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Non-standard fractional Lagrangians

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Abstract

Two mathematical physics’ approaches have recently gained increasing importance both in mathematical and in physical theories: (i) the fractional action-like variational approach which founds its significance in dissipative and non-conservative systems and (ii) the theory of non-standard Lagrangians which exist in some group of dissipative dynamical systems and are entitled “non-natural” by Arnold. Both approaches are discussed independently in the literature; nevertheless, we believe that the combination of both theories will help identifying more hidden solutions in certain classes of dynamical systems. Accordingly, we generalize the fractional action-like variational approach for the case of non-standard power-law Lagrangians of the form L 1+γ \((\gamma\in\mathbb{R})\) recently introduced by the author (Qual. Theory Dyn. Syst. doi:10.1007/s12346-012-0074-0, 2012). Many interesting features are discussed in some details.

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References

  1. Baleanu, D., Guvenc Ziya, B., Tenreiro, J.A.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin (2009)

    Google Scholar 

  2. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press/World Scientific, London/Singapore (2012)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Dreisigmeyer, D.W., Young, P.M.: Nonconservative Lagrangian mechanics: a generalized function approach. J. Phys. A 36, 8297–8310 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dreisigmeyer, D.W., Young, P.M.: Extending Bauer’s corollary to fractional derivatives. J. Phys. A, Math. Gen. 37(11), L117–121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Rabei, E.M., Alhalholy, T.S., Taani, A.A.: On Hamiltonian formulation of non-conservative systems. Turk. J. Phys. 28, 213–221 (2004)

    Google Scholar 

  8. Cresson, J., Inizan, P.: Irreversibility, least action principle and causality. arXiv:0812.3529

  9. Inizan, P.: Compatibility between fractional Hamiltonian systems. Int. J. Ecol. Econ. Stat. 9(F07), 83–91 (2007)

    MathSciNet  Google Scholar 

  10. Almeida, R., Pooseh, S., Torres, D.F.M.: Fractional variational problems depending on indefinite integrals. Nonlinear Anal. 75(3), 1009–1025 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. El-Nabulsi, R.A.: A periodic functional approach to the calculus of variations and the problem of time dependent damped harmonic oscillators. Appl. Math. Lett. 24(10), 1647–1653 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Malinowska, A.B., Ammi, M.R.S., Torres, D.F.M.: Composition functionals in fractional calculus of variations. Commun. Frac. Calc. 1, 32–40 (2010)

    Google Scholar 

  14. Malinowska, A.B., Torres, D.F.M.: Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 14(4), 523–537 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Malinowska, A.B., Torres, D.F.M.: Multiobjective fractional variational calculus in terms of a combined Caputo derivative. Appl. Math. Comput. 218(9), 5099–5111 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Atanacković, T.M., Konjik, S., Pilipović, S., Simić, S.: Variational problems with fractional derivatives: invariance conditions and Noether’s theorem. Nonlinear Anal., Theory Methods Appl. 71, 1504–1517 (2009)

    Article  MATH  Google Scholar 

  17. Atanackovic, T.M., Konjik, S., Pilipovic, S.: Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A 41(9), 095201–095213 (2008)

    Article  MathSciNet  Google Scholar 

  18. Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29(2), 417–437 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Discrete-time fractional variational problems. Signal Process. 91(3), 513–524 (2011)

    Article  MATH  Google Scholar 

  20. El-Nabulsi, R.A.: A fractional approach to nonconservative Lagrangian dynamical systems. Fizika A 14(4), 289–298 (2005)

    Google Scholar 

  21. 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)

    MathSciNet  MATH  Google Scholar 

  22. El-Nabulsi, R.A., Torres, D.F.M.: Fractional actionlike variational problems. J. Math. Phys. 49(5), 053521 (2008)

    Article  MathSciNet  Google Scholar 

  23. El-Nabulsi, R.A.: Gravitons in fractional action cosmology. Int. J. Theor. Phys. 51(12), 3978–3992 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Odzijewicz, T., Malinowska, A.B., Torres, D.F.M.: Fractional calculus of variations in terms of a generalized fractional integral and applications to physics. Abstr. Appl. Anal. 2012, 871912 (2012)

    Article  MathSciNet  Google Scholar 

  25. El-Nabulsi, R.A.: Fractional quantum Euler–Cauchy equation in the Schrodinger picture, complexified harmonic oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics. Mod. Phys. Lett. B 23(28), 3369–3386 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. El-Nabulsi, R.A.: Non-standard Lagrangians with differential operators. J. Nonlinear Anal. Optim. (in press)

