Skip to main content
SpringerLink
Log in

Structure-preserving properties of Störmer-Verlet scheme for mathematical pendulum

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conservation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Störmer-Verlet scheme is first constructed in a Hamiltonian framework. The conservation law of the Störmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Störmer-Verlet scheme associated with the conservation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the Störmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the Störmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Störmer-Verlet scheme.

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. Qin, Y. Y., Deng, Z. C., and Hu, W. P. Structure-preserving properties of three differential schemes for oscillator system. Applied Mathematics and Mechanics (English Edition), 35, 783–790 (2014) DOI 10.1007/s10483-014-1828-6

    Article  MathSciNet  MATH  Google Scholar 

  2. Balakirev, V. A., Buts, V. A., Tolstoluzhsky, A. P., and Turkin, Y. A. Nonlinear dynamics of a mathematical pendulum with a vibrating hanger (in Russian). Ukrainskii Fizicheskii Zhurnal, 32, 1270–1274 (1987)

    Google Scholar 

  3. Moauro, V. and Negrini, P. Chaotic trajectories of a double mathematical pendulum. Journal of Applied Mathematics and Mechanics, 62, 827–830 (1998)

    Article  MathSciNet  Google Scholar 

  4. Martynyuk, A. A. and Nikitina, N. V. The theory of motion of a double mathematical pendulum. International Applied Mechanics, 36, 1252–1258 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Martynyuk, A. A. and Nikitina, N. V. Regular and chaotic motions of mathematical pendulums. International Applied Mechanics, 37, 407–413 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Shaikhet, L. Stability of difference analogue of linear mathematical inverted pendulum. Discrete Dynamics in Nature and Society, 3, 215–226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hatvani, L. Stability problems for the mathematical pendulum. Periodica Mathematica Hungarica, 56, 71–82 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dittrich, W. The mathematical pendulum from Gauss via Jacobi to Riemann. Annalen der Physik, 18, 381–390 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jerman, B. and Hribar, A. Dynamics of the mathematical pendulum suspended from a moving mass. Tehnički Vjesnik, 20, 59–64 (2013)

    Google Scholar 

  10. Feng, K. On difference schemes and symplectic geometry. Proceeding of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 42–58 (1984)

    Google Scholar 

  11. Feng, K. Difference-schemes for Hamiltonian-formalism and symplectic-geometry. Journal of Computational Mathematics, 4, 279–289 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Hairer, E., Lubich, C., and Wanner, G. Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, Berlin (2002)

    Book  MATH  Google Scholar 

  13. Bridges, T. J. Multi-symplectic structures and wave propagation. Mathematical Proceedings of the Cambridge Philosophical Society, 121, 147–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu, W. P., Deng, Z. C., Han, S. M., and Zhang, W. R. Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs. Journal of Computational Physics, 235, 394–406 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Störmer, C. Sur les trajectoires des corpuscules électrisés dans l’espace sous l’action du magnétisme terrestre, avec application aux aurores boréales. Le Radium, 9, 395–399 (1912)

    Article  MATH  Google Scholar 

  16. Verlet, L. Computer experiments on classical fluids I: thermodynamical properties of Lennard-Jones molecules. Physical Review, 159, 98–103 (1967)

    Article  Google Scholar 

  17. Rivlin, L. A. Acceleration of neutrons in a scheme of a tautochronous mathematical pendulum (physical principles). Quantum Electronics, 40, 933–934 (2010)

    Article  Google Scholar 

  18. Budd, C. J. and Piggott, M. D. Geometric integration and its applications. Handbook of Numerical Analysis, 11, 35–139 (2003)

    MathSciNet  MATH  Google Scholar 

  19. Terze, Z., Muller, A., and Zlatar, D. An angular momentum and energy conserving Lie-group integration scheme for rigid body rotational dynamics originating from Störmer-Verlet algorithm. Journal of Computational and Nonlinear Dynamics, 10, 051005 (2015)

    Article  Google Scholar 

  20. Hairer, E. and Lubich, C. Long-term analysis of the Störmer-Verlet method for Hamiltonian systems with a solution-dependent high frequency. Numerische Mathematik, 134, 119–138 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xing, Y. F. and Yang, R. Phase errors and their correction in symplect implicit single-step algorithm (in Chinese). Acta Mechanica Sinica, 39, 668–671 (2007)

    Google Scholar 

  22. Görtz, P. Backward error analysis of symplectic integrators for linear separable Hamiltonian systems. Journal of Computational mathematics, 20, 449–460 (2002)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, 710072, China

    Weipeng Hu, Mingzhe Song & Zichen Deng

  2. State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116023, Liaoning Province, China

    Weipeng Hu & Zichen Deng

Authors
  1. Weipeng Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingzhe Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zichen Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weipeng Hu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11672241, 11372253, and 11432010), the Astronautics Supporting Technology Foundation of China (No. 2015-HT-XGD), and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (Nos.GZ1312 and GZ1605)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, W., Song, M. & Deng, Z. Structure-preserving properties of Störmer-Verlet scheme for mathematical pendulum. Appl. Math. Mech.-Engl. Ed. 38, 1225–1232 (2017). https://doi.org/10.1007/s10483-017-2233-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-017-2233-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation