×

Fractional variational approach with non-standard power-law degenerate Lagrangians and a generalized derivative operator. (English) Zbl 1366.37128

Summary: We extend the fractional actionlike variational approach where we substitute the standard Lagrangian by a non-standard power-law Lagrangian holding a generalized derivative operator. We focus on degenerate Lagrangians for the constructed fractional formalism where we show that non-linear oscillators with damping solutions may be obtained from degenerate non-standard Lagrangians which are linear in velocities. We explore as well the case of \(2^{nd}\)-order derivatives non-standard Lagrangians and we study the case where Lagrangians are linear in accelerations where damping solutions are obtained as well. It was observed that these extensions give another possibility to obtain more fundamental aspects which may have interesting classical effects.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
49K10 Optimality conditions for free problems in two or more independent variables
34A08 Fractional ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Almeida and D. F. M. Torres, Calculus of variations with fractional derivatives and frac- tional integrals, Appl. Math. Lett. 22 (2009), 1816-1820.; · Zbl 1183.26005
[2] V. I. Arnold, Mathematical Methods of Classical Mechanics, New York: Springer, 1978.; · Zbl 0386.70001
[3] D. Baleanu, New applications of fractional variational principles, Rep. Math. Phys. 61 (2008), 199-206.; · Zbl 1166.58304
[4] M. Bartusek, On oscillatory solutions of third order differential equations with quasiderivatives, Fourth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 03, 1999, pp 1-11.; · Zbl 0971.34016
[5] M. Bartusek and Z. Dosla, Oscillatory criteria for nonlinear third order differential equations with quasiderivatives, Diff. Eqs. Dyn. Syst. 3 (1995), 251-268.; · Zbl 0869.34028
[6] M. Bartusek and J. Osicka, Asymptotic behaviour of solutions of a third-order nonlinear dif- ferential equation, Nonlinear Anal. 34 (1998), 653-664.; · Zbl 0944.34042
[7] J. G. Carinena, M. F. Ranada and M. Santander, Lagrangian formalism for nonlinear second- order Riccati Systems: one-dimensional integrability and two-dimensional superintegrability, J. Math. Phys. 46 (2005), 062703-062721.; · Zbl 1110.37051
[8] V. K. Chandrasekar, S. N. Pandey, M. Senthilvelan and M. Lakshmanan, Simple and unified approach to identify integrable nonlinear oscillators and systems, J. Math. Phys. 47 (2006), 023508-023545.; · Zbl 1111.34003
[9] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator, Phys. Rev. E72 (2005), 066203-066211.;
[10] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, A Nonlinear oscillator with unusual dynamical properties, in Proceedings of the Third National Systems and Dynamics, pp.1-4 (2006).;
[11] A. Dector, H. A. Morales-Tecotl, L. F. Urrutia and J. D. Vergara, An alternative canonical approach to the ghost problem in a complexified extension of the Pais-Uhlenbeck oscillator, SIGMA 5 (2009), 053, 22p.; · Zbl 1208.81111
[12] R. A. El-Nabulsi, Non-standard fractional Lagrangians, Nonlinear Dyn. 74 (2013), 381-394.; · Zbl 1281.70022
[13] R. A. El-Nabulsi, Non-standard Lagrangians in rotational dynamics and the modified Navier- Stokes equation, Nonlinear Dyn. 79 (2015), 2055-2068.; · Zbl 1318.70017
[14] R. A. El-Nabulsi, Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent, Comp. Appl. Math. 33 (2014), 163-179.; · Zbl 1348.70076
[15] R. A. El-Nabulsi, Non-standard non-local-in-time Lagrangian in classical mechanics, Qual. Theory Dyn. Syst. 13 (2014), 149-160.; · Zbl 1305.70043
[16] R. A. El-Nabulsi, T. Soulati and H. Rezazadeh, Non-standard complex Lagrangian dynamics, J. Adv. Res. Dyn. Contr. Theor. 5, No. 1 (2012), 50-62.;
[17] R. A. El-Nabulsi, Nonlinear dynamics with non-standard Lagrangians, Qual. Theory Dyn. Syst. 13(2013), 273-291.; · Zbl 1322.37037
[18] R. A. El-Nabulsi, Modified Proca equation and modified dispersion relation from a power-law Lagrangian functional, Indian J. Phys. 87, No.5 (2013), 465-470; Erratum Indian J. Phys. 87, 10, (2013) p. 1059.;