  27. El-Nabulsi, R.A.: Fractional variational problems from extended exponentially fractional integral. Appl. Math. Comput. 217(22), 9492–9496 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. El-Nabulsi, R.A., Wu, G.-C.: Fractional complexified field theory from Saxena–Kumbhat fractional integral, fractional derivative of order alfa–beta and dynamical fractional integral exponent. African Disp. J. Math. 13(2), 45–61 (2012)

    MathSciNet  MATH  Google Scholar 

  29. El-Nabulsi, R.A.: The fractional Boltzmann transport equation. Comput. Math. Appl. 65, 1568–1575 (2011)

    Article  MathSciNet  Google Scholar 

  30. El-Nabulsi, R.A.: Fractional quantum field theories on multifractal sets. Am. J. Eng. Appl. Sci. 4, 133–141 (2009)

    Google Scholar 

  31. El-Nabulsi, R.A.: Fractional field theories from multidimensional fractional variational problems. Int. J. Geom. Methods Mod. Phys. 5, 863–892 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Calcagni, G.: Quantum field theory, gravity and cosmology in a fractal universe. J. High Energy Phys. 1003, 120 (2010)

    Article  MathSciNet  Google Scholar 

  33. Calcagni, G.: Fractal universe and quantum gravity. Phys. Rev. Lett. 104, 251301 (2010)

    Article  Google Scholar 

  34. El-Nabulsi, R.A.: Extended fractional calculus of variations, complexified geodesics and Wong’s fractional equation on complex place and Lie algebroids. Ann. Univ. Maria Curie 1, 49–67 (2011)

    MathSciNet  Google Scholar 

  35. El-Nabulsi, R.A.: The fractional calculus of variations from extended Erdelyi–Kober operator. Int. J. Mod. Phys. B 23(16), 3349–3361 (2009)

    Article  MathSciNet  Google Scholar 

  36. El-Nabulsi, R.A.: Universal fractional Euler–Lagrange equation from a generalized fractional derivate operator, Central Europe. J. Phys. 9(1), 250–256 (2010)

    MathSciNet  Google Scholar 

  37. Alekseev, A.I., Arbuzov, B.A.: Classical Yang–Mills field theory with nonstandard Lagrangian. Theor. Math. Phys. 59, 372–378 (1984)

    Article  MathSciNet  Google Scholar 

  38. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    Book  MATH  Google Scholar 

  39. Carinena, J.G., Ranada, M.F., Santander, M.: Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability. J. Math. Phys. 46, 062703–062721 (2005)

    Article  MathSciNet  Google Scholar 

  40. Chandrasekar, V.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: Simple and unified approach to identify integrable nonlinear oscillators and systems. J. Math. Phys. 47, 023508–023545 (2006)

    Article  MathSciNet  Google Scholar 

  41. Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator. Phys. Rev. E 72, 066203 (2005)

    Article  Google Scholar 

  42. Musielak, Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A, Math. Theor. 41, 055205–055222 (2008)

    Article  MathSciNet  Google Scholar 

  43. Musielak, Z.E.: General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos Solitons Fractals 42(15), 2645–2652 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: A nonlinear oscillator with unusual dynamical properties. In: Proceedings of the Third National Systems and Dynamics, India, pp. 1–4 (2006)

    Google Scholar 

  45. Dimitrijevic, D.D., Milosevic, M.: About non-standard Lagrangians in cosmology. AIP Conf. Proc., 1472, 41–46 (2012)

    Article  Google Scholar 

  46. Tyurin, N.A.: Nonstandard Lagrangian tori and pseudotoric structures. Teor. Mat. Fiz. 171, 321–325 (2012)

    Article  Google Scholar 

  47. Lukkassem, D.: Reiterated homogenization of non-standard Lagrangians. C. R. Acad. Sci. Paris 332, 999–1004 (2001)

    Article  Google Scholar 

  48. El-Nabulsi, R.A.: Quantum field theory from an exponential action functional. Indian J. Phys. 87, 379–383 (2013)

    Article  Google Scholar 

  49. El-Nabulsi, R.A.: Modified Proca equation and modified dispersion relation from a power-law Lagrangian functional. Indian J. Phys. 87, 465–470 (2013); Erratum (to appear)

    Article  Google Scholar 

  50. El-Nabulsi, R.A.: Nonlinear dynamics with non-standard Lagrangians. Qual. Theory Dyn. Syst. (2012). doi:10.1007/s12346-012-0074-0

