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A type of the new exact and approximate conserved quantity deduced from Mei symmetry for a weakly nonholonomic system

Han Yue-Lin Wang Xiao-Xiao Zhang Mei-Ling Jia Li-Qun

A type of the new exact and approximate conserved quantity deduced from Mei symmetry for a weakly nonholonomic system

Han Yue-Lin, Wang Xiao-Xiao, Zhang Mei-Ling, Jia Li-Qun
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  • A type of structural equation, new exact and approximate conserved quantity which are deduced from Mei symmetry of Lagrange equations for a weakly nonholonomic system, are investigated. First, Lagrange equations of weakly nonholonomic system are established. Next, under the infinitesimal transformations of Lie groups, the definition and the criterion of Mei symmetry for Lagrange equations in weakly nonholonomic systems and its first-degree approximate holonomic system are given. And then, the expressions of new structural equation and new exact and approximate conserved quantities of Mei symmetry for Lagrange equations in weakly nonholonomic systems are obtained. Finelly, an example is given to study the question of the exact and the approximate new conserved quantities.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11142014), and the scientific research and innovation plan for College Graduates of Jiangsu province, China (Grant No. CXLX12_0720).
    [1]

    Neimark J I, Fufaev N A 1972 Providence, RI:AMS

    [2]

    Bloch A M, Krishnaprasad P S, Marsden J E, Murray R M 1996 Arch. Rat. Mech. Anal. 136 21

    [3]

    Ostrovskaya S, Angels, 1998 ASME Appl. Mech. Rev. 51 415

    [4]

    Mei F X 2000 ASME Appl. Mech. Rev. 53 283

    [5]

    Zegzhda S A, Soltakhanov S K, Yushkov M P 2005 Moscow: FIMATLIT

    [6]

    Mei F X 1989 Beijing Inst. Technol. 9 10

    [7]

    Mei F X 1992 Chin. Sci. Bull. 37 1180

    [8]

    Mei F X 1995 Beijing Inst. Technol. 15 237

    [9]

    Noether A E 1918 Nachr Akad Wiss Gottingen Math. Phys. Kl 235

    [10]

    Mei F X 2000 Beijing Inst. Technol. 9 120

    [11]

    Zheng S W, Xie J F, Chen W C 2008 Chin. Phys. Lett. 25 809

    [12]

    Ge W K 2008 Acta Phys. Sin. 57 6714 (in Chinese) [葛伟宽 2008 物理学报 57 6714]

    [13]

    Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 物理学报 58 22]

    [14]

    Fang J H 2009 Acta Phys. Sin. 58 3617 (in Chinese) [方建会 2009 物理学报 58 3617]

    [15]

    Zheng S W, Xie J F, Chen X W 2010 Acta Phys. Sin. 59 5209 (in Chinese) [郑世旺, 解加芳, 陈向伟 2010 物理学报 59 5209]

    [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 物理学报 60 111101]

    [17]

    Cai J L 2012 Nonlinear Dyn. 69 487

    [18]

    Jia L Q, Wang X X, Zhang M L Han Y L 2012 Nonlinear Dyn. 69 1807

    [19]

    Chen X W, Li Y M, Zhao Y H 2005 Phys. Lett. A 337 274

    [20]

    Luo S K 2007 Chin. Phys. Lett. 24 2463

    [21]

    Cai J L 2008 Chin. Phys. Lett. 25 1523

    [22]

    Cai J L, Mei F X 2008 Acta Phys. Sin. 57 5369 (in Chinese) [蔡建乐, 梅凤翔 2008 物理学报 57 5369]

    [23]

    Jiang W A, Li Z J, Luo S K 2011 Chin. Phys. B 20 030202

    [24]

    Xu X J, Mei F X, Qin M C 2004 Acta Phys. Sin. 53 4021 (in Chinese) [许学军, 梅凤翔, 秦茂昌 2004 物理学报 53 4021]

    [25]

    Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170

    [26]

    Cui J C, Zhang Y Y, Jia L Q 2009 Chin. Phys. B 18 1731

    [27]

