[1] Zhang Yi,. Method of variation of parametersfor solving a constrained Birkhoffian system [J]. Journal of Southeast University (English Edition), 2013, 29 (3): 342-345. [doi:10.3969/j.issn.1003-7985.2013.03.020]

Copy

Method of variation of parametersfor solving a constrained Birkhoffian system()

求解约束Birkhoff系统的参数变异法

Share：

Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

29

Issue:

2013 3

Page:

342-345

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2013-09-20

Info

Title:

Method of variation of parametersfor solving a constrained Birkhoffian system

求解约束Birkhoff系统的参数变异法

Author(s):

Zhang Yi

College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China

张毅

苏州科技学院土木工程学院, 苏州 215011

Keywords:

Birkhoffian mechanics;  method of integration;  method of variation of parameter;  constrained Birkhoffian system

Birkhoff力学;  积分方法;  参数变异法;  约束Birkhoff系统

PACS:

O316

DOI:

10.3969/j.issn.1003-7985.2013.03.020

Abstract:

For an in-depth study on the integration problem of the constrained mechanical systems, the method of integration for the Birkhoffian system with constraints is discussed, and the method of variation of parameters for solving the dynamical equations of the constrained Birkhoffian system is provided. First, the differential equations of motion for the constrained Birkhoffian system as well as for the corresponding free Birkhoffian system are established. Secondly, a system of auxiliary equations is constructed, and the general solution of the equations is found. Finally, by varying the parameters and utilizing the properties of the generalized canonical transformation of the Birkhoffian system, the solution of the problem can be obtained. The proposed method reveals the inherent relationship between the solution of a free Birkhoffian system and that of a constrained Birkhoffian system. The research results are of universal significance, which can be further used in a variety of constrained mechanical systems, such as non-conservative systems and nonholonomic systems etc.

为了深入研究约束力学系统的积分问题, 讨论了具有约束的Birkhoff系统的积分方法问题, 提出用参数变异法求解约束Birkhoff系统的动力学方程.首先, 建立约束Birkhoff系统及相应的自由Birkhoff系统的运动微分方程;其次, 构建辅助方程系统, 并找到其一般解;最后, 变异参数并利用Birkhoff系统的广义正则变换的性质, 获得问题的解.该方法揭示了自由Birkhoff系统和约束Birkhoff系统的解之间的内在联系.所提研究方法和结果具有普遍意义, 可进一步应用于各种约束力学系统, 例如, 非保守系统和非完整约束系统等.

References:

[1] Mei F X, Shi R C, Zhang Y F, et al. Dynamics of Birkhoffian systems[M]. Beijing: Beijing Institute of Technology Press, 1996.(in Chinese)
[2] Santilli R M. Foundations of theoretical mechanics Ⅱ [M]. New York: Springer-Verlag, 1983.
[3] Galiullin A S, Gafarov G G, Malaishka R P, et al. Analytical dynamics of Helmholtz, Birkhoff and Nambu systems [M]. Moscow: UFN, 1997.(in Russian)
[4] Mei F X. Noether theory of Birkhoffian system [J]. Science in China: Series A, 1993, 36(12): 1456-1467.
[5] Mei F X. Poisson’s theory of Birkhoffian system [J]. Chinese Science Bulletin, 1996, 41(8): 641-645.
[6] Mei F X, Wu H B. First integral and integral invariant of Birkhoffian system [J]. Chinese Science Bulletin, 2000, 45(5): 412-414.
[7] Mei F X. On the Birkhoffian mechanics [J]. International Journal of Non-Linear Mechanics, 2001, 36(5): 817-834.
[8] Zhang Y. The method of variation on parameters for integration of a generalized Birkhoffian system [J]. Acta Mechanica Sinica, 2011, 27(6):1059-1064.
[9] Guo Y X, Shang M, Luo S K. Poincaré-Cartan integral variants of Birkhoff system [J]. Applied Mathematics and Mechanics, 2003, 24(1): 76-82.
[10] Guo Y X, Liu C, Liu S X. Generalized Birkhoffian formulation of nonholonomic systems [J]. Communications in Mathematics, 2010, 18(1): 21-35.
[11] Luo S K, Guo Y X. Routh order reduction method of relativistic Birkhoffian systems [J]. Communication in Theoretical Physics, 2007, 47(2): 209-212.
[12] Mei F X, Wu H B. Form invariance and new conserved quantity of generalized Birkhoffian system [J]. Chinese Physics B, 2010, 19(5): 050301.
[13] Li Y M. Lie symmetries, perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system [J]. Chinese Physics Letters, 2010, 27(1): 010202.
[14] Zhang Y. Poisson theory and integration method of Birkhoffian systems in the event space [J]. Chinese Physics B, 2010, 19(8): 080301.
[15] Zhang M J, Fang J H, Lu K. Perturbation to Mei symmetry and generalized Mei adiabatic invariants for Birkhoffian systems [J]. International Journal of Theoretical Physics, 2010, 49(2): 427-437.
[16] Zhang Y, Zhou Y. Symmetries and conserved quantities for fractional action-like Pfaff variational problems [J]. Nonlinear Dynamics, 2013, 73(1/2): 783-793.
[17] Zhang Y. A new method for integration of a Birkhoffian system [J]. Journal of Southeast University: English Edition, 2011, 27(2): 188-191.
[18] Wu H B, Mei F X. Type of integral and reduction for a generalized Birkhoffian system [J]. Chinese Physics B, 2011, 20(10): 104501.

Memo

Memo:

Biography: Zhang Yi(1964—), male, doctor, professor, weidiezh@gmail.com.
Foundation item: The National Natural Science Foundation of China(No.10972151, 11272227).
Citation: Zhang Yi.Method of variation of parameters for solving a constrained Birkhoffian system[J].Journal of Southeast University(English Edition), 2013, 29(3):342-345.[doi:10.3969/j.issn.1003-7985.2013.03.020]

Common functions