    Google Scholar 

  51. El-Nabulsi, R.A., Soulati, T., Rezazadeh, H.: Non-standard complex Lagrangian dynamics. J. Adv. Res. Dyn. Cont. Syst. 5(1), 50–62 (2013)

    MathSciNet  Google Scholar 

  52. Dirac, P.A.M.: Generalized Hamiltonian dynamics. Can. J. Math. II, 129–148 (1950)

    Article  MathSciNet  Google Scholar 

  53. Cisneros-Parra, J.U.: On singular Lagrangians and Dirac’s method. Rev. Mex. Fis. 58, 61–68 (2012)

    MathSciNet  Google Scholar 

  54. Havelkova, M.: A geometric analysis of dynamical systems with singular Lagrangians. Math. Commun. 19, 169–178 (2011)

    MathSciNet  MATH  Google Scholar 

  55. Zhao, L., Yu, P., Xu, W.: Hamiltonian description of singular Lagrangian systems with spontaneously broken time translation symmetry. Mod. Phys. Lett. A 28(5), 1350002–13500015 (2013)

    Article  MathSciNet  Google Scholar 

  56. Carinena, J.F., Fermandez-Nunez, J., Ranada, M.F.: On singular Lagrangians affine in velocities. J. Phys. A 36, 3789–3808 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Carinena, J.F., Lopez, C., Ranada, M.F.: Geometric Lagrangian approach to first order systems and applications. J. Math. Phys. 29, 1134–1142 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  58. Carinena, F.: Singular Lagrangians. Fortschr. Phys. 38, 641–680 (1990)

    Article  MathSciNet  Google Scholar 

  59. Carinena, J.F., Ranada, M.F., Santander, F.: Lagrangian formalism for nonlinear second-order Riccati equations: one-dimensional integrability and two-dimensional super integrability. J. Math. Phys. 46, 062703–062721 (2005)

    Article  MathSciNet  Google Scholar 

  60. Saha, A., Talukdar, B.: On the non-standard Lagrangian equations. arXiv:1301.2667

  61. Cieslinski, J.I., Nikiciuk, T.: A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients. J. Phys. A, Math. Theor. 43, 175205–175220 (2010)

    Article  MathSciNet  Google Scholar 

  62. Garousi, M.R., Sami, M., Tsujikawa, S.: Constraints on Dirac–Born–Infeld type dark energy models from varying alpha. Phys. Rev. D 71, 083005 (2005)

    Article  Google Scholar 

  63. Copeland, E.J., Garousi, M.R., Sami, M., Tsujikawa, S.: What is needed of a tachyon if it is to be the dark energy? Phys. Rev. D 71, 043003 (2005)

    Article  Google Scholar 

  64. Dimitrijevic, D.D., Milosevic, M.: About non-standard Lagrangians in cosmology. AIP Conf. Proc., 1472, 41–46 (2012)

    Article  Google Scholar 

  65. Alekseev, A.I., Vshivtsev, A.S., Tatarintsev, A.V.: Classical non-Abelian solutions for non-standard Lagrangians. Theor. Math. Phys. 77(2), 1189–1197 (1988)

    Article  Google Scholar 

  66. Ghosh, S., Ghoudhuri, A., Talukdar, B.: On the quantization of damped harmonic oscillator. Acta Phys. Pol. B 40(1), 49–57 (2009)

    Google Scholar 

  67. Emden, R.: Gaskugeln, Anwendungen der mechanischen Warmen-theorie auf Kosmologie und meteorologische Probleme, Teubner, Leipzig (1907)

    Google Scholar 

  68. Wrubel, H.M.: Stellar interiors. In: Flugge, S. (ed.) Encyclopedia of Physics. Springer, Berlin (1958)

    Google Scholar 

  69. Muatjetjeja, B., Khalique, C.M.: Exact solutions of the generalized Lane–Emden equations of the first and second kind. Pramana J. Phys. 77(3), 545–554 (2011)

    Article  Google Scholar 

  70. Dixon, J.M., Tuszynski, J.A.: Solutions of a generalized Emden equation and their physical significance. Phys. Rev. A 41, 4166–4173 (1990)

    Article  MathSciNet  Google Scholar 

  71. Mukherjee, S., Roy, B., Chatterjee, P.: Solution of modified equations of Emden-type by differential transform method. J. Mod. Phys. 2(6), 559–563 (2011)

