Skip to main content
SpringerLink
Log in

Special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system

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

Abstract

The weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system are studied. Appell equations for a weakly nonholonomic system are established and the definition and the criterion of the special Mei symmetry of the system are given. The structure equation of the special Mei symmetry for a weakly nonholonomic system and approximate conserved quantity deduced from the special Mei symmetry of the system are obtained. Finally, special approximate conserved quantity issues of Appell equations for a two freedom degrees weakly nonholonomic system are investigated using the results of this paper.

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. Hertz, H.R.: Die Prinzipien der Mechanik. Gesammelte Werke, Leibzing (1894)

    Google Scholar 

  2. Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. AMS, Providence (1972)

    MATH  Google Scholar 

  3. Mei, F.X.: Foundations of Mechanics of Nonholonomic Systems. Beijing Institute of Technology Press, Beijing (1985)

    Google Scholar 

  4. Ostrovskaya, S., Angels, J.: Nonholonomic systems revisited within the frame work of analytical mechanics. Appl. Mech. Rev. 51, 415–433 (1998)

    Article  Google Scholar 

  5. Mei, F.X.: Nonholonomic mechanics. Appl. Mech. Rev. 53, 283–305 (2000)

    Article  Google Scholar 

  6. Zegzhda, S.A., Soltakhanov, S.K., Yushkov, M.P.: Equations of Motion of Nonholonomic Systems and Variational Principles of Mechanics. New Kind of Control Problems. FIMATLIT, Moscow (2005)

    Google Scholar 

  7. Luo, S.K., Zhang, Y.F.: Advances in the Study of Dynamics of Constrained Systems. Science Press, Beijing (2008)

    Google Scholar 

  8. Zhang, J.W., Li, J.F., Wu, R.H.: Global attractor of strongly damped nonlinear thermoelastic coupled rod system. Acta Phys. Sin. 60, 070205 (2011)

    Google Scholar 

  9. Noether, A.E.: Invariant variations problem. Nachr. Akad. Wiss. Gött. Math.-Phys. 2, 235–257 (1918)

    Google Scholar 

  10. Djukić, D.J.S., Vujanvić, B.D.: Noether’s theory in classical nonconservative mechanics. Acta Mech. 23, 17–27 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Vujanović, B.: Conservation laws of dynamical systems via d’Alembert’s principle. Int. J. Non-Linear Mech. 13, 185–197 (1978)

    Article  MATH  Google Scholar 

  12. Lutzky, M.: Dynamical symmetries and conserved quantities. J. Phys. A, Math. Gen. 12, 973–981 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lutzky, M.: Non-invariance symmetries and constants of the motion. Phys. Lett. A 72, 86–88 (1979)

    Article  MathSciNet  Google Scholar 

  14. Lutzky, M.: Origin of non-Noether invariants. Phys. Lett. A 75, 8–10 (1979)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Vujanović, B.: A study of conservation laws of dynamical systems by means of the differential variational principles of Jourdain and Gauss. Acta Mech. 65, 63–80 (1986)

    Article  Google Scholar 

  17. Luo, S.K.: Generalized Noether theorem of variable mass higher-order nonholonomic mechanical system in noninertial reference frame. Chin. Sci. Bull. 36, 1930–1932 (1991)

    Google Scholar 

  18. Hojman, S.A.: A new conservation law constructed without using either Lagrangians or Hamiltonians. J. Phys. A, Math. Gen. 25, 291–295 (1992)

    Article  MathSciNet  Google Scholar 

  19. 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 

  20. Lutzky, M.: Conserved quantities and velocity dependent symmetries in Lagrangian dynamics. Int. J. Non-Linear Mech. 33, 393–396 (1999)

    Article  MathSciNet  Google Scholar 

  21. Lutzky, M.: Conserved quantities from non-Noether symmetries without alternative Lagrangians. Int. J. Non-Linear Mech. 34, 387–390 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  23. Luo, S.K.: Appell equations and form invariance of rotational relativistic systems. Acta Phys. Sin. 51, 712–717 (2002)

    MATH  Google Scholar 

  24. 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  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  26. Ge, W.K.: Mei symmetry and conserved quantity of a holonomic system. Acta Phys. Sin. 57, 6714–6717 (2008)

