Skip to main content
SpringerLink
Log in

Two forms of the discrete equations and the Noether theorems for nonautonomous Birkhoffian systems

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

Two different ways of constructing the discrete equations and the corresponding physical laws of continuous Birkhoffian systems are respectively proposed in this paper. The corresponding mathematical methods and geometric structures are formulated and compared. The determining equations of the Noether symmetries are obtained via the Lie point transformations acting on the difference equations. Two types of the discrete conserved quantities of the systems are presented using the structure equation satisfied by the gauge functions. The algorithms can be developed based on these two approaches applied to the nonholonomic systems with symmetries. As a result, the geometric structure and the Noether invariants are numerically preserved. The numerical simulations based on the two approaches demonstrate the high precision and the long-time stability of the algorithms compared with the standard Runge–Kutta method.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Birkhoff, G.D.: The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations. Proc. Am. Acad. Arts Sci. 49, 521–568 (1913)

    Article  Google Scholar 

  2. Cadzow, J.A.: Discrete calculus of variations. Int. J. Control. 11, 393–407 (1970)

    Article  Google Scholar 

  3. Cortés, J., Martínez, S.: Non-holonomic integrators. Nonlinearity 14, 1365–1392 (2001)

    Article  MathSciNet  Google Scholar 

  4. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)

    Article  MathSciNet  Google Scholar 

  5. Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear nonlinear PDE’s. Commun. Math. Phys. 199, 351–395 (1998)

    Article  MathSciNet  Google Scholar 

  6. He, L., Wu, H.B., Mei, F.X.: Variational integrators for fractional Birkhoffian systems. Nonlinear Dyn. 87, 1–10 (2016)

    MATH  Google Scholar 

  7. Ferraro, S., Iglesias, D., Diego, D.M.: Momentum and energy preserving integrators for nonholonomic dynamics. Nonlinearity 21, 1911–1928 (2008)

    Article  MathSciNet  Google Scholar 

  8. Liu, S.X., Hua, W., Guo, Y.X.: Research on the discrete variational method for a Birkhoffian system. Chin. Phys. B 23, 064501 (2014)

    Article  Google Scholar 

  9. Dorodnitsyn, V.: Applications of Lie Groups to Difference Equations. Chapman & Hall/CRC, Boca Raton (2011)

    MATH  Google Scholar 

  10. Dorodnitsyn, V.: Transformation groups in net spaces. J. Soviet Math. 55, 1490–1517 (1991)

    Article  Google Scholar 

  11. Levi, D., Winternitz, P.: Continuous symmetries of discrete equations. Phys. Lett. A 152, 335–338 (1991)

    Article  MathSciNet  Google Scholar 

  12. Fu, J.L., Li, X.W., Li, C.R., Zhao, W.J., Chen, B.Y.: Symmetries and exact solutions of discrete nonconservative systems. Sci. China Phys. Mech. 53, 1699–1706 (2010)

    Article  Google Scholar 

  13. Fu, J.L., Chen, B.Y., Chen, L.Q.: Noether symmetries of discrete nonholonomic dynamical systems. Phys. Lett. A 373, 409–412 (2009)

    Article  MathSciNet  Google Scholar 

  14. Fu, J.L., Fu, H., Liu, R.W.: Hojman conserved quantities of discrete mechanico–electrical systems constructed by continuous symmetries. Phys. Lett. A 374, 1812–1818 (2010)

    Article  MathSciNet  Google Scholar 

  15. Bourlioux, A., Cyr-Gagnon, C., Winternitz, P.: Difference schemes with point symmetries and their numerical tests. J. Phys. A Gen. Phys. 39, 6877–6896 (2006)

    Article  MathSciNet  Google Scholar 

  16. Zhang, H.B., Lv, H.S., Gu, S.L.: The Lie point symmetry-preserving difference scheme of holonomic constrained mechanical systems. Acta Phys. Sin. 59, 5213–5218 (2010)

    Google Scholar 

  17. Xia, L.L., Chen, L.Q.: Mei symmetries and conserved quantities for non-conservative Hamiltonian difference systems with irregular lattices. Nonlinear Dyn. 70, 1223–1230 (2012)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  19. Birkhoff, G.D.: Dynamical Systems. AMS College Publication, Providence (1927)

    MATH  Google Scholar 

  20. Mei, F.X., Shi, R.C., Zhang, Y.F., Wu, H.B.: Dynamics of Birkhoffian System. Institute of Technology Press, Beijing (1996).. (in Chinese)

    Google Scholar 

  21. Mei, F.X.: On the Birkhoffian Mechanics. Int. J. Nonlinear Mech. 36, 817–834 (2001)

    Article  MathSciNet  Google Scholar 

  22. Guo, Y.X., Liu, C., Liu, S.X.: Generalized Birkhoffian realization of nonholonomic systems. Commun. Math. 18, 21–35 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Zhang, H.B.: Generalized variational problems and Birkhoff equations. Nonlinear Dyn. 83, 347–354 (2016)

    Article  MathSciNet  Google Scholar 

  26. Kong, X.L., Wu, H.B., Mei, F.X.: Variational discretization for the planar Lotka–Volterra equations in the Birkhoffian sense. Nonlinear Dyn. 84, 733–742 (2016)

    Article  MathSciNet  Google Scholar 

  27. Su, H.L., Qin, M.C., Wang, Y.S., Scherer, R.: Multi-symplectic Birkhoffian structure for PDEs with dissipation terms. Phys. Lett. A 374, 2410–2416 (2010)

    Article  MathSciNet  Google Scholar 

  28. Boronenko, T.S.: On the use of the autonomous Birkhoff equations in Lie series perturbation theory. Celest. Mech. Dyn. Astr. 127, 139–161 (2017)

    Article  MathSciNet  Google Scholar 

  29. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)

    Book  Google Scholar 

  30. Broucke, R.: On Pfaffs equations of motion in dynamic; applications to satellite theory. Celest. Mech. Dyn. Astron. 18, 207–222 (1978)

    Article  MathSciNet  Google Scholar 

  31. Liu, C., Liu, S.X., Guo, Y.X.: Inverse problem for Chaplygin’s nonholonomic systems. Sci. Chin. G 54, 2100–2106 (2010)

    MATH  Google Scholar 

  32. Miller, K.S.: On linear difference equations. Am. Math. Mon. 75, 630–632 (1968)

    Article  Google Scholar 

  33. Rosenbery, R.M.: Analytical Dynamics of Discrete Systems. Plenum Press, New York (1977)

    Book  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12072041, 11732005), the spetial research project and Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University.

Funding

National Natural Science Foundation of China (Grant No. 11502071).

Author information

Authors and Affiliations

  1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Lili Xia

  2. Mechanical and Electrical Engineering School, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Xinsheng Ge

  3. School of Science, Harbin Institute of Technology, Shenzhen, 518055, People’s Republic of China

    Liqun Chen

Authors
  1. Lili Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinsheng Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liqun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liqun Chen.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xia, L., Ge, X. & Chen, L. Two forms of the discrete equations and the Noether theorems for nonautonomous Birkhoffian systems. Anal.Math.Phys. 11, 159 (2021). https://doi.org/10.1007/s13324-021-00594-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-021-00594-1

Keywords

Search

Navigation