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Noether-Lie symmetry and conserved quantities of the Rosenberg problem

Liu Xiao-Wei Li Yuan-Cheng

Noether-Lie symmetry and conserved quantities of the Rosenberg problem

Liu Xiao-Wei, Li Yuan-Cheng
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  • The Noether-Lie symmetry and conserved quantities of the Rosenberg problem are studied. From the study of the Rosenberg problem,the Noether symmetry and the Lie symmetry for the equation are obtained, thereby the conserved quantities are deduced. Then the definition and the criterion for Noether-Lie symmetry of the Rosenberg problem are derived. Finally,the Noether conserved quantity and the Hojman conserved quantity are deduced from the Noether-Lie symmetry.
    [1]

    Noether A E 1918 Nachr. Akad. Wiss. Gttingen.Math. Phys. KI II 235

    [2]

    Lutzky M 1979 J. Phys. A: Math. Gen. 12 973

    [3]

    Mei F X 2000 J. Beijing Inst. Technol. 9 120

    [4]

    Mei F X 2001 Chin. Phys. 10 177

    [5]

    Li Z P 1993 Classical and quantal dynamics of constrained systems and Their symmetrical properties (Beijing: Beijing Polytechnic University press) (in Chinese) [李子平 1993 经典和量子约束系统及其对称性质 (北京:北京工业大学出社)]

    [6]

    Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (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)]

    [7]

    Bahar L Y,Kwatny H G 1987 Int. J. Non-Linear Mech. 22 125

    [8]

    Mei F X 2000 Acta Mech. Sin. 32 466 (in Chinese)[梅凤翔 2000 力学学报32 466]

    [9]

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

    [10]

    Zhang Y 2003 Acta Phys. Sin. 52 1832 (in Chinese) [张 毅 2003 物理学报 52 1832]

    [11]

    Wang S Y, Mei F X 2001 Chin. Phys. 10 373

    [12]

    Lou Z M 2004 Acta Phys. Sin. 53 2046 (in Chinese) [楼智美 2004 物理学报53 2046]

    [13]

    Luo S K,Guo Y X,Mei F X 2004 Acta Phys. Sin. 53 2413 (in Chinese) [罗绍凯、郭永新、梅凤翔 2004 物理学报 53 2413]

    [14]

    Hojman S A 1992 J. Phys. A: Math. Gen. 25 L291

    [15]

    Xu X J,Mei F X,Qin M C 2004 Chin. Phys. 13 1999

    [16]

    Mei F X 2005 Transactions of Beijing Institute of Technology 25 283(in Chinese) [梅凤翔 2005 北京理工大学学报 25 283]

    [17]

    Li Y C,Xia L L,Wang X M,Liu X W 2010 Acta Phys. Sin. 59 3639 (in Chinese) [李元成、夏丽莉、王小明、刘晓巍 2010 物理学报 59 3639]

    [18]

    Rosenberg R M 1977 Analytical Dynamics of Discrete Systems (New York: Plenum Press)

    [19]

    Ge W H,Zhang Y,Xue Y 2010 Acta Phys. Sin. 59 4434 (in Chinese) [葛伟宽、张 毅、薛 纭 2010 物理学报 59 4434]

    [20]

    Novoselov V S 1966 Variational Priciples in Mechanics (Leningrad: LGV Press) (in Russian)

    [21]

    Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)[梅凤翔 1985 非完整力学基础 (北京:北京工业学院出版社)]

  • [1]

    Noether A E 1918 Nachr. Akad. Wiss. Gttingen.Math. Phys. KI II 235

    [2]

    Lutzky M 1979 J. Phys. A: Math. Gen. 12 973

    [3]

    Mei F X 2000 J. Beijing Inst. Technol. 9 120

    [4]

    Mei F X 2001 Chin. Phys. 10 177

    [5]

    Li Z P 1993 Classical and quantal dynamics of constrained systems and Their symmetrical properties (Beijing: Beijing Polytechnic University press) (in Chinese) [李子平 1993 经典和量子约束系统及其对称性质 (北京:北京工业大学出社)]

    [6]

    Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (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)]

    [7]

    Bahar L Y,Kwatny H G 1987 Int. J. Non-Linear Mech. 22 125

    [8]

    Mei F X 2000 Acta Mech. Sin. 32 466 (in Chinese)[梅凤翔 2000 力学学报32 466]

    [9]

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

    [10]

    Zhang Y 2003 Acta Phys. Sin. 52 1832 (in Chinese) [张 毅 2003 物理学报 52 1832]

    [11]

    Wang S Y, Mei F X 2001 Chin. Phys. 10 373

    [12]

    Lou Z M 2004 Acta Phys. Sin. 53 2046 (in Chinese) [楼智美 2004 物理学报53 2046]

    [13]

    Luo S K,Guo Y X,Mei F X 2004 Acta Phys. Sin. 53 2413 (in Chinese) [罗绍凯、郭永新、梅凤翔 2004 物理学报 53 2413]

    [14]

    Hojman S A 1992 J. Phys. A: Math. Gen. 25 L291

    [15]

    Xu X J,Mei F X,Qin M C 2004 Chin. Phys. 13 1999

    [16]

    Mei F X 2005 Transactions of Beijing Institute of Technology 25 283(in Chinese) [梅凤翔 2005 北京理工大学学报 25 283]

    [17]

    Li Y C,Xia L L,Wang X M,Liu X W 2010 Acta Phys. Sin. 59 3639 (in Chinese) [李元成、夏丽莉、王小明、刘晓巍 2010 物理学报 59 3639]

    [18]

    Rosenberg R M 1977 Analytical Dynamics of Discrete Systems (New York: Plenum Press)

    [19]

    Ge W H,Zhang Y,Xue Y 2010 Acta Phys. Sin. 59 4434 (in Chinese) [葛伟宽、张 毅、薛 纭 2010 物理学报 59 4434]

    [20]

    Novoselov V S 1966 Variational Priciples in Mechanics (Leningrad: LGV Press) (in Russian)

    [21]

    Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)[梅凤翔 1985 非完整力学基础 (北京:北京工业学院出版社)]

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    [2] Jia Li-Qun, Zheng Shi-Wang. Mei symmetry and conserved quantity of Tzénoff equations for nonholonomic systems. Acta Physica Sinica, 2007, 56(2): 661-665. doi: 10.7498/aps.56.661
    [3] Zhang Bin, Fang Jian-Hui, Zhang Ke-Jun. Symmetry and conserved quantity of Lagrangians for nonholonomic variable mass system. Acta Physica Sinica, 2012, 61(2): 021101. doi: 10.7498/aps.61.021101
    [4] Xu Chao, Li Yuan-Cheng. Noether-Lie symmetry and conserved quantities of Nielsen equations for a singular variable mass nonholonomic system with unilateral constraints. Acta Physica Sinica, 2013, 62(17): 171101. doi: 10.7498/aps.62.171101
    [5] Mei Feng-Xiang, Guo Yong-Xin, Luo Shao-Kai. Noether symmetry and Hojman conserved quantity for nonholonomic mechanical systems. Acta Physica Sinica, 2004, 53(5): 1270-1275. doi: 10.7498/aps.53.1270
    [6] Jia Li-Qun, Zhang Yao-Yu, Luo Shao-Kai. Mei symmetry and Mei conserved quantity of Nielsen equation for a nonholonomic system. Acta Physica Sinica, 2008, 57(4): 2006-2010. doi: 10.7498/aps.57.2006
    [7] Zhang Yi, Ge Wei-Kuan, Xue Yun. Symmetries and conserved quantities of the Rosenberg problem. Acta Physica Sinica, 2010, 59(7): 4434-4436. doi: 10.7498/aps.59.4434
    [8] Hu Chu-Le. Lie symmetries and Hojman conserved quantities of one kind of differential equations of motion of nonholonomic systems. Acta Physica Sinica, 2007, 56(7): 3675-3677. doi: 10.7498/aps.56.3675
    [9] Fang Jian-Hui, Ding Ning, Wang Peng. Noether-Lie symmetry of non-holonomic mechanical system. Acta Physica Sinica, 2006, 55(8): 3817-3820. doi: 10.7498/aps.55.3817
    [10] Jia Li-Qun, Zhang Yao-Yu, Zheng Shi-Wang. Mei symmetry and Mei conserved quantity of nonholonomic systems of non-Chetaev’s type in event space. Acta Physica Sinica, 2007, 56(10): 5575-5579. doi: 10.7498/aps.56.5575
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  • Received Date:  08 October 2010
  • Accepted Date:  16 October 2010
  • Published Online:  15 July 2011

Noether-Lie symmetry and conserved quantities of the Rosenberg problem

  • 1. College of Physics Science and Technology, China University of Petroleum (East China)Qingdao 266555,China

Abstract: The Noether-Lie symmetry and conserved quantities of the Rosenberg problem are studied. From the study of the Rosenberg problem,the Noether symmetry and the Lie symmetry for the equation are obtained, thereby the conserved quantities are deduced. Then the definition and the criterion for Noether-Lie symmetry of the Rosenberg problem are derived. Finally,the Noether conserved quantity and the Hojman conserved quantity are deduced from the Noether-Lie symmetry.

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