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Perturbation to Noether symmetries and adiabatic invariants for disturbed Hamiltonian systems based on El-Nabulsi nonconservative dynamics model

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Abstract

This paper focuses on studying the perturbation to the Noether symmetries and the adiabatic invariants for nonconservative dynamic systems in phase space under nonconservative dynamics model presented by El-Nabulsi. First of all, the El-Nabulsi dynamics model for a nonconservative system is introduced and the El-Nabulsi–Hamilton canonical equations are established. Secondly, the basic formulae for the variation of El-Nabulsi–Hamilton action in phase space are deduced, the definition and criterion of the Noether quasi-symmetric transformation are given, and the exact invariant led directly by the Noether symmetry is obtained. Finally, based upon the concept of high-order adiabatic invariant of a mechanical system, the relationship between the perturbation to the Noether symmetry and the adiabatic invariant after the action of a small disturbance is studied and the conditions that the perturbation of symmetry leads to the adiabatic invariant and its formulation are given. At the end of the paper, two examples are given to illustrate the application of the method and results.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivative. Czech. J. Phys. 51(12), 1348–1354 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. 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 

  6. Agrawa, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40(24), 6287–6303 (2007)

    Article  MathSciNet  Google Scholar 

  7. Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48(3), 033504 (2007)

    Article  MathSciNet  Google Scholar 

  8. Atanacković, T.M.: Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A 41(9), 095201 (2008)

    Article  MathSciNet  Google Scholar 

  9. Baleanu, D., Trujillo, J.I.: A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 15(5), 1111–1115 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imperial College Press, London (2012)

    MATH  Google Scholar 

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

    Google Scholar 

  12. El-Nabulsi, A.R.: A fractional action-like variational approach of some classical, quantum and geometrical dynamics. Int. J. Appl. Math. 17(3), 299–317 (2005)

    MATH  MathSciNet  Google Scholar 

  13. El-Nabulsi, A.R., Torres, D.F.M.: Fractional action-like variational problems. J. Math. Phys. 49, 053521 (2008)

    Article  MathSciNet  Google Scholar 

  14. El-Nabulsi, A.R.: Fractional action-like variational problems in holonomic, non-holonomic and semi-holonomic constrained and dissipative dynamical systems. Chaos Solitons Fract. 42, 52–61 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang, Y.: Noether symmetries and conserved quantities for fractional action-like variational problems in phase space. Acta Sci. Nat. Univ. Sunyatsen 52(4), 20–25 (2013)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Long, Z.X., Zhang, Y.: Noether’s theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space. Acta Mech. 225(1), 77–90 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Noether, A.E.: Invariante variationsprobleme. Nachr. Akad. Wiss. Gött. Math. Phys. KI. II 2, 235–257 (1918)

    Google Scholar 

  19. Djukic, Dj.S. Vujanovic, B.: Noether’s theory in classical nonconservative mechanics. Acta Mech. 23(1–2), 17–27 (1975)

    Google Scholar 

  20. Li, Z.P.: The transformation properties of constrained system. Acta Phys. Sin. 30(12), 1659–1671 (1981)

    Google Scholar 

  21. Bahar, L.Y., Kwatny, H.G.: Extension of Noether’s theorem to constrained nonconservative dynamical systems. Int. J. Non-Linear Mech. 22(2), 125–138 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu, D.: Noether theorem and its converse for non-holonomic conservative dynamical systems. Sci. China Ser. A 20(11), 1189–1197 (1991)

    Google Scholar 

  23. Luo, S.K.: Generalized Noether theorem of nonholonomic nonpotential system in noninertial reference frame. Appl. Math. Mech. 12(9), 927–934 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mei, F.X.: Noether theory of Birkhoffian system. Sci. China Ser. A 23(7), 709–717 (1993)

    Google Scholar 

  25. Zhang, Y., Mei, F.X.: Noether’s theory of mechanical systems with unilateral constraints. Appl. Math. Mech. 21(1), 59–66 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. 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  MATH  MathSciNet  Google Scholar 

  27. 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 

  28. Malinowska, A.B.: A formulation of the fractional Noether-Type theorem for multidimensional Lagrangians. Appl. Math. Lett. 25(11), 1941–1946 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  30. Frederico, G.S.F., Torres, D.F.M.: Constants of motion for fractional action-like variational problems. Int. J. Appl. Math. 19(1), 97–104 (2006)