    Article  Google Scholar 

  72. Khalique, C.M., Mahomed, F.M., Muatjetjeja, B.: Lagrangian formulation of a generalized Lane–Emden equation and double reduction. J. Nonlinear Math. Phys. 15(2), 152–161 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  73. Alexanian, G., MacKenzie, R., Paranjape, M.B., Ruel, J.: Path integration and perturbation theory with complex Euclidean actions. Phys. Rev. D 77, 105014 (2008)

    Article  MathSciNet  Google Scholar 

  74. Alexanian, G., MacKenzie, R., Paranjape, M.B., Ruel, J.: Problems with complex actions. hep-th/0609146

  75. Goldfarb, Y., Tannor, D.J.: Interference in Bohmian mechanics with complex action. arXiv:0706.3507

  76. Goldfarb, Y., Degani, I., Tannor, D.J.: Bohmian mechanics with complex action: a new trajectory-based formulation of quantum mechanics. quant-ph/0604150

  77. Hayward, S.A.: Complex lapse, complex action and path integrals. Phys. Rev. D 53, 5664–5669 (1996)

    Article  MathSciNet  Google Scholar 

  78. Wright, E.M.: Path integral approach to the Schrödinger equation with a complex potential. Phys. Lett. A 104(3), 119–122 (1984)

    Article  MathSciNet  Google Scholar 

  79. Sexty, D.: Complex actions and stochastic quantization. Talk given at University of Heidelberg, St. Goar, September 2009

  80. Nielsen, H.B., Ninomiya, M.: What comes beyond the standard model. In: The Proceedings of Bled, Slovenia, pp. 144–185 (2007)

    Google Scholar 

  81. Nielsen, H.B.: Initial condition model from imaginary part of action and the information loss. arXiv:0911.3859

  82. Nagao, K., Nielsen, H.B.: Formulation of complex action theory. Prog. Theor. Phys. 126, 1021–1049 (2011)

    Article  MATH  Google Scholar 

  83. El-Nabulsi, R.A.: Lagrangian and Hamiltonian dynamics with imaginary time. J. Anal. Appl. 18, 283–295 (2012)

    MathSciNet  MATH  Google Scholar 

  84. Leon, D., Rodrigues, M.P.R.: Generalized Classical Mechanics and Field Theory, North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  85. Simon, J.Z.: Higher-derivatives Lagrangians, nonlocality, problems, and solutions. Phys. Rev. D 41, 3720–3733 (1990)

    Article  MathSciNet  Google Scholar 

  86. Vitagliano, L.: On higher derivatives as constraints in field theory: a geometric perspective. Int. J. Geom. Methods Mod. Phys. 08, 1687–1693 (2011)

    Article  MathSciNet  Google Scholar 

  87. Vitagliano, L.: The Lagrangian–Hamiltonian formalism for higher order fields theories. J. Geom. Phys. 60, 857–873 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  88. Aldaya, V., de Azcarraga, J.: Higher order Hamiltonian formalism in field theory. J. Phys. A, Math. Gen. 13, 2545–2551 (1982)

    Article  Google Scholar 

  89. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. NBS Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1964)

    MATH  Google Scholar 

  90. Gladwin Pradeep, R., Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: Non-standard conserved Hamiltonian structures in dissipative/damped systems: nonlinear generalizations of damped harmonic oscillator. J. Math. Phys. 50, 052901–052916 (2009)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  92. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives. Gordon and Breach, Amsterdam (1993)

    MATH  Google Scholar 

  93. Razminia, A., Majid, V.J., Dizaji, A.F.: An extended formulation of calculus of variations for incommensurate fractional derivatives with fractional performance index. Nonlinear Dyn. 69, 1263–1284 (2012)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  95. Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53, 215–222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  96. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  97. Herzallah, M.A.E., Baleanu, D.: Fractional Euler–Lagrange equations revisited. Nonlinear Dyn. 69, 977–982 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  98. Diaz, G., Coimbra, C.F.M.: Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dyn. 56, 145–157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  99. Herzallah, M.A.E., Baleanu, D.: Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn. 58, 385–391 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  100. El-Nabulsi, R.A.: Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent. Comput. Appl. Math. (2013). doi:10.1007/s40314-013-0053-3

    Google Scholar 

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  1. College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan, 641112, China

    Rami Ahmad El-Nabulsi

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El-Nabulsi, R.A. Non-standard fractional Lagrangians. Nonlinear Dyn 74, 381–394 (2013). https://doi.org/10.1007/s11071-013-0977-6

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