedosov, B.V.: Deformation quantization and asymptotic operator representation. Funkc. Anal. Prilozh. 25, 1984–1994 (1990)

    Google Scholar 

  36. Fedosov, B.V.: A simple geometric construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994)

    MathSciNet  MATH  Google Scholar 

  37. Karabegov, A.V., Schlichenmaier, M.: Almost Kähler deformation quantization. Lett. Math. Phys. 57, 135–148 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Cartan, E.: Les Espaces de Finsler. Herman, Paris (1935)

    Google Scholar 

  39. Miron, R., Anastasiei, M.: The Geometry of Lagrange Spaces: Theory and Applications. FTPH, vol. 59. Kluwer Academic, Dordrecht (1994)

    Google Scholar 

  40. Matsumoto, M.: Foundations of Finsler Geometry and Special Finsler Spaces. Kaisisha, Shingaken (1986)

    MATH  Google Scholar 

  41. Oproiu, V.A.: A Riemannian structure in Lagrange geometry. Rend. Semin. Fac. Sci. Univ. Cagliari 55, 1–20 (1985)

    MathSciNet  MATH  Google Scholar 

  42. Anastasiei, M., Vacaru, S.: Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co)tangent bundles. J. Math. Phys. 50, 013510 (2009). arXiv:0710.3079

    Article  MathSciNet  Google Scholar 

  43. Vacaru, S.: Branes and quantization for an A-model complexification of Einstein gravity in almost Kahler variables. Int. J. Geom. Methods Mod. Phys. 6, 873–909 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, M.-F., Ren, J.-R., Zhu, T.: Series expansion in fractional calculus and fractional differential equations. arXiv:0910.4819

  45. Kern, J.: Lagrange geometry. Arch. Math. 25, 438–443 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  46. Vacaru, S.: Finsler and Lagrange geometries in Einstein and string gravity. Int. J. Geom. Methods Mod. Phys. 5, 473–511 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Vacaru, S., Stavrinos, P., Gaburov, E., Gonţa, D.: Clifford and Riemann–Finsler Structures in Geometric Mechanics and Gravity, Selected Works. Differential Geometry—Dynamical Systems, Monograph 7. Geometry Balkan Press (2006). www.mathem.pub.ro/dgds/mono/va-t.pdf and arXiv:gr-qc/0508023

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Çankaya University, 06530, Ankara, Turkey

    Dumitru Baleanu

  2. Science Department, University “Al. I. Cuza” Iaşi, 54, Lascar Catargi street, Iaşi, 700107, Romania

    Sergiu I. Vacaru

Authors
  1. Dumitru Baleanu
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  2. Sergiu I. Vacaru
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Correspondence to Dumitru Baleanu.

Additional information

D. Baleanu is on leave of absence from Institute of Space Sciences, P.O. Box, MG-23, R 76900, Magurele–Bucharest, Romania.

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Baleanu, D., Vacaru, S.I. Fractional almost Kähler–Lagrange geometry. Nonlinear Dyn 64, 365–373 (2011). https://doi.org/10.1007/s11071-010-9867-3

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  • DOI: https://doi.org/10.1007/s11071-010-9867-3

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