[19] R. A. El-Nabulsi, Quantum field theory from an exponential action functional, Indian J. Phys. 87, No. 4 (2013), 379-383.;
[20] R. A. El-Nabulsi, Generalizations of the Klein-Gordon and the Dirac Equations from non- standard Lagrangians, Proc. Nat. Acad. Sci. India Sec. A: Phys. Sci. 83 (2013), 383-387.; · Zbl 1330.35364
[21] R. A. El-Nabulsi, A generalized nonlinear oscillator from non-standard degenerate Lagrangians and its consequent Hamiltonian formalism, Proc. Nat. Acad. Sci. India Sec. A: Phys. Sci 84(4) (2014), 563-569.; · Zbl 1314.34077
[22] R. A. El-Nabulsi, Electrodynamics of relativistic particles through non-standard Lagrangian, J. At. Mol. Sci. 5(3) (2014), 268-279.;
[23] R. A. El-Nabulsi, Non-standard power-law Lagrangians in classical and quantum dynamics, Appl. Math. Lett. 43 (2015), 120-127.; · Zbl 1317.81139
[24] R. A. El-Nabulsi, Classical string field mechanics with non-standard Lagrangians, Math. Sci. 9 (2015), 173-179.; · Zbl 1407.70039
[25] R. A. El-Nabulsi, From classical to discrete gravity through exponential non-standard La- grangians in general relativity, Mathematics 3(3) (2015), 727-745.; · Zbl 1332.83081
[26] R. A. El-Nabulsi and D. F. M. Torres, Fractional actionlike variational problems, J. Math. Phys. 49 (2008), 053521-053529.; · Zbl 1152.81422
[27] R. A. El-Nabulsi, A fractional approach of nonconservative Lagrangian dynamics, Fiz. A14, No. 4 (2005), 289-298.;
[28] R. A. El-Nabulsi, The fractional calculus of variations from extended Erdelyi-Kober operator, Int. J. Mod. Phys. B23 (2009), 3349-3361.;
[29] R. A. El-Nabulsi, Higher-order fractional field equations in (0+1) dimensions and physics beyond the standard model, Fiz. A19 (2010), 55-72.;
[30] R. A. El-Nabulsi, Lagrangian and Hamiltonian dynamics with imaginary time, J. Appl. Anal. 18(2) (2012), 283-295.; · Zbl 1276.70013
[31] G. S. Frederico and D. F. M. Torres, Necessary optimality conditions for fractional action-like problems with intrinsic and observer times, WSEAS Trans. Math. 7(1) (2008), 6-11.;
[32] I. M. Gelfand, S. V. Fomin and R. A. Silverman, Calculus of variations, Englewood Cliffs, N.J.: Prentice-Hall Inc., 1963.;
[33] S. Ghosh and S. K. Modak, Classical oscillator with position-dependent mass in a complex domain, Phys. Lett. A373 (2009), 1212-1217.; · Zbl 1228.70010
[34] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific Publishing Company, 2011.; · Zbl 1232.26006
[35] R. S. Kaushal, Classical and quantum mechanics of complex Hamiltonian systems: An extended complex phase space approach, PRAMANA J. Phys. 73(2) (2009), 287-297.;
[36] A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, UK, 2012.; · Zbl 1258.49001
[37] Z. E. Musielak, Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients, J. Phys. A: Math. Theor. 41 (2008), 055205-055222.; · Zbl 1136.37044
[38] Z. E. Musielak, General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems, Chaos Solitons and Fractals 42, No. 15 (2009), 2645-2652.; · Zbl 1198.34057
[39] T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Fractional calculus of variations in terms of a generalized fractional integral with applications to physics, Abs. Appl. Anal. 2012 (2012), Article ID 871912.; · Zbl 1242.49019
[40] T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, A generalized fractional calculus of variations, Control and Cybernetics 42(2) (2013), 443-458.; · Zbl 1318.49033
[41] A. Saha and B. Talukdar, On the non-standard Lagrangian equations, arXiv: 1301.2667.;
[42] A. Saha and B. Talukdar, Inverse variational problem for non-standard Lagrangians, Rep. Math. Phys. 73, (2014) 299-309.; · Zbl 1326.49059
[43] V. I. Sbitnev, Bohmian trajectories and the path integral paradigm. Complexified Lagrangian mechanics, Int. J. Bifur. Chaos 19(9) (2009), 2335-2346.; · Zbl 1176.81003
[44] G. S. Taverna and D. F. M. Torres, Generalized fractional operators for nonstandard La- grangians, Math. Meth. Appl. Sci. 38 (2015), 1808-1812.; · Zbl 1322.49040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.