    Xie Y L, Jia L Q 2010 Chin Phys. Lett. 27 120201

    [28]

    Zheng S W, Xie J F, Chen X W 2010 Acta Phys. Sin. 59 5209 (in Chinese) [郑世旺, 解加芳, 陈向伟 2010 物理学报 59 5209]

    [29]

    Li Y C, Wang X M, Xia L L 2010 Acta Phys. Sin. 59 2935 (in Chinese) [李元成, 王小明, 夏丽莉 2010 物理学报 59 2935]

    [30]

    Jia L Q, Sun X T, Zhang M L, Wang X X, Xie Y L 2011 Acta Phys. Sin. 60 084501 (in Chinese) [贾利群, 孙现亭, 张美玲, 王肖肖, 解银丽 2011 物理学报 60 084501]

    [31]

    Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 475

    [32]

    Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117

    [33]

    Luo S K, Li Z J, Li L 2012 Acta Mech. 223 2621

    [34]

    Luo S K, Li Z J, Peng W, Li L 2013 Acta Mech. 224 71

    [35]

    Han Y L, Wang X X, Zhang M L, Jia L Q 2013 Nonlinear Dyn. 71 401

    [36]

    Mei F X 2003 Acta Phys. Sin. 52 1048 (in Chinese) [梅凤翔 2003 物理学报 52 1048]

    [37]

    Zhang Y, Fan C X, Ge W K 2004 Acta Phys. Sin. 53 3644 (in Chinese) [张毅, 范存新, 葛伟宽 2008 物理学报 53 3644]

    [38]

    Fang J H, Liu Y K, Zhang X N 2008 Chin. Phys. B 17 1962

    [39]

    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 2010 物理学报 59 7552]

    [40]

    Zhao L, Fu J L, Chen B Y 2011 Chin. Phys. B 20 040201

    [41]

    Han Y L, Sun X T, Wang X X, Zhang M L, Jia L Q 2012 Chin. Phys. B 21 120201

    [42]

    Mei F X 2004 Beijing Institute of Technology Press, Beijing

  • [1]

    Neimark J I, Fufaev N A 1972 Providence, RI:AMS

    [2]

    Bloch A M, Krishnaprasad P S, Marsden J E, Murray R M 1996 Arch. Rat. Mech. Anal. 136 21

    [3]

    Ostrovskaya S, Angels, 1998 ASME Appl. Mech. Rev. 51 415

    [4]

    Mei F X 2000 ASME Appl. Mech. Rev. 53 283

    [5]

    Zegzhda S A, Soltakhanov S K, Yushkov M P 2005 Moscow: FIMATLIT

    [6]

    Mei F X 1989 Beijing Inst. Technol. 9 10

    [7]

    Mei F X 1992 Chin. Sci. Bull. 37 1180

    [8]

    Mei F X 1995 Beijing Inst. Technol. 15 237

    [9]

    Noether A E 1918 Nachr Akad Wiss Gottingen Math. Phys. Kl 235

    [10]

    Mei F X 2000 Beijing Inst. Technol. 9 120

    [11]

    Zheng S W, Xie J F, Chen W C 2008 Chin. Phys. Lett. 25 809

    [12]

    Ge W K 2008 Acta Phys. Sin. 57 6714 (in Chinese) [葛伟宽 2008 物理学报 57 6714]

    [13]

    Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 物理学报 58 22]

    [14]

    Fang J H 2009 Acta Phys. Sin. 58 3617 (in Chinese) [方建会 2009 物理学报 58 3617]

    [15]

    Zheng S W, Xie J F, Chen X W 2010 Acta Phys. Sin. 59 5209 (in Chinese) [郑世旺, 解加芳, 陈向伟 2010 物理学报 59 5209]

    [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 物理学报 60 111101]

    [17]

    Cai J L 2012 Nonlinear Dyn. 69 487

    [18]

    Jia L Q, Wang X X, Zhang M L Han Y L 2012 Nonlinear Dyn. 69 1807

    [19]

