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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.
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Keywords:
- generalized Birkhoff system /
- combined gradient system /
- integration /
- stability
[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 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
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[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 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
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