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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.
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Keywords:
- holonomic system /
- event space /
- gradient /
- fractional dynamics
[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 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 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 63 110202]
[21] Mei F X, Wu H B 2015 Chin. Phys. B 24 054501
-
[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 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 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 63 110202]
[21] Mei F X, Wu H B 2015 Chin. Phys. B 24 054501
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