Skip to main content
Log in

Non-Standard Non-Local-in-Time Lagrangians in Classical Mechanics

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

We introduce in this communication a new type of non-standard nonlocal-in-time Lagrangian functional and discuss their implications in classical dynamical systems and Newton’s law. Several illustrations were proposed and discussed accordingly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

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

    Book  MATH  Google Scholar 

  2. 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 

  3. 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 

  4. Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator. Phys. Rev. E72, 066203–066211 (2005)

    Google Scholar 

  5. 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 

  6. 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  MATH  MathSciNet  Google Scholar 

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

  8. 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 

  9. El-Nabulsi, R.A.: Non-standard fractional Lagrangians. Nonlinear Dyn. 74, 381–394 (2013)

    Google Scholar 

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

  11. El-Nabulsi, R.A.: Nonlinear dynamics with non-standard Lagrangians. Qual. Theory Dyn. Syst. 13, 273–291 (2013)

    Google Scholar 

  12. El-Nabulsi, R.A., Soulati, T., Rezazadeh, H.: Non-standard complex Lagrangian dynamics. J. Adv. Res. Dyn. Control. Theor. 5(1), 50–62 (2012)

    MathSciNet  Google Scholar 

  13. El-Nabulsi, R.A.: Some consequences of nonstandard Lagrangians with time-dependent coefficients in general relativity. J. Theor. Appl. Phys. 7, 60 (2013). doi:10.1186/2251-7235-7-60

  14. Saha A., Talukdar B.: On the non-standard Lagrangian equations. (2013). arXiv:1301.2667

  15. Saha A., Talukdar B.: Inverse variational problem for non-standard Lagrangians. (2013). arXiv:1305.6386

  16. El-Nabulsi, R.A.: Modified Proca equation and modified dispersion relation from a power-law Lagrangian functional. Indian J. Phys. 87(5), 465–470 (2013). (Erratum Indian J. Phys. 87(10), 1059 (2013))

  17. El-Nabulsi, R.A.: Quantum field theory from an exponential action functional. Indian J. Phys. 87(4), 379–383 (2013)

    Article  MathSciNet  Google Scholar 

  18. El-Nabulsi, R.A.: Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity. Canad. J. Phys. 91(8), 618–622 (2013). doi:10.1139/cjp-2013-0145

    Article  Google Scholar 

  19. Dimitrijevic, D.D., Milosevic, M.: About non-standard Lagrangians in cosmology. In: Proceedings of the Physics Conference TIM-11. AIP Conference Proceedings, vol. 1472, pp 41–46 (2012)

  20. 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 

  21. Kamalov, T.F.: A model of extended mechanics and nonlocal hidden variables for quantum theory. J. Russ. Laser Res. 30, 466–471 (2009)

    Article  Google Scholar 

  22. Choudhary S.K., Goyal S.K., Konrad T.H., Ghosh S.: Persistence. (2013). arXiv:1302.5296

  23. Kamalov T.F.: How to complement the description of physical Universe? (2009). arXiv:0906.3539

  24. Kamalov T.F.: How to complete the description of physical reality by non-local hidden variables? arXiv:0907.5303

  25. Suykens, J.A.K.: Extending Newton’s law from nonlocal-in-time kinetic energy. Phys. Lett. A373, 1201–1211 (2009)

    Article  Google Scholar 

  26. Feynman, R.P.: Space-time approach to relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  MathSciNet  Google Scholar 

  27. Pais, A., Uhlenbeck, G.E.: On Field theories with nonlocalized action. Phys. Rev. 79, 145–165 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  28. Li, Z.-Y., Fu, J.-L., Chen, L.-Q.: Euler–Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system. Phys. Lett. A374, 106–109 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rami Ahmad El-Nabulsi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

El-Nabulsi, R.A. Non-Standard Non-Local-in-Time Lagrangians in Classical Mechanics. Qual. Theory Dyn. Syst. 13, 149–160 (2014). https://doi.org/10.1007/s12346-014-0110-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-014-0110-3

Keywords

Mathematics Subject Classification (2000)

Navigation