    Chen X W, Li Y M, Zhao Y H 2005 Phys. Lett. A 337 274

    [20]

    Luo S K 2007 Chin. Phys. Lett. 24 2463

    [21]

    Cai J L 2008 Chin. Phys. Lett. 25 1523

    [22]

    Cai J L, Mei F X 2008 Acta Phys. Sin. 57 5369 (in Chinese) [蔡建乐, 梅凤翔 2008 物理学报 57 5369]

    [23]

    Jiang W A, Li Z J, Luo S K 2011 Chin. Phys. B 20 030202

    [24]

    Xu X J, Mei F X, Qin M C 2004 Acta Phys. Sin. 53 4021 (in Chinese) [许学军, 梅凤翔, 秦茂昌 2004 物理学报 53 4021]

    [25]

    Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170

    [26]

    Cui J C, Zhang Y Y, Jia L Q 2009 Chin. Phys. B 18 1731

    [27]

    Xie Y L, Jia L Q 2010 Chin Phys. Lett. 27 120201

    [28]

    Zheng S W, Xie J F, Chen X W 2010 Acta Phys. Sin. 59 5209 (in Chinese) [郑世旺, 解加芳, 陈向伟 2010 物理学报 59 5209]

    [29]

    Li Y C, Wang X M, Xia L L 2010 Acta Phys. Sin. 59 2935 (in Chinese) [李元成, 王小明, 夏丽莉 2010 物理学报 59 2935]

    [30]

    Jia L Q, Sun X T, Zhang M L, Wang X X, Xie Y L 2011 Acta Phys. Sin. 60 084501 (in Chinese) [贾利群, 孙现亭, 张美玲, 王肖肖, 解银丽 2011 物理学报 60 084501]

    [31]

    Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 475

    [32]

    Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117

    [33]

    Luo S K, Li Z J, Li L 2012 Acta Mech. 223 2621

    [34]

    Luo S K, Li Z J, Peng W, Li L 2013 Acta Mech. 224 71

    [35]

    Han Y L, Wang X X, Zhang M L, Jia L Q 2013 Nonlinear Dyn. 71 401

    [36]

    Mei F X 2003 Acta Phys. Sin. 52 1048 (in Chinese) [梅凤翔 2003 物理学报 52 1048]

    [37]

    Zhang Y, Fan C X, Ge W K 2004 Acta Phys. Sin. 53 3644 (in Chinese) [张毅, 范存新, 葛伟宽 2008 物理学报 53 3644]

    [38]

    Fang J H, Liu Y K, Zhang X N 2008 Chin. Phys. B 17 1962

    [39]

    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 2010 物理学报 59 7552]

    [40]

    Zhao L, Fu J L, Chen B Y 2011 Chin. Phys. B 20 040201

    [41]

    Han Y L, Sun X T, Wang X X, Zhang M L, Jia L Q 2012 Chin. Phys. B 21 120201

    [42]

    Mei F X 2004 Beijing Institute of Technology Press, Beijing

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  • Received Date:  28 December 2012
  • Accepted Date:  23 January 2013
  • Published Online:  05 June 2013

A type of the new exact and approximate conserved quantity deduced from Mei symmetry for a weakly nonholonomic system

  • 1. School of Science, Jiangnan University, Wuxi 214122, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11142014), and the scientific research and innovation plan for College Graduates of Jiangsu province, China (Grant No. CXLX12_0720).

Abstract: A type of structural equation, new exact and approximate conserved quantity which are deduced from Mei symmetry of Lagrange equations for a weakly nonholonomic system, are investigated. First, Lagrange equations of weakly nonholonomic system are established. Next, under the infinitesimal transformations of Lie groups, the definition and the criterion of Mei symmetry for Lagrange equations in weakly nonholonomic systems and its first-degree approximate holonomic system are given. And then, the expressions of new structural equation and new exact and approximate conserved quantities of Mei symmetry for Lagrange equations in weakly nonholonomic systems are obtained. Finelly, an example is given to study the question of the exact and the approximate new conserved quantities.

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