相空间中相对运动完整力学系统的共形不变性与守恒量

王廷志; 孙现亭; 韩月林

doi:10.7498/aps.63.104502

摘要

研究了相空间中相对运动完整力学系统的共形不变性与守恒量. 给出了该系统共形不变性的定义，并推导出相空间中相对运动完整力学系统的运动微分方程具有共形不变性并且是Lie对称性的充分必要条件. 利用规范函数满足的结构方程导出该系统相应的守恒量，并给出应用算例.

关键词:

Conformal invariance and conserved quantity of relative motion holonomic dynamical system in phase space are studied. The definition of conformal invariance of relative motion holonomic dynamical system in phase space is provided. The necessary and sufficient conditions that conformal invariance of the system would be Lie symmetry are deduced. By use of a structure equation that the gauge function satisfies, the corresponding conserved quantity of the system is derived. Finally an illustrative example is given to verify 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 467000

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 467000, China

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

参考文献

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[2]	Djukic D S, Vujanovi B D 1975 Acta Mech. 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) p43 (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) 第43页]
[6]	Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz Birkhoff and Nambu Systems (Moscow: UFN) p183 (in Russian)
[7]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) p366 (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366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]	Luo S K, Li Z J, Li L 2012 Acta Mech. 223 2621
[11]	Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117
[12]	Han Y L, Wang X X, Zhang M L, Jia L Q 2013 Nonlinear Dyn. 71 401
[13]	Han Y L, Wang X X, Zhang M L, Jia L Q 2013 Nonlinear Dyn. 73 357
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[15]	Xu C, Li Y C 2013 Acta Phys. Sin. 62 171101 (in Chinese) [徐超, 李元成 2013 62 171101]
[16]	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]
[17]	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]
[18]	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]
[19]	Zhang M L, Wang X X, Han Y L, Jia L Q 2012 Chin. Phys. B 21 100203
[20]	Jia L Q, Wang X X, Zhang M L, Han Y L 2012 Nonlinear Dyn. 69 1807
[21]	Zhang Y, Xue Y 2009 Chin. Q. Mech. 30 216 (in Chinese) [张毅, 薛纭 2009 力学季刊 30 216]
[22]	Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[23]	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]
[24]	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]
[25]	Chen R, Xu X J 2012 Acta Phys. Sin. 61 021102 (in Chinese) [陈蓉, 许学军 2012 61 021102]
[26]	Chen X W, Zhao Y H, Li Y M 2009 Chin. Phys. B 18 3139
[27]	Cai J L, Shi S S, Fang H J, Xu J 2012 Mechanica 47 63
[28]	Huang W L, Cai J L 2012 Acta Mech. 223 433
[29]	Cai J L 2012 Nonlinear Dyn. 69 487
[30]	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]
[31]	Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170
[32]	Fang J H, Zhang P Y 2004 Acta Phys. Sin. 53 4041 (in Chinese) [方建会, 张鹏玉 2004 53 4041]
[33]	Fang J H, Zhang B, Zhang W W, Xu R L 2012 Chin. Phys. B 21 050202

施引文献

[1]	Noether A E 1918 Nachr. Akad. Wiss. Göttingen Math. Phys. 2 235
[2]	Djukic D S, Vujanovi B D 1975 Acta Mech. 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) p43 (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) 第43页]
[6]	Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz Birkhoff and Nambu Systems (Moscow: UFN) p183 (in Russian)
[7]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) p366 (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366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]	Luo S K, Li Z J, Li L 2012 Acta Mech. 223 2621
[11]	Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117
[12]	Han Y L, Wang X X, Zhang M L, Jia L Q 2013 Nonlinear Dyn. 71 401
[13]	Han Y L, Wang X X, Zhang M L, Jia L Q 2013 Nonlinear Dyn. 73 357
[14]	Lou Z M, Mei F X, Chen Z D 2012 Acta Phys. Sin. 61 110204 (in Chinese) [楼智美, 梅凤翔, 陈子栋 2012 61 110204]
[15]	Xu C, Li Y C 2013 Acta Phys. Sin. 62 171101 (in Chinese) [徐超, 李元成 2013 62 171101]
[16]	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]
[17]	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]
[18]	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]
[19]	Zhang M L, Wang X X, Han Y L, Jia L Q 2012 Chin. Phys. B 21 100203
[20]	Jia L Q, Wang X X, Zhang M L, Han Y L 2012 Nonlinear Dyn. 69 1807
[21]	Zhang Y, Xue Y 2009 Chin. Q. Mech. 30 216 (in Chinese) [张毅, 薛纭 2009 力学季刊 30 216]
[22]	Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[23]	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]
[24]	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]
[25]	Chen R, Xu X J 2012 Acta Phys. Sin. 61 021102 (in Chinese) [陈蓉, 许学军 2012 61 021102]
[26]	Chen X W, Zhao Y H, Li Y M 2009 Chin. Phys. B 18 3139
[27]	Cai J L, Shi S S, Fang H J, Xu J 2012 Mechanica 47 63
[28]	Huang W L, Cai J L 2012 Acta Mech. 223 433
[29]	Cai J L 2012 Nonlinear Dyn. 69 487
[30]	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]
[31]	Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170
[32]	Fang J H, Zhang P Y 2004 Acta Phys. Sin. 53 4041 (in Chinese) [方建会, 张鹏玉 2004 53 4041]
[33]	Fang J H, Zhang B, Zhang W W, Xu R L 2012 Chin. Phys. B 21 050202

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