Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian

Ola Jarab'ah; Khaled Nawafleh

doi:doi:10.11648/j.ajpa.20180604.12

| Peer-Reviewed

Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian

Ola Jarab'ah, Khaled Nawafleh

Published in American Journal of Physics and Applications (Volume 6, Issue 4)

Received: 5 July 2018 Accepted: 19 July 2018 Published: 24 August 2018

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, the nonconservative systems with second order Lagrangian are investigated using fractional derivatives. The fractional Euler Lagrange equations for these systems are obtained. Then, fractional Hamiltonian for these systems is constructed, which is used to find the Hamilton's equations of motion in the same manner as those obtained by using the formulation of Euler Lagrange equations from variational problems, and it is observed that the Hamiltonian formulation is in exact agreement with the Lagrangian formulation. The passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. We have examined one example to illustrate the formalism.

Published in	American Journal of Physics and Applications (Volume 6, Issue 4)
DOI	10.11648/j.ajpa.20180604.12
Page(s)	85-88
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Fractional Derivatives, Lagrangian Formulation, Hamiltonian Formulation, Nonconservative Systems, Euler Lagrange Equations, Second Order Lagrangian

References

[1]	R. Metzler, J. Klafter, Physics Reports, 339, 1-77 (2000).
[2]	S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Linghorne, PA, Switzerland; Philadelphia, PA, USA (1993).
[3]	G. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford, (2005).
[4]	B. J. West, M. Bologna, and P. Grigolini. Physics of Fractal Operators. Springer-Verlag, New York, (2003).
[5]	F. Riewe, Physical Review E 53, 1890 (1996).
[6]	F. Riewe, Phys. Rev. E 55, 3581 (1997).
[7]	E. M. Rabei, T. S. Al halholy, and A, A. Taani, Turkish Journal of Physics. 28, (1) (2004).
[8]	E. M. Rabei, T. S. Al halholy, and A. Rousan. International Journal of Modern Physics A, 19, 3083 (2004).
[9]	E. M. Rabei, K. I. Nawafleh, R. S. Hijjawi, S. I. Muslih, and D. Baleanu, Journal of Mathematical Analysis and Applications 327, 891 (2007).
[10]	O. P. Agrawal, An Analytical Scheme for Stochastic Dynamics Systems Containing Fractional Derivatives. ASME Design Engineering Technical Conferences. (1999).
[11]	O. P. Agrawal, Journal of Applied Mechanics 68, 339 (2001).
[12]	O. P. Agrawal, Journal of Mathematical Analysis and Applications 272, 368 (2002).
[13]	D. Baleanu, S. Muslih, Physica Scripta 27, 105 (2005).
[14]	D. W. Dreisi-gmeyer, P. M. Yoang, Journal of Physica A 36, 3297 (2003).
[15]	M. Klimek, Journal of Physics 52, 1247, (2002).
[16]	D. Baleanu, K. Tas, and S. Muslih, Journal of Mathematical Physics 47, 103503 (2006).
[17]	E. H. Hasan, J. Asad, Journal of Advanced Physics 6 (3), 430–433 (2017).
[18]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, (1999).
[19]	O. Jarab'ah, K. Nawafleh, and H. Ghassib, European Scientific Journal 9 (36), 132-154 (2013).
[20]	H. Goldstein. Classical Mechanics (2nd edition). Addison-Wesley, Reading- Massachusetts (1980).
[21]	E. M. Rabei, E. H. Hasan and H. B. Ghassib, International Journal of Theoretical Physics 43, 1073 (2004).
[22]	E. H. Hasan, Quantization of Higher-Order Constrained Lagrangian Systems Using the WKB Approximation, Ph.D. Thesis, University of Jordan (2004).
[23]	O. Jarab'ah, E. H. Hasan, K. I. Nawafleh. European Scientific Journal 10 (9), 135-142 (2014).

Cite This Article

Plain Text BibTeX RIS

APA Style

Ola Jarab'ah, Khaled Nawafleh. (2018). Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian. American Journal of Physics and Applications, 6(4), 85-88. https://doi.org/10.11648/j.ajpa.20180604.12

Copy | Download

ACS Style

Ola Jarab'ah; Khaled Nawafleh. Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian. Am. J. Phys. Appl. 2018, 6(4), 85-88. doi: 10.11648/j.ajpa.20180604.12

Copy | Download

AMA Style

Ola Jarab'ah, Khaled Nawafleh. Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian. Am J Phys Appl. 2018;6(4):85-88. doi: 10.11648/j.ajpa.20180604.12

