事件空间中完整力学系统的梯度表示

吴惠彬; 梅凤翔

doi:10.7498/aps.64.234501

摘要

本文研究事件空间中完整力学系统的梯度表示和分数维梯度表示, 建立系统的微分方程并将其表示为一阶形式, 给出系统成为梯度系统的条件以及成为分数维梯度系统的条件. 最后, 举例说明结果的应用.

关键词:

The dynamics research in the event space has important geometric and mechanical meanings, and great progress has been made in this field. A gradient system is a kind of important systems in differential equations and dynamical systems, and is receiving more and more attention. In this paper, a gradient representation and a fractional gradient representation of a holonomic system in the event space are studied. First, the differential equations of motion for the system are established and expressed in the first order form. Second, we have obtained the condition under which the system can be considered as a gradient system and also the condition under which the system can be considered as a fractional gradient system. When a constrained mechanical system is transformed into a gradient system or a fractional gradient system, one can use the properties of the gradient system or the fractional gradient system to study the integration and the stability of a constrained mechanical system. Finally, two examples are given to illustrate the application of the results. The event space is known as more extensive than the configuration space, therefore, the result in the configuration space is a special case of this paper.

Keywords:

作者及机构信息

吴惠彬¹,
梅凤翔²

1.
北京理工大学数学学院, 北京 100081;

2.
北京理工大学宇航学院, 北京 100081

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.

Authors and contacts

Wu Hui-Bin¹,
Mei Feng-Xiang²

1.
School of Mathematics, Beijing Institute of Technology, Beijing 100081, China;

2.
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Corresponding author: Wu Hui-Bin, huibinwu@bit.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos 10932002, 11272050).

参考文献

[1]	Synge J L 1960 Classical Dynamics (Berlin: Springer-Verlag)
[2]	Rumyatsev V V 1984 P. M. M. 48 540 (in Russian)
[3]	Mei F X 1990 Acta Mech. Sin. 6 160
[4]	Li Y C, Zhang Y, Liang J H 2000 Appl. Math. Mech. 21 543
[5]	Fang J H 2002 Appl. Math. Mech. 23 89
[6]	Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 888
[7]	Jia L Q, Zhang Y Y, Luo S K 2007 Chin. Phys. 16 3168
[8]	Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量(北京: 北京理工大学出版社)]
[9]	Zhang Y 2008 Chin. Phys. B 17 4365
[10]	Zhang Y 2008 Acta Phys. Sin. 57 2643 (in Chinese) [张毅 2008 物理学报 57 2643]
[11]	Zhang Y 2010 Chin. Phys. B 19 080301
[12]	Zhang B, Fang J H, Zhang W W 2012 Chin. Phys. B 21 070208
[13]	Zhang X W, Li Y Y, Zhao X X, Luo W F 2014 Chin. Phys. B 23 104501
[14]	Tarasov V E 2010 Fractional Dynamics (Beijing: Higher Education Press)
[15]	Hirsch M W, Smale S, Devaney R L 2008 Differential Equations, Dynamical Systems and an Introduction to Chaos (Singapore: Elsevier)
[16]	Quispel G RW, Capel H W 1996 Physics Letters A 218 223
[17]	Quispel G RW, Turner G S 1996 J. Phys. A: Math. Gen. 29 L341
[18]	Hong J L, Zhai S X, Zhang J J 2011 S IA M J. Numer. Anal. 49 2017
[19]	Mei F X, Wu H B 2013 Acta Phys. Sin. 62 214501 (in Chinese) [梅凤翔, 吴惠彬 2013 物理学报 62 214501]
[20]	Ge W K, Xue Y, Lou Z M 2014 Acta Phys. Sin. 63 110202 (in Chinese) [葛伟宽, 薛纭, 楼智美 2014 物理学报 63 110202]
[21]	Mei F X, Wu H B 2015 Chin. Phys. B 24 054501

施引文献

[1]	Synge J L 1960 Classical Dynamics (Berlin: Springer-Verlag)
[2]	Rumyatsev V V 1984 P. M. M. 48 540 (in Russian)
[3]	Mei F X 1990 Acta Mech. Sin. 6 160
[4]	Li Y C, Zhang Y, Liang J H 2000 Appl. Math. Mech. 21 543
[5]	Fang J H 2002 Appl. Math. Mech. 23 89
[6]	Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 888
[7]	Jia L Q, Zhang Y Y, Luo S K 2007 Chin. Phys. 16 3168
[8]	Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量(北京: 北京理工大学出版社)]
[9]	Zhang Y 2008 Chin. Phys. B 17 4365
[10]	Zhang Y 2008 Acta Phys. Sin. 57 2643 (in Chinese) [张毅 2008 物理学报 57 2643]
[11]	Zhang Y 2010 Chin. Phys. B 19 080301
[12]	Zhang B, Fang J H, Zhang W W 2012 Chin. Phys. B 21 070208
[13]	Zhang X W, Li Y Y, Zhao X X, Luo W F 2014 Chin. Phys. B 23 104501
[14]	Tarasov V E 2010 Fractional Dynamics (Beijing: Higher Education Press)
[15]	Hirsch M W, Smale S, Devaney R L 2008 Differential Equations, Dynamical Systems and an Introduction to Chaos (Singapore: Elsevier)
[16]	Quispel G RW, Capel H W 1996 Physics Letters A 218 223
[17]	Quispel G RW, Turner G S 1996 J. Phys. A: Math. Gen. 29 L341
[18]	Hong J L, Zhai S X, Zhang J J 2011 S IA M J. Numer. Anal. 49 2017
[19]	Mei F X, Wu H B 2013 Acta Phys. Sin. 62 214501 (in Chinese) [梅凤翔, 吴惠彬 2013 物理学报 62 214501]
[20]	Ge W K, Xue Y, Lou Z M 2014 Acta Phys. Sin. 63 110202 (in Chinese) [葛伟宽, 薛纭, 楼智美 2014 物理学报 63 110202]
[21]	Mei F X, Wu H B 2015 Chin. Phys. B 24 054501

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