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广义Birkhoff系统与一类组合梯度系统

梅凤翔 吴惠彬

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广义Birkhoff系统与一类组合梯度系统

梅凤翔, 吴惠彬

Generalized Birkhoff system and a kind of combined gradient system

Mei Feng-Xiang, Wu Hui-Bin
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  • 提出一类组合梯度系统, 即将梯度系统与斜梯度系统相加而组成的一个系统, 并研究组合梯度系统的重要性质. 将广义Birkhoff系统在一定条件下化成组合梯度系统, 并利用组合梯度系统的性质来研究广义Birkhoff系统的积分和稳定性.
    A combined gradient system which is obtained by adding a gradient system to a skew-gradient system is proposed. The property of the system is studied. Three results on its solution are obtained. Moreover, a criterion on the stability of the system is presented. The generalized Birkhoff system is a more extensive constrained mechanical system than Lagrange system, Hamilton system and Birkhoff system. The conditions under which a generalized Birkhoff system can be considered as a combined gradient system are given. When a generalized Birkhoff system is transformed into a combined gradient system, its integration and stability can be discussed by using the property. Finally, some examples are given to illustrate the application of our results.
      通信作者: 吴惠彬, huibinwu@bit.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 10932002, 11272050)资助的课题.
      Corresponding author: Wu Hui-Bin, huibinwu@bit.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 10932002, 11272050).
    [1]

    Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York: Springer-Verlag)

    [2]

    Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 Birkhoff系统动力学 (北京: 北京理工大学出版社)]

    [3]

    Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz, Birkhoff, Nambu Systems (Moskow: UFN) (in Russian)

    [4]

    Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) (in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与不变量 (北京: 科学出版社)]

    [5]

    Mei F X 1993 Sci. China Ser. A 36 1456

    [6]

    Zhang Y 2002 Chin. Phys. 11 437

    [7]

    Xu X J, Mei F X, Qin M C 2004 Chin. Phys. 13 1999

    [8]

    Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282

    [9]

    Zhang H B, Gu S L 2002 Chin. Phys. 11 765

    [10]

    Luo S K, Chen X W, Fu J L 2001 Chin. Phys. 10 271

    [11]

    Fu J L, Chen X W, Luo Y 2003 Chin. Phys. 12 351

    [12]

    Mei F X, Zhang Y F, He G, Gang T Q, Xie J F 2007 Trans. Beijing Inst. Tech. 27 1035 (in Chinese) [梅凤翔, 张永发, 何光, 冮铁强, 解加芳 2007 北京理工大学学报 27 1035]

    [13]

    Mei F X, Xie J F, Gang T Q 2008 Acta Mech. Sin. 24 583

    [14]

    Shang M, Mei F X 2009 Chin. Phys. B 18 3155

    [15]

    Mei F X, Wu H B 2010 Chin. Phys. B 19 050301

    [16]

    Li Y M, Mei F X 2010 Acta Phys. Sin. 59 5219(in Chinese) [李彦敏, 梅凤翔 2010 物理学报 59 5219]

    [17]

    Li Y M, Mei F X 2010 Chin. Phys. B 19 080302

    [18]

    Hirsch M W, Smale S, Devaney R L 2008 Differential Equations, Dynamical Systems and an Introduction to Chaos (Singapore: Elsevier)

    [19]

    Mei F X 2012 Mech. Eng. 34 89 (in Chinese) [梅凤翔 2012 力学与实践 34 89]

    [20]

    McLachlan R I, Quispel G R W, Robidoux N 1999 Phil. Tran. R. Soc. Lond. A 357 1021

  • [1]

    Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York: Springer-Verlag)

    [2]

    Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 Birkhoff系统动力学 (北京: 北京理工大学出版社)]

    [3]

    Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz, Birkhoff, Nambu Systems (Moskow: UFN) (in Russian)

    [4]

    Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) (in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与不变量 (北京: 科学出版社)]

    [5]

    Mei F X 1993 Sci. China Ser. A 36 1456

    [6]

    Zhang Y 2002 Chin. Phys. 11 437

    [7]

    Xu X J, Mei F X, Qin M C 2004 Chin. Phys. 13 1999

    [8]

    Chen X W, Zhang R C, Mei F X 2000 Acta Mech. Sin. 16 282

    [9]

    Zhang H B, Gu S L 2002 Chin. Phys. 11 765

    [10]

    Luo S K, Chen X W, Fu J L 2001 Chin. Phys. 10 271

    [11]

    Fu J L, Chen X W, Luo Y 2003 Chin. Phys. 12 351

    [12]

    Mei F X, Zhang Y F, He G, Gang T Q, Xie J F 2007 Trans. Beijing Inst. Tech. 27 1035 (in Chinese) [梅凤翔, 张永发, 何光, 冮铁强, 解加芳 2007 北京理工大学学报 27 1035]

    [13]

    Mei F X, Xie J F, Gang T Q 2008 Acta Mech. Sin. 24 583

    [14]

    Shang M, Mei F X 2009 Chin. Phys. B 18 3155

    [15]

    Mei F X, Wu H B 2010 Chin. Phys. B 19 050301

    [16]

    Li Y M, Mei F X 2010 Acta Phys. Sin. 59 5219(in Chinese) [李彦敏, 梅凤翔 2010 物理学报 59 5219]

    [17]

    Li Y M, Mei F X 2010 Chin. Phys. B 19 080302

    [18]

    Hirsch M W, Smale S, Devaney R L 2008 Differential Equations, Dynamical Systems and an Introduction to Chaos (Singapore: Elsevier)

    [19]

    Mei F X 2012 Mech. Eng. 34 89 (in Chinese) [梅凤翔 2012 力学与实践 34 89]

    [20]

    McLachlan R I, Quispel G R W, Robidoux N 1999 Phil. Tran. R. Soc. Lond. A 357 1021

计量
  • 文章访问数:  4349
  • PDF下载量:  114
  • 被引次数: 0
出版历程
  • 收稿日期:  2015-02-24
  • 修回日期:  2015-05-06
  • 刊出日期:  2015-09-05

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