非完整系统的共形不变性导致的新型守恒量

王廷志; 孙现亭; 韩月林

doi:10.7498/aps.63.090201

摘要

研究了非完整系统的共形不变性与新型守恒量. 提出了该系统共形不变性的概念；得出了非完整系统的运动微分方程具有共形不变性并且是Lie对称性的充要条件. 利用规范函数满足的新型结构方程，导出系统相应的新型守恒量. 最后给出应用算例.

关键词:

Conformal invariance and a new type of conserved quantity in nonholonomic mechanical system are studied. The definition and determining equation of conformal invariance for the nonholonomic mechanical system are provided; and the necessary and sufficient conditions that the conformal invariance for a nonholonomic mechanical system should be of Lie symmetry are deduced. With the aid of a new structure equation that the gauge function satisfies, the system's corresponding new conserved quantity is obtained. Finally an example is given to illustrate the application of the results.

Keywords:

作者及机构信息

1.
江南大学理学院, 无锡 214122;

2.
平顶山学院电气信息工程学院, 平顶山 467002

基金项目: 国家自然科学基金（批准号：11142014）资助的课题.

Authors and contacts

1.
School of Science, Jiangnan University, Wuxi 214122, China;

2.
School of Electric and Information Engineering, Pingdingshan University, Pingdingshan 467002, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11142014).

参考文献

[1]	Noether A E 1918 Nachr. Akad. Math. 2 235
[2]	Djukić D S, Vujanovi B D 1975 Acta Mechanica 23 17
[3]	Lutzky M 1979 J. Phys. A Math. Gen. 12 973
[4]	Hojman S A 1992 J. Phys. A Math. Gen. 25 L291
[5]	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)]
[6]	Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz Birkhoff and Nambu Systems (Moscow: UFN) (in Russian)
[7]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) p383 (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用(北京: 科学出版社)第383页]
[8]	Zhang M L, Wang X X, Han Y L, Jia L Q 2012 Chin. Phys. B 21 100203
[9]	Wang X X, Han Y L, Zhang M L, Jia L Q 2013 Chin. Phys. B 22 020201
[10]	Yang X F, Sun X T, Wang X X, Zhang M L, Jia L Q 2011 Acta Phys. Sin. 60 111101 (in Chinese) [杨新芳, 孙现亭, 王肖肖, 张美玲, 贾利群 2011 60 111101]
[11]	Wang X X, Zhang M L, Han Y L, Jia L Q 2012 Acta Phys. Sin. 61 200203 (in Chinese) [王肖肖, 张美玲, 韩月林, 贾利群 2012 61 200203]
[12]	Sun X T, Han Y L, Wang X X, Zhang M L, Jia L Q 2012 Acta Phys. Sin. 61 200204 (in Chinese) [孙现亭, 韩月林, 王肖肖, 张美玲, 贾利群 2012 61 200204]
[13]	Zhang Y, Xue Y 2009 Chinese Quarterly of Mechanics 30 216 (in Chinese) [张毅, 薛纭 2009 力学季刊 30 216]
[14]	Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[15]	Chen X W, Zhao Y H, Liu C 2009 Acta Phys. Sin. 58 5150 (in Chinese) [陈向炜, 赵永红, 刘畅 2009 58 5150]
[16]	Chen R, Xu X J 2012 Acta Phys. Sin. 61 021102 (in Chinese) [陈蓉, 许学军 2012 61 021102]
[17]	Chen X W, Zhao Y H, Li Y M 2009 Chin Phys. B 18 3139
[18]	Huang W L, Cai J L 2012 Applied Mathematics and Mechanics. 33 1294 [黄卫立, 蔡建乐 2012 应用数学和力学 33 1294 ]
[19]	Han Y L, Sun X T, Zhang Y Y, Jia L Q 2013 Acta Phys. Sin. 62 160201 (in Chinese) [韩月林, 孙现亭, 张耀宇, 贾利群 2013 62 160201]
[20]	Wang T Z, Han Y L 2013 Journal of Jiangnan University (Natural Science Edition). 2 234 (in Chinese) [王廷志, 韩月林 2013 江南大学学报(自然科学版) 2 234]
[21]	Wang T Z, Sun X T, Han Y L 2013 Acta Phys. Sin. 62 231101 (in Chinese) [王廷志, 孙现亭, 韩月林 2013 62 231101]
[22]	Jia L Q, Sun X T, Zhang M L, Zhang Y Y, Han Y L 2014 Acta Phys. Sin. 63 010201 (in Chinese) [贾利群, 孙现亭, 张美玲, 张耀宇, 韩月林 2014 63 010201]
[23]	Luo S K, Li Z J, Li L 2012 Acta Mechanica 223 2621
[24]	Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117
[25]	Li L, Luo S K 2013 Acta Mechanica 224 1757
[26]	Li Z J, Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 445
[27]	Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 475
[28]	Jia L Q, Xie Y L, Zhang Y Y, Cui J C, Yang X F 2010 Acta Phys. Sin. 59 7552 (in Chinese) [贾利群, 解银丽, 张耀宇, 崔金超, 杨新芳 2012 59 7552]
[29]	SHen Y, Zhang L T 2012 ACTA ARMAMENTARII. 11 1342 (in Chinese) [沈跃, 张令坦 2012 兵工学报 11 1342]
[30]	Jia L Q, Han Y L, Zhang M L, Wang X X 2013 Journal of Pingdingsban University. 2 27 (in Chinese) [贾利群, 韩月林, 张美玲, 王肖肖 2013 平顶山学院学报 2 27]