Copy | Download

@article{10.11648/j.ajpa.20180604.12,
  author = {Ola Jarab'ah and Khaled Nawafleh},
  title = {Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian},
  journal = {American Journal of Physics and Applications},
  volume = {6},
  number = {4},
  pages = {85-88},
  doi = {10.11648/j.ajpa.20180604.12},
  url = {https://doi.org/10.11648/j.ajpa.20180604.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20180604.12},
  abstract = {In this paper, the nonconservative systems with second order Lagrangian are investigated using fractional derivatives. The fractional Euler Lagrange equations for these systems are obtained. Then, fractional Hamiltonian for these systems is constructed, which is used to find the Hamilton's equations of motion in the same manner as those obtained by using the formulation of Euler Lagrange equations from variational problems, and it is observed that the Hamiltonian formulation is in exact agreement with the Lagrangian formulation. The passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. We have examined one example to illustrate the formalism.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian
AU  - Ola Jarab'ah
AU  - Khaled Nawafleh
Y1  - 2018/08/24
PY  - 2018
N1  - https://doi.org/10.11648/j.ajpa.20180604.12
DO  - 10.11648/j.ajpa.20180604.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 85
EP  - 88
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20180604.12
AB  - In this paper, the nonconservative systems with second order Lagrangian are investigated using fractional derivatives. The fractional Euler Lagrange equations for these systems are obtained. Then, fractional Hamiltonian for these systems is constructed, which is used to find the Hamilton's equations of motion in the same manner as those obtained by using the formulation of Euler Lagrange equations from variational problems, and it is observed that the Hamiltonian formulation is in exact agreement with the Lagrangian formulation. The passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. We have examined one example to illustrate the formalism.
VL  - 6
IS  - 4
ER  -

Copy | Download

Author Information

Ola Jarab'ah

Applied Physics Department, Faculty of Science, Tafila Technical University, Tafila, Jordan
Khaled Nawafleh

Department of Physics, Faculty of Science, Mu'tah University, AL-Karak, Jordan

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Ola Jarab'ah, Khaled Nawafleh. (2018). Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian. American Journal of Physics and Applications, 6(4), 85-88. https://doi.org/10.11648/j.ajpa.20180604.12

Copy | Download

ACS Style

Ola Jarab'ah; Khaled Nawafleh. Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian. Am. J. Phys. Appl. 2018, 6(4), 85-88. doi: 10.11648/j.ajpa.20180604.12

Copy | Download

AMA Style

Ola Jarab'ah, Khaled Nawafleh. Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian. Am J Phys Appl. 2018;6(4):85-88. doi: 10.11648/j.ajpa.20180604.12

Copy | Download

@article{10.11648/j.ajpa.20180604.12,
  author = {Ola Jarab'ah and Khaled Nawafleh},
  title = {Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian},
  journal = {American Journal of Physics and Applications},
  volume = {6},
  number = {4},
  pages = {85-88},
  doi = {10.11648/j.ajpa.20180604.12},
  url = {https://doi.org/10.11648/j.ajpa.20180604.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajpa.20180604.12},
  abstract = {In this paper, the nonconservative systems with second order Lagrangian are investigated using fractional derivatives. The fractional Euler Lagrange equations for these systems are obtained. Then, fractional Hamiltonian for these systems is constructed, which is used to find the Hamilton's equations of motion in the same manner as those obtained by using the formulation of Euler Lagrange equations from variational problems, and it is observed that the Hamiltonian formulation is in exact agreement with the Lagrangian formulation. The passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. We have examined one example to illustrate the formalism.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Fractional Hamiltonian of Nonconservative Systems with Second Order Lagrangian
AU  - Ola Jarab'ah
AU  - Khaled Nawafleh
Y1  - 2018/08/24
PY  - 2018
N1  - https://doi.org/10.11648/j.ajpa.20180604.12
DO  - 10.11648/j.ajpa.20180604.12
T2  - American Journal of Physics and Applications
JF  - American Journal of Physics and Applications
JO  - American Journal of Physics and Applications
SP  - 85
EP  - 88
PB  - Science Publishing Group
SN  - 2330-4308
UR  - https://doi.org/10.11648/j.ajpa.20180604.12
AB  - In this paper, the nonconservative systems with second order Lagrangian are investigated using fractional derivatives. The fractional Euler Lagrange equations for these systems are obtained. Then, fractional Hamiltonian for these systems is constructed, which is used to find the Hamilton's equations of motion in the same manner as those obtained by using the formulation of Euler Lagrange equations from variational problems, and it is observed that the Hamiltonian formulation is in exact agreement with the Lagrangian formulation. The passage from the Lagrangian containing fractional derivatives to the Hamiltonian is achieved. We have examined one example to illustrate the formalism.
VL  - 6
IS  - 4
ER  -

Copy | Download