Lagrangian and Hamiltonian Formulations of the Damped Harmonic Oscillator Using Caputo Fractional Derivative

Tarawneh, Khaldoun M.; Rabei, Eqab M.; Ghassib, Humam B.

doi:10.1080/1726037X.2010.10698578

Open Access Original Articles

Lagrangian and Hamiltonian Formulations of the Damped Harmonic Oscillator Using Caputo Fractional Derivative

Khaldoun M. Tarawnehkhaldoun@psut.edu.joDepartment of Arts and SciencePrincess Sumaya University for TechnologyAmman, JordanView full profile → , Eqab M. Rabeieqabrabei@gmail.comDepartment of PhysicsAl-Al-Bayt UniversityMafraq, 25113, P. O. BOX 130040, JordanView full profile → , Humam B. GhassibDepartment of PhysicsUniversity of JordanAmman, JordanView full profile →

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pp. 59–70Vol. 8Issue 1June 2013DOI: 10.1080/1726037X.2010.10698578XML

Published Online:: 03 Jun 2013
Article type:: Original Articles
Language:: EN
Article no.:: 1726037X.2010.10698578
Pages:: 59–70

Abstract

In this work, the Caputo fractional derivative is used to construct the Lagrangian and the Hamiltonian formulations for nonconservative mechanical systems. The Euler-Lagrange equations are derived using a variational principle. Hamilton’s equations of motion are then obtained by introducing a set of momenta and a Hamiltonian in a manner that is more consistent with that used in conventional classical mechanics. The proposed formalism is applied to the damped harmonic oscillator, as an example.

Keywords

Caputo fractional derivative nonconservative systems Euler-Lagrange equations

References

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Rabei, E. M., Ajlouni, A. W. and Ghassib, H. B.2006 . Quantization of Brownian Motion . , 45 ( 9 ) : 1613 – 1623 .
Rabei, E. M., Alhalholy, T. and Rousan, A.2004 . Potantial of Arbitrary Forces with Fractional Derivatives . , 19 : 3083 – 3092 .
Rabei, E. M., Nawafleh, K. I., Hijjawi, r. S., Muslih, S. I. and Baleanu, D.2007 . Hamilton Formalism with Fractional Derivatives . , 327 : 891 – 897 .
Raspini, A.2001 . Simple Solutions of the Fractional Dirac Equation of Order 2/3 . , 64 : 20 – 22 .
Ray, J. R.1979 . Lagrangians and Systems They Describe: How Not to Treat Dissipation in Quantum Mechanics . , 47 : 626 – 629 .
Riewe, F.1997 . Mechanics With Fractional Derivatives . , 55 ( 3 ) : 3581 – 3592 .
Riewe, F.1996 . Nonconservative Lagrangian and Hamiltonian Mechanics . , 53 ( 2 ) : 1890 – 1899 .
Samko, S. G., Kilbas, A. A. and Marichev, O. I.1993 . , Longhorne: Gordon and Breach .
Sentizky, I. R.1960 . Dissipation in Quantum Mechanics: The Harmonic Oscillator . , 119 : 670 – 679 .
Vance, W. and Ross, J.2000 . Master Equation Approach to Synchronization in Diffusion-Coupled Nonlinear Oscillators . , 62 ( 3 ) : 3303 – 3310 .

Views: 47Downloads: 81Citations: 3

[1] Agrawal, O. P.2002 . Formulation of Euler-Lagrange Equation for Fractional Variational Problems . , 272 : 368 – 379 .

[2] Agrawal, O. P.2007 . Generalized Euler-Lagrange Equations and Transversality Conditions for FVPs in Terms of the Caputo Derivative . , 13 : 1217 – 1237 .

[3] Baleanu, D. and Agrawal, O. P.2006 . Fractional Hamilton Formalism within Caputo’s Derivative . , 56 : 1087 – 1092 .

[4] Callen, H. B. and Welton, T. A.1951 . Irrevesibility and Generalized Noise . , 83 : 34 – 40 .

[5] Celeghini, M., Rasetti, M., Tarlini, M. and Vitiello, G.1989 . Squeezed States as Damped Oscillators . , 3 ( 16 ) : 1213 – 1220 .

[6] Chen, W. and Holm, S.2004 . Fractional Laplacian Time-Space models for Linear and Nonlinear Lossy Media Exhibiting Arbitrary Frequency Power-Law Dependency . , 115 : 1424 – 1430 .

[7] Goldstein, H.1980 . , 2nd , Massachusetts: Addison-Wesley .

[8] Gruver, J. L., Aliaga, J., Cerdeira, H. A. and Proto, A. N.1995 . , 51 : 6263 – 6266 .

[9] Hasse, R. W.1975 . On The Quantum Mechanical Treatment of Dissipative Systems . , 16 ( 10 ) : 2005 – 2012 .

[10] Hilfer, R.2000 . , Singapore: World Scientific .

[11] Klimek, M.2001 . Fractional Sequential Mechanics-Models with Symmetric Fractional Derivatives . , 51 : 1348 – 1354 .

[12] Klimek, M.2001 . Lagrangian and Hamiltonian Fractional Sequential Mechanics . , 52 : 1247 – 11253 .

[13] Mainardi, F., Luchko, Y. and Pagnini, G.2001 . The Fundamental Solutions of the Spacetime Fractional Diffusion Equation . , 4 : 153 – 192 .

[14] Miller, K. S. and Ross, B.1993 . , NewYork: Wiley .

[15] Naber, M.2004 . Time Fractional Schroodinger Equation . , 45 ( 8 ) : 3339 – 3353 .

[16] Oldham, K. B. and Spanier, J.1974 . , New York: Academic Press .

[17] Podlubny, I.1999 . , New York: Academic Press .

[18] Rabei, E. M., Ajlouni, A. W. and Ghassib, H. B.2006 . Quantization of Brownian Motion . , 45 ( 9 ) : 1613 – 1623 .

[19] Rabei, E. M., Alhalholy, T. and Rousan, A.2004 . Potantial of Arbitrary Forces with Fractional Derivatives . , 19 : 3083 – 3092 .

[20] Rabei, E. M., Nawafleh, K. I., Hijjawi, r. S., Muslih, S. I. and Baleanu, D.2007 . Hamilton Formalism with Fractional Derivatives . , 327 : 891 – 897 .

[21] Raspini, A.2001 . Simple Solutions of the Fractional Dirac Equation of Order 2/3 . , 64 : 20 – 22 .

[22] Ray, J. R.1979 . Lagrangians and Systems They Describe: How Not to Treat Dissipation in Quantum Mechanics . , 47 : 626 – 629 .

[23] Riewe, F.1997 . Mechanics With Fractional Derivatives . , 55 ( 3 ) : 3581 – 3592 .

[24] Riewe, F.1996 . Nonconservative Lagrangian and Hamiltonian Mechanics . , 53 ( 2 ) : 1890 – 1899 .

[25] Samko, S. G., Kilbas, A. A. and Marichev, O. I.1993 . , Longhorne: Gordon and Breach .

[26] Sentizky, I. R.1960 . Dissipation in Quantum Mechanics: The Harmonic Oscillator . , 119 : 670 – 679 .

[27] Vance, W. and Ross, J.2000 . Master Equation Approach to Synchronization in Diffusion-Coupled Nonlinear Oscillators . , 62 ( 3 ) : 3303 – 3310 .

Journal of Dynamical Systems and Geometric Theories

WoS JIF 2026 : 0.5 (Q3)

Lagrangian and Hamiltonian Formulations of the Damped Harmonic Oscillator Using Caputo Fractional Derivative

Abstract

Keywords

References