Skip to main content
SpringerLink
Log in

Noether symmetries for non-conservative Lagrange systems with time delay based on fractional model

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The fractional Noether symmetries and conserved quantities for non-conservative Lagrange systems with time delay are proposed and studied. Firstly, the fractional Hamilton variational principles for non-conservative Lagrange systems with time delay are established, and the fractional differential equations of motion with time delay are obtained. Secondly, based upon the invariance of the fractional Hamilton action with time delay under the group of infinitesimal transformations which depends on the generalized velocities, the generalized coordinates and the time, the fractional Noether symmetric transformations, the fractional Noether quasi-symmetric transformations and the fractional generalized Noether quasi-symmetric transformations with time delay are defined, and the criteria of the fractional symmetries are obtained. Finally, the relationship between the fractional symmetries and the fractional conserved quantities with time delay are studied, and the fractional Noether theories are established. At the end of the paper, two examples are given to illustrate the application of the results.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Debnath, L.: Recent application of fraction calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54), 3413–3442 (2003)

    Article  MATH  Google Scholar 

  2. Hilfer, R.: Application of Fractional Calculus in Physics. Word Scientific, Singapore (2000)

    Book  Google Scholar 

  3. Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kulish, V.V., Lage, J.L.: Applications of fractional calculus to fluid mechanics. Fluids Eng. 124(3), 803–806 (2002)

    Article  Google Scholar 

  5. Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32(1), 1–104 (2004)

    Article  Google Scholar 

  6. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley Inc., New York (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Kilbass, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  9. West, B.J., Bologna, M., Grigolini, P.: Physics of Fractional Operators. Springer, Berlin (2003)

    Book  Google Scholar 

  10. Chen, Y.Q., Vinagre, B.M.: A new IIR-type digital fractional order differentiator. Signal Process 83(11), 2359–2365 (2003)

    Article  MATH  Google Scholar 

  11. Riewe, F.: Nonconservation Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  12. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581–3592 (1997)

    Article  MathSciNet  Google Scholar 

  13. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272(1), 368–379 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38(1–4), 323–337 (2004)

    Article  MATH  Google Scholar 

  15. Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A Math. Gen. 39(33), 10375–10384 (2006)

    Article  MATH  Google Scholar 

  16. Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52(11), 1247–1253 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Baleanu, D., Muslih, S.I.: Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives. Phys. Scripta 72(2–3), 119–121 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Baleanu, D., Muslih, S.I., Tas, K.: Fractional Hamiltonian analysis of higher order derivatives systems. J. Math. Phys. 47(10), 103503 (2006)

    Article  MathSciNet  Google Scholar 

  19. Herallah, M.A.E., Baleanu, D.: Fractional-order Euler–Lagrange equations and formulation of Hamiltonian equations. Nonlinear Dyn. 58(1–2), 385–391 (2009)

    Article  Google Scholar 

  20. Herallah, M.A.E., Baleanu, D.: Fractional Euler–Lagrange equations revisited. Nonlinear Dyn. 69(3), 977–982 (2012)

    Article  Google Scholar 

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

  22. El-Nabulsi, R.A: Universal fractional Euler–Lagrange equation from a generalized fractional derivate operator. Cent. Eur. J. Phys. 9(1), 250–256 (2011)

  23. Frederico, G.S.F., Torres, D.F.M.: Nonconservative Noethers theorem in optimal control. Int. J. Tomogr. Stat. 5(W07), 109–114 (2007)

    MathSciNet  Google Scholar 

  24. Frederico, G.S.F., Torres, D.F.M.: A formulation of Noethers theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334(2), 834–846 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Frederico, G.S.F., Torres, D.F.M.: Noethers theorem for fractional optimal control problems. In: Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, pp. 142–147, July 19–21 (2006)

  26. Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Atanacković, T.M., Konjik, S., Simić, S.: Variational problems with fractional derivatives: invariance conditions and Noethers theorem. Nonlinear Anal. 71(5–6), 1504–1517 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhang, Y.: Fractional differential equations of motion in terms of combined Riemann–Liouville derivatives. Chin. Phys. B 21(8), 084502 (2012)

    Article  Google Scholar 

  29. Long, Z.X., Zhang, Y.: Fractional action-like variational problem and its symmetry for a nonholonomic system. Acta Mech. 225(1), 77–90 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Zhang, Y., Zhou, Y.: Symmetries and conserved quantities for fractional action-like Pfaffian variational problems. Nonlinear Dyn. 73(1–2), 783–793 (2013)

  31. Zhang, Y., Mei, F.X.: Fractional differential equations of motion in terms of Riesz fractional derivatives. J. Beijing Inst. Technol. 32(7), 766–770 (2013) (in Chinese)

  32. Elsgolc, L.E: Qualitative Methods in Mathematical Analysis. American Mathematical Society, Providence (1964)

  33. Abdeljawad, T., Baleanu, D., Jarad, F.: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J. Math. Phys. 49(8), 083507 (2008)

  34. Jarad, F., Abdeljawad, T., Baleanu, D.: Fractional variational principles with delay within Caputo derivatives. Math. Phys. 65(1), 17–28 (2010)

    MATH  MathSciNet  Google Scholar 

  35. Jarad, F., Abdeljawad, T., Baleanu, D.: Fractional variational optimal control problems with delayed argument. Nonlinear Dyn. 62(3), 609–614 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. Jarad, F., Abdeljawad, T., Baleanu, D.: Higher order fractional variational optimal control problems with time delay. Appl. Math. Comput. 218(2), 9234–9240 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Frederico, G.S.F., Torres, D.F.M.: Noethers symmetry theorem for variational and optimal control problem with time delay. Numer. Algebra Control Optim. 2(3), 619–630 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zhang, Y., Jin, S.X.: The Noether symmetries of dynamics for non-conservative system with time delay. Acta Phys. Sin. 62(23), 214502 (2013). (in Chinese)

    MathSciNet  Google Scholar 

  39. Vujanovic, B.D., Jones, S.E.: Variational Methods in Nonconservative Phenomena. Academic Press, San Diego (1989)

    MATH  Google Scholar 

Download references

Acknowledgments

Project was supported by the National Natural Science Foundation of China (Grant Nos.10972151 and 11272227) and the Innovation Program for Scientific Research of Suzhou University of Science and Technology (No.SKCX12S_039). And we would like to thank the anonymous referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, Jiangsu, People’s Republic of China

    Shi-Xin Jin

  2. College of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, People’s Republic of China

    Shi-Xin Jin

  3. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou, 215011, Jiangsu, People’s Republic of China

    Yi Zhang

Authors
  1. Shi-Xin Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jin, SX., Zhang, Y. Noether symmetries for non-conservative Lagrange systems with time delay based on fractional model. Nonlinear Dyn 79, 1169–1183 (2015). https://doi.org/10.1007/s11071-014-1734-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1734-1

Keywords

Search

Navigation