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On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative

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Abstract

Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics, the equivalent Lagrangians play an important role because they admit the same Euler–Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant, the Lagrangian we obtain are the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann–Liouville fractional derivatives. As a consequence of applying this procedure, the classical results are reobtained as a special case.

The fractional generalization of Faà di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler–Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated, and two examples are analyzed in detail.

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References

  1. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach, Linghorne (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Tenreiro Machado, J.A.: A probabilistic interpretation of the fractional-order differentiation. Frac. Calc. Appl. Anal. 8, 73–80 (2003)

    MathSciNet  Google Scholar 

  5. Tenreiro Machado, J.A.: Discrete-time fractional-order controllers. Frac. Calc. Appl. Anal. 4, 47–66 (2001)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Agrawal, O.P.: Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dyn. 38, 191–206 (2004)

    Article  MATH  Google Scholar 

  8. 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 

  9. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MATH  Google Scholar 

  10. 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 

  11. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7(9), 1461–1477 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mainardi, F., Pagnini, G., Gorenflo, R.: Mellin transform and subordination laws in fractional diffusion processes. Frac. Calc. Appl. Anal. 6(4), 441–459 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Raspini, A.: Simple solutions of the fractional Dirac equation of order \(\frac{2}{3}\) . Phys. Scr. 64, 20–22 (2001)

    Article  MATH  Google Scholar 

  14. Rabei, E., Alhalholy, T., Rousan, A.: Potential of arbitrary forces with fractional derivatives. Int. J. Theor. Phys. A 19(17–18), 3083–3096 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Naber, M.: Time fractional Schrödinger equation. J. Math. Phys. 45(8), 3339–3352 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lim, S.C., Muniady, S.V.: Stochastic quantization of nonlocal fields. Phys. Lett. A 324, 396–405 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Magin, R.L.: Fractional calculus in bioengineering. Critic. Rev. Biomed. Eng. 32(1), 1–104 (2004)

    Article  Google Scholar 

  18. Magin, R.L.: Fractional calculus in bioengineering, part 2. Critic. Rev. Biomed. Eng. 32(2), 105–193 (2004)

    Article  Google Scholar 

  19. Magin, R.L.: Fractional calculus in bioengineering, part 3. Critic. Rev. Biomed. Eng. 32(3/4), 194–377 (2004)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  22. 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 

  23. Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivatives. Czech. J. Phys. 51, 1348–1354 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52, 1247–1253 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. Preprint (2006)

  26. Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives. Phys. Scr. 72(2–3), 119–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Muslih, S., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 304(3), 599–606 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I., Baleanu, D.: The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327(2), 891–897 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Baleanu, D., Muslih, S.I.: Formulation of Hamiltonian equations for fractional variational problems. Czech. J. Phys. 55(6), 633–642 (2005)

    Article  MathSciNet  Google Scholar 

  30. Muslih, S., Baleanu, D., Rabei, E.: Hamiltonian formulation of classical fields within Riemann–Liouville. Phys. Scr. 73(6), 436–438 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004)

    Google Scholar 

  32. Baleanu, D.: Constrained systems and Riemann–Liouville fractional derivative. In: Proceedings of 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, July 19–21, pp. 597–602 (2004)

  33. Baleanu, D.: About fractional calculus of singular Lagrangians. In: Proceedings of IEEE International Conference on Computational Cybernetics ICCC, Wien, Austria, August 30–September 1, pp. 379–385 (2004)

  34. Baleanu, D., Muslih, S.: New trends in fractional quantization method, 1st edn. Intelligent Systems at the Service of Mankind, vol. 2. U-Books, Augsburg (2005)

    Google Scholar 

  35. Baleanu, D.: Constrained systems and Riemann–Liouville fractional derivatives. In: Le Mehaute, A., Tenreiro Machado, J.A., Trigeassou, J.C., Sabatier, J. (eds.) Fractional Differentiation and Its Applications, pp. 69–80. U-Books, Augsburg (2005)

    Google Scholar 

  36. Baleanu, D., Muslih, S.I.: About fractional supersymmetric quantum mechanics. Czech. J. Phys. 55(9), 1063–1066 (2005)

    Article  MathSciNet  Google Scholar 

  37. Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1993)

    Google Scholar 

  38. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Nauka, Moscow (1979)

    MATH  Google Scholar 

  39. Hardy, G.H.: Riemann’s form of Taylor’s series. J. Lond. Math. Soc. 20, 48–57 (1954)

    Article  MathSciNet  Google Scholar 

  40. Trujillo, J.J.: On a Riemann–Liouville generalized Taylor’s formula. J. Math. Anal. Appl. 231, 255–265 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Llosa, J., Vives, J.: Hamiltonian formalism for nonlocal Lagrangians. J. Math. Phys. 35, 2856–2877 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gitman, D.M., Tytin, I.V.: Quantization of Fields with Constraints. Springer, Berlin (1990)

    MATH  Google Scholar 

  43. Nesterenko, V.V.: Singular Lagrangians with higher derivatives. J. Phys. A: Math. Gen. 22, 1673–1687 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  44. Bering, K.: A note of non-locality and Ostrogradski’s construction. Preprint hep-th/0007192

  45. Gomis, J., Kamimura, K., Llosa, J.: Hamiltonian formalism for space-time noncummutative theories. Phys. Rev. D 63, 045003 (2001)

    MathSciNet  Google Scholar 

  46. Carathéodory, C.: Calculus of Variations and Partial Differential Equations of First Order, Part II. Holden-Day, Oackland (1967)

    MATH  Google Scholar 

  47. Negri, L.J., da Silva Joseph, E.G.: s-equivalent Lagrangians in generalized mechanics. Phys. Rev. D 27, 2227–2232 (1986)

    Google Scholar 

  48. Santilli, M.: Foundations of Theoretical Mechanics I, II. Springer, New York (1977)

    Google Scholar 

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Author information

Authors and Affiliations

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

    Dumitru Baleanu

  2. Department of Physics, Al-Azhar University, Gaza, Palestine

    Sami I. Muslih

  3. International Center for Theoretical Physics, Trieste, Italy

    Sami I. Muslih

  4. Department of Science, Jerash Private University, Jerash, Jordan

    Eqab M. Rabei

  5. Department of Physics, Mutah University, Karak, Jordan

    Eqab M. Rabei

Authors
  1. Dumitru Baleanu
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  2. Sami I. Muslih
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  3. Eqab M. Rabei
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Correspondence to Dumitru Baleanu.

Additional information

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

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Baleanu, D., Muslih, S.I. & Rabei, E.M. On fractional Euler–Lagrange and Hamilton equations and the fractional generalization of total time derivative. Nonlinear Dyn 53, 67–74 (2008). https://doi.org/10.1007/s11071-007-9296-0

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  • DOI: https://doi.org/10.1007/s11071-007-9296-0

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