    Google Scholar 

  31. Frederico, G.S.F., Torres, D.F.M.: Non-conservative Noether’s theorem for fractional action-like variational problems with intrinsic and observer times. Int. J. Ecol. Econ. Stat. 9(F07), 74–82 (2007)

    Google Scholar 

  32. Wu, D.Q.: Classical Dynamics. Science Press, Beijing (1983). (in Chinese)

    Google Scholar 

  33. Mei, F.X., Liu, D., Luo, Y.: Advanced Analytical Mechanics. Beijing Institute of Technology Press, Beijing (1991). (in Chinese)

    Google Scholar 

  34. Djukic, D.S.: Adiabatic invariants for dynamical systems with one degree of freedom. Int. J. Non-linear Mech. 16, 489–498 (1981)

    Article  MATH  Google Scholar 

  35. Bulanov, S.V., Shasharina, S.G.: Behavior of adiabatic invariants near the sparatrix in a stellarator. Nucl. Fusion 32, 1531–1543 (1992)

    Article  Google Scholar 

  36. Notte, J., Fajans, J., Chu, R., Wurtele, J.S.: Experimental breaking of an adiabatic invariants. Phys. Rev. Lett. 70, 3900–3903 (1993)

    Article  Google Scholar 

  37. Zhao, Y.Y.: Perturbation to symmetries and adiabatic invariants for mechanics system. J. Hunan Univ. 23(1), 45–50 (1996). (in Chinese)

    Google Scholar 

  38. Zhao, Y.Y., Mei, F.X.: Exact invariant and adiabatic invariant of a general dynamical system. Acta Mech. Sin. 28(2), 207–216 (1996)

    MathSciNet  Google Scholar 

  39. Chen, X.W., Mei, F.X.: Perturbation to the symmetries and adiabatic invariants of holonomic variable mass systems. Chin. Phys. 9(10), 721–725 (2000)

    Article  MathSciNet  Google Scholar 

  40. Chen, X.W., Zhao, Y.H., Li, Y.M.: Perturbation to symmetries and adiabatic invariants of nonholonomic systems in terms of quasi-coordinates. Commun. Theor. Phys. 44(5), 773–778 (2005)

    Article  MathSciNet  Google Scholar 

  41. Zhang, Y.: Perturbation to symmetries and adiabatic invariant of Birkhoffian systems with unilateral constraints. Acta Phys. Sin. 51(08), 1666–1670 (2002)

    MATH  Google Scholar 

  42. Xia, L.L., Li, Y.C.: Perturbation to symmetries and adiabatic invariants for nonholonomic controllable mechanical system in phase space. Acta Phys. Sin. 56(11), 6183–6187 (2007)

    MATH  MathSciNet  Google Scholar 

  43. Zhang, Y., Fan, C.X., Mei, F.X.: Perturbation to symmetries and Hojman adiabatic invariants for Lagrange system. Acta Phys. Sin. 55(07), 3237–3240 (2006)

    MATH  MathSciNet  Google Scholar 

  44. Chen, X.W., Li, Y.M.: Exact invariants and adiabatic invariants of the singular Lagrange system. Chin. Phys. 12(9), 936–939 (2003)

    Article  Google Scholar 

  45. Zhang, Y.: Exact invariants and adiabatic invariants of constrained Hamiltonian systems in phase space. Acta Phys. Sin. 51(11), 2417–2422 (2002)

    MATH  Google Scholar 

  46. Fu, J.L., Chen, L.Q.: Perturbation of symmetries of rotational relativistic Birkhoffian systems and its inverse problems. Phys. Lett. A 324, 95–103 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  47. Zhang, Y.: A new type of adiabatic invariants for Birkhoffian system. Acta Phys. Sin. 55(8), 3833–3837 (2006)

    MATH  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 10972151 and 11272227), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (No. CXLX13_855), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (No. SKCX13S_050).

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

    Ju Chen

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

    Yi Zhang

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  1. Ju Chen
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Correspondence to Yi Zhang.

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Chen, J., Zhang, Y. Perturbation to Noether symmetries and adiabatic invariants for disturbed Hamiltonian systems based on El-Nabulsi nonconservative dynamics model. Nonlinear Dyn 77, 353–360 (2014). https://doi.org/10.1007/s11071-014-1298-0

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