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Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems

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Abstract

For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation. Then, a new type of exact invariants of non-Noether type is given, furthermore adiabatic invariants and exact invariants of non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate the application of the method and results.

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References

  1. Burgers, J.M.: Die adiabatischen invarianten bedingt periodischenr systems. Ann. Phys. (Leipz.) 52, 195–202 (1917)

    Article  Google Scholar 

  2. 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 

  3. Bulanov, S.V., Shasharina, S.G.: Behaviour of adiabatic invariant near the separatrix in a stellarator. Nucl. Fusion 32, 1531–1543 (1992)

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Ibragimov, N.H.: CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3. CRC, New York (1996)

    MATH  Google Scholar 

  6. Zhao, Y.Y., Mei, F.X.: Exact invariants and adiabatic invariants of general mechanical systems. Acta Mech. Sin. 28, 207–216 (1996)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Chen, X.W., Li, Y.M., Zhao, Y.H.: Lie symmetries, perturbation to symmetries and adiabatic invariants of Lagrange system. Phys. Lett. A 337, 274–278 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Zhang, Y.: A new type of adiabatic invariants for nonconservative systems of generalized classical mechanics. Chin. Phys. 15, 1935–1940 (2006)

    Article  Google Scholar 

  10. Zhang, Y., Fan, C.X.: Perturbation of symmetries and Hojman adiabatic invariants for mechanical systems with unilateral holonomic constraints. Commun. Theor. Phys. 47, 607–610 (2007)

    Article  Google Scholar 

  11. Luo, S.K.: Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for disturbed nonholonomic systems. Chin. Phys. Lett. 24, 3017–3020 (2007)

    Article  Google Scholar 

  12. Luo, S.K.: Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type of relativistic Birkhoffian. Commun. Theor. Phys. 47, 25–30 (2007)

    Article  Google Scholar 

  13. Xia, L.L., Li, Y.C.: Perturbation to symmetries and Hojman adiabatic invariants for nonholonomic controllable mechanical systems with non-Chetaev type constraints. Chin. Phys. 16, 1516–1520 (2007)

    Article  Google Scholar 

  14. Ding, N., Fang, J.H., Wang, P., Zhang, X.N.: Perturbation to Lie symmetries and adiabatic invariants for general holonomic mechanical system. Commun. Theor. Phys. 48, 19–22 (2007)

    Article  MathSciNet  Google Scholar 

  15. Jiang, W.A., Luo, S.K.: A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9996-3

    Google Scholar 

  16. Li, Z.J., Jiang, W.A., Luo, S.K.: Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems. Nonlinear Dyn. (2011). doi:10.1007/s11071-011-9993-6

    Google Scholar 

  17. Birkhoff, G.D.: Dynamical Systems. AMS College Publisher, Providence (1927)

    MATH  Google Scholar 

  18. Santilli, R.M.: Foundations of Theoretical Mechanics I. Springer, New York (1978)

    MATH  Google Scholar 

  19. Santilli, R.M.: Foundations of Theoretical Mechanics II. Springer, New York (1983)

    MATH  Google Scholar 

  20. Mei, F.X., Shi, R.C., Zhang, Y.F., Wu, H.B.: Dynamics of Birkhoff Systems. Beijing Institute of Technology, Beijing (1996)

    Google Scholar 

  21. Chen, X.W.: Closed orbits and limit cycles of second-order autonomous Birkhoff systems. Chin. Phys. 12, 586–589 (2003)

    Article  Google Scholar 

  22. Chen, X.W.: Chaos in the second order autonomous Birkhoff system with a heteroclinic circle. Chin. Phys. 11, 441–444 (2002)

    Article  Google Scholar 

  23. Luo, S.K.: First integrals and integral invariants of relativistic Birkhoffian systems. Commun. Theor. Phys. 40, 133–136 (2003)

    MATH  Google Scholar 

  24. Luo, S.K., Cai, J.L.: A set of the Lie symmetrical conservation laws for the rotational relativistic Birkhoffian system. Chin. Phys. 12, 357–360 (2003)

    Article  Google Scholar 

  25. Zhang, Y.: A geometrical approach to Hojman theorem of a rotational relativistic Birkhoffian system. Commun. Theor. Phys. 42, 669–671 (2004)

    MATH  Google Scholar 

  26. Su, H.L., Qin, M.Z.: Symplectic schemes for Birkhoffian system. Commun. Theor. Phys. 41, 329–334 (2004)

    MATH  MathSciNet  Google Scholar 

  27. Su, H.L.: Birkhoffian symplectic scheme for a quantum system. Commun. Theor. Phys. 53, 476–480 (2010)

    Article  MATH  Google Scholar 

  28. Li, Y.M.: Lie symmetries, perturbation to symmetries and adiabatic invariants of generalized Birkhoff systems. Chin. Phys. Lett. 27, 010202 (2010)

    Article  Google Scholar 

  29. Li, Y.M., Mei, F.X.: Stability for manifolds of equilibrium states of generalized Birkhoff system. Chin. Phys. B 19, 080302 (2010)

    Google Scholar 

  30. Wang, C.D., Liu, S.X., Mei, F.X.: Generalized Pfaff-Birkhoff-dAlembert principle and form invariance of generalized Birkhoff equations. Acta Phys. Sin. 59, 8322 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou, 310018, P.R. China

    Wenan Jiang, Lin Li, Zhuangjun Li & Shaokai Luo

Authors
  1. Wenan Jiang
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  2. Lin Li
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  3. Zhuangjun Li
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  4. Shaokai Luo
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Corresponding author

Correspondence to Shaokai Luo.

Additional information

This work is partly supported by National Natural Science Foundation of China (grant Nos. 10972127 and 10372053).

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Jiang, W., Li, L., Li, Z. et al. Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems. Nonlinear Dyn 67, 1075–1081 (2012). https://doi.org/10.1007/s11071-011-0051-1

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