施引文献

[1]	Noether A E 1918 Nachr. Akad. Math. 2 235
[2]	Djukić D S, Vujanovi B D 1975 Acta Mechanica 23 17
[3]	Lutzky M 1979 J. Phys. A Math. Gen. 12 973
[4]	Hojman S A 1992 J. Phys. A Math. Gen. 25 L291
[5]	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)]
[6]	Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz Birkhoff and Nambu Systems (Moscow: UFN) (in Russian)
[7]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) p383 (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用(北京: 科学出版社)第383页]
[8]	Zhang M L, Wang X X, Han Y L, Jia L Q 2012 Chin. Phys. B 21 100203
[9]	Wang X X, Han Y L, Zhang M L, Jia L Q 2013 Chin. Phys. B 22 020201
[10]	Yang X F, Sun X T, Wang X X, Zhang M L, Jia L Q 2011 Acta Phys. Sin. 60 111101 (in Chinese) [杨新芳, 孙现亭, 王肖肖, 张美玲, 贾利群 2011 60 111101]
[11]	Wang X X, Zhang M L, Han Y L, Jia L Q 2012 Acta Phys. Sin. 61 200203 (in Chinese) [王肖肖, 张美玲, 韩月林, 贾利群 2012 61 200203]
[12]	Sun X T, Han Y L, Wang X X, Zhang M L, Jia L Q 2012 Acta Phys. Sin. 61 200204 (in Chinese) [孙现亭, 韩月林, 王肖肖, 张美玲, 贾利群 2012 61 200204]
[13]	Zhang Y, Xue Y 2009 Chinese Quarterly of Mechanics 30 216 (in Chinese) [张毅, 薛纭 2009 力学季刊 30 216]
[14]	Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[15]	Chen X W, Zhao Y H, Liu C 2009 Acta Phys. Sin. 58 5150 (in Chinese) [陈向炜, 赵永红, 刘畅 2009 58 5150]
[16]	Chen R, Xu X J 2012 Acta Phys. Sin. 61 021102 (in Chinese) [陈蓉, 许学军 2012 61 021102]
[17]	Chen X W, Zhao Y H, Li Y M 2009 Chin Phys. B 18 3139
[18]	Huang W L, Cai J L 2012 Applied Mathematics and Mechanics. 33 1294 [黄卫立, 蔡建乐 2012 应用数学和力学 33 1294 ]
[19]	Han Y L, Sun X T, Zhang Y Y, Jia L Q 2013 Acta Phys. Sin. 62 160201 (in Chinese) [韩月林, 孙现亭, 张耀宇, 贾利群 2013 62 160201]
[20]	Wang T Z, Han Y L 2013 Journal of Jiangnan University (Natural Science Edition). 2 234 (in Chinese) [王廷志, 韩月林 2013 江南大学学报(自然科学版) 2 234]
[21]	Wang T Z, Sun X T, Han Y L 2013 Acta Phys. Sin. 62 231101 (in Chinese) [王廷志, 孙现亭, 韩月林 2013 62 231101]
[22]	Jia L Q, Sun X T, Zhang M L, Zhang Y Y, Han Y L 2014 Acta Phys. Sin. 63 010201 (in Chinese) [贾利群, 孙现亭, 张美玲, 张耀宇, 韩月林 2014 63 010201]
[23]	Luo S K, Li Z J, Li L 2012 Acta Mechanica 223 2621
[24]	Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117
[25]	Li L, Luo S K 2013 Acta Mechanica 224 1757
[26]	Li Z J, Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 445
[27]	Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 475
[28]	Jia L Q, Xie Y L, Zhang Y Y, Cui J C, Yang X F 2010 Acta Phys. Sin. 59 7552 (in Chinese) [贾利群, 解银丽, 张耀宇, 崔金超, 杨新芳 2012 59 7552]
[29]	SHen Y, Zhang L T 2012 ACTA ARMAMENTARII. 11 1342 (in Chinese) [沈跃, 张令坦 2012 兵工学报 11 1342]
[30]	Jia L Q, Han Y L, Zhang M L, Wang X X 2013 Journal of Pingdingsban University. 2 27 (in Chinese) [贾利群, 韩月林, 张美玲, 王肖肖 2013 平顶山学院学报 2 27]

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