    MATH  Google Scholar 

  27. Xie, Y.L., Jia, L.Q.: Special Lie–Mei symmetry and conserved quantities of Appell equations expressed by Appell fun. Chin. Phys. Lett. 27, 120201 (2010)

    Article  Google Scholar 

  28. Cai, J.L.: Conformal invariance and conserved quantity for the nonholonomic system of Chetaev’s type. Int. J. Theor. Phys. 49, 201–211 (2010)

    Article  MATH  Google Scholar 

  29. Cai, J.L.: Conformal invariance and conserved quantity of Hamilton system under second-class Mei symmetry. Acta Phys. Pol. A 117, 445–448 (2010)

    Google Scholar 

  30. Jia, L.Q., Xie, Y.L., Zhang, Y.Y., Yang, X.F.: A type of new conserved quantity deduced from Mei symmetry for Appell equations in a holonomic system with unilateral constraints. Chin. Phys. B 19, 110301 (2010)

    Article  Google Scholar 

  31. Ding, N., Fang, J.H.: Mei adiabatic invariants induced by perturbation of Mei symmetry for nonholonomic controllable mechanical systems. Commun. Theor. Phys. 54, 785–791 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zheng, S.W., Xie, J.F., Chen, X.W., Du, X.L.: Another kind of conserved quantity induced directly from Mei symmetry of Tzénoff equations for holonomic systems. Acta Phys. Sin. 59, 5209–5212 (2010)

    MATH  Google Scholar 

  33. Zhang, M.L., Sun, X.T., Wang, X.X., Xie, Y.Li., Jia, L.Q.: Lie symmetry and generalized Hojman conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion. Chin. Phys. B 20, 110202 (2011)

    Article  Google Scholar 

  34. Li, Z.J., Jiang, W.A., Luo, S.K.: Lie symmetries, symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems. Nonlinear Dyn. 67, 445–455 (2012)

    Article  Google Scholar 

  35. Jiang, W.A., Luo, S.K.: A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems. Nonlinear Dyn. 67, 475–482 (2012)

    Article  Google Scholar 

  36. Jiang, W.A., Li, L., Li, Z.J., Luo, S.K.: Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems. Nonlinear Dyn. 67, 1075–1081 (2012)

    Article  Google Scholar 

  37. Jiang, W.N., Li, Z.J., Luo, S.K.: Mei symmetries and Mei conserved quantities for higher-order nonholonomic systems. Chin. Phys. B 20, 030202 (2011)

    Article  Google Scholar 

  38. Jiang, W.N., Luo, S.K.: Mei symmetry leading to Mei conserved quantity of generalized Hamilton systems. Acta Phys. Sin. 60, 060201 (2011)

    MATH  Google Scholar 

  39. Mei, F.X.: Form invariance of Lagrange system. J. Beijing Inst. Technol. 9(2), 120–124 (2000)

    MATH  Google Scholar 

  40. Mei, F.X.: Equations of motion for weak nonholonomic systems and their approximate solution. J. Beijing Inst. Technol. 9(3), 10–17 (1989)

    MATH  Google Scholar 

  41. Mei, F.X.: Canonical transformation of weak nonholonomic systems. Chin. Sci. Bull. 37, 1180–1183 (1992)

    Google Scholar 

  42. Mei, F.X.: On the stability of one type of weakly nonholonomic systems. J. Beijing Inst. Technol. 15, 237–242 (1995)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Jiangnan University, Wuxi, 214122, P.R. China

    Liqun Jia, Xiaoxiao Wang, Meiling Zhang & Yuelin Han

Authors
  1. Liqun Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoxiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meiling Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yuelin Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liqun Jia.

Additional information

Project supported by the National Natural Science Foundation of China (Grant No. 11142014 and 61178032).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jia, L., Wang, X., Zhang, M. et al. Special Mei symmetry and approximate conserved quantity of Appell equations for a weakly nonholonomic system. Nonlinear Dyn 69, 1807–1812 (2012). https://doi.org/10.1007/s11071-012-0387-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0387-1

Keywords

Search

Navigation