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Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations

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Abstract

This paper presents the fractional order Euler–Lagrange equations and the transversality conditions for fractional variational problems with fractional integral and fractional derivatives defined in the sense of Caputo and Riemann–Liouville. A fractional Hamiltonian formulation was developed and some illustrative examples were treated in detail.

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References

  1. Agrawal, O.P.: A new Lagrangian and a new Lagrange equation of motion for fractionally damped systems. J. Appl. Mech. 68, 339–341 (2001)

    Article  MATH  Google Scholar 

  2. 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 

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

    Article  MATH  Google Scholar 

  4. Agrawal, O.P.: A general finite element formulation for fractional variational problems. J. Math. Anal. Appl. 337, 1–12 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Agrawal, O.P., Machado, J.A.T., Sabatier, J.: Nonlinear Dynamics, Special Issue: Fractional Derivatives and Their Applications. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  6. Baleanu, D., Agrawal, O.P.: Fractional Hamilton formalism within Caputo’s derivative. Czech. J. Phys. 56(10–11), 1087–1092 (2006)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. Carpinteri, A., Mainardi, F.: Fractals and Fractional Calculus in Continuum Mechanics. Springer, Vienna (1997)

    MATH  Google Scholar 

  9. El-Nabulsi, R.A.: A fractional approach of nonconservative Lagrangian dynamics. Fizika A 14(4), 289–298 (2005)

    Google Scholar 

  10. El-Nabulsi, R.A.: A fractional action-like variational approach of some classical, quantum and geometrical dynamics. Int. J. Appl. Math. 17(3), 299–317 (2005)

    MATH  MathSciNet  Google Scholar 

  11. El-Nabulsi, R.A., Torres, D.F.M.: Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β). Math. Methods Appl. Sci. 30(15), 1931–1939 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  13. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematical Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  14. Klimek, M.: Fractional sequential mechanics—models with symmetric fractional derivative. Czech. J. Phys. 51, 1348–1354 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Klimek, M.: Stationary conservation laws for fractional differential equations with variable coefficients. J. Phys. A, Math. Gen. 35, 6675–6693 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lorenzo, C.F., Hartley, T.T.: Fractional trigonometry and the spiral functions. Nonlinear Dyn. 38(1–4), 23–60 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Magin, R.L.: Fractional calculus in bioengineering. Parts 1–3, Crit. Rev. Biomed. Eng. 32(1), 1–377 (2004)

    Article  Google Scholar 

  19. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  20. Muslih, S.I., 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 

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

    Article  MathSciNet  Google Scholar 

  22. Ortigueira, M.D.: Fractional central differences and derivatives. J. Vib. Control 14(9–10), 1255–1266 (2008)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  24. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  25. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3582–3592 (1997)

    Article  MathSciNet  Google Scholar 

  26. Suarez, J.I., Vinagre, B.M., Chen, Y.Q.: A fractional adaptation scheme for lateral control of an AGV. J. Vib. Control 14(9–10), 1499–1511 (2008)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  28. van Brunt, B.: The Calculus of Variations. Springer, New York (2004)

    MATH  Google Scholar 

  29. Weinstock, R.: Calculus of Variations with Applications to Physics and Engineering. Dover, New York (1974)

    MATH  Google Scholar 

  30. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, Berlin (2003)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Science, Zagazig University, Zagazig, Egypt

    Mohamed A. E. Herzallah

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

    Dumitru Baleanu

Authors
  1. Mohamed A. E. Herzallah
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  2. Dumitru Baleanu
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Correspondence to Dumitru Baleanu.

Additional information

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

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Herzallah, M.A.E., Baleanu, D. Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn 58, 385–391 (2009). https://doi.org/10.1007/s11071-009-9486-z

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  • DOI: https://doi.org/10.1007/s11071-009-9486-z

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