Research on the discrete variational method for a Birkhoffian system

doi:10.1088/1674-1056/23/6/064501

中国物理B ›› 2014, Vol. 23 ›› Issue (6): 64501-064501.doi: 10.1088/1674-1056/23/6/064501

• ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS • 上一篇下一篇

Research on the discrete variational method for a Birkhoffian system

刘世兴^a, 花巍^b, 郭永新^a

a College of Physics, Liaoning University, Shenyang 110036, China;
b College of Physics and Technology, Shenyang Normal University, Shenyang 110034, China

收稿日期:2013-06-25 修回日期:2013-11-10 出版日期:2014-06-15 发布日期:2014-06-15
基金资助:
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).

Research on the discrete variational method for a Birkhoffian system

Liu Shi-Xing (刘世兴)^a, Hua Wei (花巍)^b, Guo Yong-Xin (郭永新)^a

a College of Physics, Liaoning University, Shenyang 110036, China;
b College of Physics and Technology, Shenyang Normal University, Shenyang 110034, China

Received:2013-06-25 Revised:2013-11-10 Online:2014-06-15 Published:2014-06-15
Contact: Guo Yong-Xin E-mail:yxguo@lnu.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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.

关键词: Birkhoff', s equations, discrete variational methods, general symplectic structure, discrete Birkhoff', s equations

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.

Key words: Birkhoff's equations, discrete variational methods, general symplectic structure, discrete Birkhoff's equations

中图分类号: (Lagrangian and Hamiltonian mechanics)

45.20.Jj

45.10.-b (Computational methods in classical mechanics)

刘世兴, 花巍, 郭永新. Research on the discrete variational method for a Birkhoffian system[J]. 中国物理B, 2014, 23(6): 64501-064501.

Liu Shi-Xing (刘世兴), Hua Wei (花巍), Guo Yong-Xin (郭永新). Research on the discrete variational method for a Birkhoffian system[J]. Chin. Phys. B, 2014, 23(6): 64501-064501.

参考文献 17

[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)

Research on the discrete variational method for a Birkhoffian system

Research on the discrete variational method for a Birkhoffian system

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