Please wait a minute...
Chin. Phys. B, 2014, Vol. 23(6): 064501    DOI: 10.1088/1674-1056/23/6/064501

Research on the discrete variational method for a Birkhoffian system

Liu Shi-Xinga, Hua Weib, Guo Yong-Xina
a College of Physics, Liaoning University, Shenyang 110036, China;
b College of Physics and Technology, Shenyang Normal University, Shenyang 110034, China
Abstract  In this paper, we present a new integration algorithm based on the discrete Pfaff-Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structures of Birkhoffian systems. A numerical experiment for a damping oscillator system is conducted. The result shows that the new algorithm can better simulate the energy dissipation than the R-K method, which illustrates that we can numerically solve the dynamical equations by the discrete variational method in a Birkhoffian framework for the systems with a general symplectic structure. Furthermore, it is demonstrated that the results of the numerical experiments are determined not by the constructing methods of Birkhoffian functions but by whether the numerical method can preserve the inherent nature of the dynamical system.
Keywords:  Birkhoff's equations      discrete variational methods      general symplectic structure      discrete Birkhoff's equations  
Received:  25 June 2013      Revised:  10 November 2013      Published:  15 June 2014
PACS:  45.20.Jj (Lagrangian and Hamiltonian mechanics)  
  45.10.-b (Computational methods in classical mechanics)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11301350, 11172120, and 11202090) and the Liaoning University Pre-reporting Fund Natural Projects (Grant No. 2013LDGY02).
Corresponding Authors:  Guo Yong-Xin     E-mail:

Cite this article: 

Liu Shi-Xing, Hua Wei, Guo Yong-Xin Research on the discrete variational method for a Birkhoffian system 2014 Chin. Phys. B 23 064501

[1] Birkhoff G D 1927 Dynamic Systems, AMS College Pub l., (Providence, RL De Donder TH: Theories des Invariants Integraux, Gauthier-Villars, Paris)
[2] Santilli R M 1983 Foundations of Theoretical Mechanics II: Birkhoffian Generalization of Hamiltonian Mechanics (New York: Springer-Verlag)
[3] Mei F X, Shi R C, Zhang Y F and Wu H B 1996 Dynamics of Birkhoff Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)
[4] Badía-Majós A, Carińena J F and López C 2006 J. Phys A: Math. Gen. 39 14699
[5] Ionescu D 2006 J. Geom. Phys. 56 2545
[6] Mei F X, Gang T Q and Xie J F 2006 Chin. Phys. 15 1678
[7] Guo Y X, Liu C and Liu S X 2010 Comm. Math. 18 21
[8] Wang C D, Liu S X and Mei F X 2010 Acta Phys. Sin. 59 8322 (in Chinese)
[9] Su H L, Sun Y J, Qin M Z and Scherer R 2007 Int. J. Pur. Appl. Math. 40 341
[10] Sun Y J and Shang Z J 2005 Phys. Lett. A 336 358
[11] Zhang H B, Chen L Q, Gu S L and Liu C Z 2007 Chin. Phys. 16 582
[12] Liu S X, Liu C and GuoY X 2011 Chin. Phys. B 20 034501
[13] Liu S X, Liu C and Guo Y X 2011 Acta Phys. Sin. 60 064501 (in Chinese)
[14] Zhang X W, Wu J K, Zhu H P and Huang K F 2002 Appl. Math. Mech. 23 1029
[15] Marsden J E and West M 2001 Acta Numer. 10 357
[16] Hairer E, Lubich C and Wanner G 2002 Geometric Numerical Integration: Structure- Preserving Algorithms for Ordinary Differential Equations (Berlin: Springer-Verlag)
[17] Cortes J 2002 Geometric, Control and Numerical Aspects of Nonholonomic Systems (Berlin: Springer-Verlag)
[1] On superintegrable systems with a position-dependent mass in polar-like coordinates
Hai Zhang(章海)†. Chin. Phys. B, 2020, 29(10): 100201.
[2] Quasi-canonicalization for linear homogeneous nonholonomic systems
Yong Wang(王勇), Jin-Chao Cui(崔金超), Ju Chen(陈菊), Yong-Xin Guo(郭永新). Chin. Phys. B, 2020, 29(6): 064501.
[3] Discrete symmetrical perturbation and variational algorithm of disturbed Lagrangian systems
Li-Li Xia(夏丽莉), Xin-Sheng Ge(戈新生), Li-Qun Chen(陈立群). Chin. Phys. B, 2019, 28(3): 030201.
[4] Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales
Jing Song(宋静), Yi Zhang(张毅). Chin. Phys. B, 2017, 26(8): 084501.
[5] Fully nonlinear (2+1)-dimensional displacement shallow water wave equation
Feng Wu(吴锋), Zheng Yao(姚征), Wanxie Zhong(钟万勰). Chin. Phys. B, 2017, 26(5): 054501.
[6] Methods of reduction for Lagrange systems on time scaleswith nabla derivatives
Shi-Xin Jin(金世欣), Yi Zhang(张毅). Chin. Phys. B, 2017, 26(1): 014501.
[7] Stability analysis of a simple rheonomic nonholonomic constrained system
Chang Liu(刘畅), Shi-Xing Liu(刘世兴), Feng-Xing Mei(梅凤翔). Chin. Phys. B, 2016, 25(12): 124501.
[8] Generalized Birkhoffian representation of nonholonomic systems and its discrete variational algorithm
Shixing Liu(刘世兴), Chang Liu(刘畅), Wei Hua(花巍), Yongxin Guo(郭永新). Chin. Phys. B, 2016, 25(11): 114501.
[9] An application of a combined gradient system to stabilize a mechanical system
Xiang-Wei Chen(陈向炜), Ye Zhang(张晔), Feng-Xiang Mei(梅凤翔). Chin. Phys. B, 2016, 25(10): 100201.
[10] Two kinds of generalized gradient representationsfor holonomic mechanical systems
Feng-Xiang Mei(梅凤翔) and Hui-Bin Wu(吴惠彬). Chin. Phys. B, 2016, 25(1): 014502.
[11] Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms
Xin-Lei Kong(孔新雷), Hui-Bin Wu(吴惠彬), Feng-Xiang Mei(梅凤翔). Chin. Phys. B, 2016, 25(1): 010203.
[12] Dynamics of two polarized nanoparticles
Duan Xiao-Yong, Wang Zhi-Guo. Chin. Phys. B, 2015, 24(11): 118106.
[13] Skew-gradient representation of generalized Birkhoffian system
Mei Feng-Xiang, Wu Hui-Bin. Chin. Phys. B, 2015, 24(10): 104502.
[14] Bifurcation for the generalized Birkhoffian system
Mei Feng-Xiang, Wu Hui-Bin. Chin. Phys. B, 2015, 24(5): 054501.
[15] Fractional cyclic integrals and Routh equations of fractional Lagrange system with combined Caputo derivatives
Wang Lin-Li, Fu Jing-Li. Chin. Phys. B, 2014, 23(12): 124501.
No Suggested Reading articles found!