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Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians

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Abstract

The Noether theorem and its inverse theorem for the nonlinear dynamical systems with nonstandard Lagrangians are studied. In this paper, two kinds of nonstandard Lagrangians, namely exponential Lagrangians and power-law Lagrangians, are discussed. For each case, the Hamilton principle based on the action with nonstandard Lagrangians is established, the differential equations of motion for the dynamical systems with nonstandard Lagrangians are obtained, and two basic formulae for the variation in Hamilton action with nonstandard Lagrangians are derived. The definitions and the criteria of the Noether symmetric transformations and the Noether quasi-symmetric transformations are given. The Noether theorem and its inverse theorem are established, which reveal the intrinsic relation between the symmetry and the conserved quantity for the systems with nonstandard Lagrangians. Two examples are given to illustrate the application of the results.

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References

  1. Noether, A.E.: Invariante variationsprobleme. Nachr. Akad. Wiss. Gott. Math. Phys. 2, 235–237 (1918)

    Google Scholar 

  2. Djukić, Dj.S., Vujanović, B.D.: Noether theory in classical nonconservative mechanics. Acta Mech. 23, 17–27 (1975)

  3. Mei, F.X.: Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems. Science Press, Beijing (1999)

    Google Scholar 

  4. Li, Z.P.: The transformation properties of constrained system. Acta Phys. Sin. 20(12), 1659–1671 (1981)

    MathSciNet  Google Scholar 

  5. Liu, D.: Noether’s theorem and its inverse of nonholonomic nonconservative dynamical systems. Sci. China Ser. A 34(4), 419–429 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Borisov, A.V., Mamaev, I.S.: Symmetries and reduction in nonholonomic mechanics. Regul. Chaotic Dyn. 20(5), 553–604 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Atanacković, T.M., Konjik, S., Pilipović, S., Simić, S.: Variational problems with fractional derivatives: invariance conditions and Noether’s theorem. Nonlinear Anal. 71, 1504–1517 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhou, S., Fu, H., Fu, J.L.: Symmetry theories of Hamiltonian systems with fractional derivatives. Sci. Chin. Phys. Mech. Astron. 54(10), 1847–1853 (2011)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, Y., Zhai, X.H.: Noether symmetries and conserved quantities for fractional Birkhoffian systems. Nonlinear Dyn. 81, 469–480 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhai, X.H., Zhang, Y.: Noether symmetries and conserved quantities for Birkhoffian systems with time delay. Nonlinear Dyn. 77, 73–86 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  15. Jin, S.X., Zhang, Y.: Noether symmetries for non-conservative Lagrange systems with time delay based on fractional model. Nonlinear Dyn. 79(2), 1169–1183 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  17. Carinena, J.F., 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 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chandrasekar, V.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: A simple and unified approach to identify integrable nonlinear oscillators and systems. J. Math. Phys. 47, 023508 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator. J. Math. Phys. 48, 032701 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Udwadia, F.E., Cho, H.: First integral and solutions of Duffing–van der Pol type equations. J. Appl. Mech. 81(3), 034501 (2014)

    Article  Google Scholar 

  21. Udwadia, F.E., Cho, H.: Lagrangians for damped linear multi-degree-of-freedom systems. J. Appl. Mech. 80(4), 041023 (2013)

    Article  Google Scholar 

  22. Alekseev, A.I., Arbuzov, B.A.: Classical Yang–Mills field theory with nonstandard Lagrangian. Theor. Math. Phys. 59(1), 372–378 (1984)

    Article  MathSciNet  Google Scholar 

  23. Musielak, Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A Math. Theor. 41(5), 055205 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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  MathSciNet  MATH  Google Scholar 

  25. El-Nabulsi, A.R.: Non-standard fractional Lagrangians. Nonlinear Dyn. 74(1), 381–394 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. El-Nabulsi, A.R.: Nonlinear dynamics with nonstandard Lagrangians. Qual. Theory Dyn. Syst. 12(2), 273–291 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. El-Nabulsi, A.R.: Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent. Comput. Appl. Math. 33(1), 163–179 (2014)

  28. El-Nabulsi, A.R.: Non-standard Lagrangians in rotational dynamics and the modified Navier–Stokes equation. Nonlinear Dyn. 79(3), 2055–2068 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  31. El-Nabulsi, A.R., Soulati, T., Rezazadeh, H.: Nonstandard complex Lagrangian dynamics. J. Adv. Res. Dyn. Control Syst. 5(1), 50–62 (2013)

    MathSciNet  Google Scholar 

  32. El-Nabulsi, A.R.: Non-standard non-local-in-time Lagrangians in classical mechanics. J. Qual. Theory Dyn. Syst. 13(1), 149–160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Saha, A., Talukdar, B.: On the non-standard Lagrangian equations. arXiv:1301.2667

  34. Saha, A., Talukdar, B.: Inverse variational problem for non-standard Lagrangians. arXiv:1305.6386

  35. Dimitrijevic, D.D., Milosevic, M.: About non-standard Lagrangians in cosmology. AIP Conf. Proc. 1472, 41 (2012)

    Article  Google Scholar 

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11272227 and 11572212).

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Authors and Affiliations

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

    Yi Zhang

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

    Xiao-San Zhou

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  1. Yi Zhang
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  2. Xiao-San Zhou
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Correspondence to Yi Zhang.

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Zhang, Y., Zhou, XS. Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians. Nonlinear Dyn 84, 1867–1876 (2016). https://doi.org/10.1007/s11071-016-2611-x

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