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For a nonholonomic system of Chetaev's type, the conformal invariance and the conserved quantity are studied. By the infinitesimal one-parameter transformation group and the infinitesimal generator vector, the definition of conformal invariance of Mei symmetry and the determining equation for the holonomic system which corresponds to a nonholonomic system are provided, and the relationship between the system conformal invariance and Mei symmetry is discussed. Using the restriction equations and the additional restriction equations, the conformal invariances of weak and strong Mei symmetrys for the system are given. With the aid of a structure equation that gauge function satisfies, the system corresponding conserved quantity is derived. Finally, an example is given to illustrate the application of the result.
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Keywords:
- nonholonomic system /
- Mei symmetry /
- conformal invariance /
- conformal factor
[1] Noether A E 1918 Nachr. Akad. Math. 2 235
[2] Djukić D S, Vujanović B D 1975 Acta Mechanica 23 17
[3] Mei F X 1999 Applications of Lie Groups and Lie Algebras to ConstrainedMechanical 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)]
[4] Zhao Y Y, Mei F X 1999 Symmetries and Invariants of MechanicalSystems (Beijing: Science Press)(in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与不变量(北京:科学出版社)]
[5] Mei F X, Zheng G H 2002 Acta Mech. Sin. 18 414
[6] Ge W K 2002 Acta Phys. Sin. 51 1156 (in Chinese) [葛伟宽 2002 51 1156]
[7] Fang J H 2003 Commun. Theor. Phys. 40 269
[8] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of TechnologyPress) (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] Jiang W A, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075
[10] Li Z J, Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 445
[11] Luo S K 2007 Chin. Phys. Lett. 24 2463
[12] Jia L Q, Zhang Y Y, Zheng S W 2007 Acta Phys. Sin. 56 649 (in Chinese) [贾利群,张耀宇,郑世旺 2007 56 649]
[13] Luo S K 2007 Chin. Phys. Lett. 24 3017
[14] Luo S K , Zhang Y F 2008 Advances in the Study of Dynamics ofConstrained Systems (Beijing: Science Press) (in Chinese) [罗绍凯,张永发 2008 约束系统动力学研究进展(北京:科学出版社)]
[15] Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems(Moscow: UFN) (in Russian)
[16] Cai J L 2008 Chin. Phys. Lett. 25 1523
[17] Mei F X, Xie J F, Gang T Q 2008 Acta Mech. Sin. 24 583
[18] He G, Mei F X 2008 Chin. Phys. B 17 2764
[19] Cai J L, Mei F X 2008 Acta Phys. Sin. 57 5369 (in Chinese) [蔡建乐,梅凤翔 2008 57 5369]
[20] Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170
[21] Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[22] Cai J L 2009 Acta Phys. Pol. A 115 854
[23] Xia L L, Cai J L, Li Y C 2009 Chin. Phys. B 18 3158
[24] Luo Y P 2009 Int. J. Theor. Phys. 48 2665
[25] Li Y C, Xia L L, Wang X M 2009 Chin. Phys. B 18 4643
[26] Zhang M J, Fang J H, Lin P, Lu K, Pang T 2009 Commun. Theor.Phys. 52 561
[27] Chen X W, Li Y M, Zhao Y H 2009 Chin. Phys. B 18 3139
[28] Zhang Y 2009 Chin. Phys. B 18 4636
[29] Cai J L 2010 Acta Phys. Pol. A 117 445
[30] Xia L L, Cai J L 2010 Chin. Phys. B 19 040302
[31] Zhang Y 2010 Commun. Theor. Phys. 53 166
[32] Luo Y P, Fu J L 2010 Chin. Phys. B 19 090303
[33] Luo Y P, Fu J L 2010 Chin. Phys. B 19 090304
[34] Luo Y P, Fu J L 2011 Chin. Phys. B 20 021102
[35] Li Y, Fang J H, Zhang K J 2011 Chin. Phys. B 20 030201
[36] Cai J L 2010 Int. J. Theor. Phys. 49 201
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[1] Noether A E 1918 Nachr. Akad. Math. 2 235
[2] Djukić D S, Vujanović B D 1975 Acta Mechanica 23 17
[3] Mei F X 1999 Applications of Lie Groups and Lie Algebras to ConstrainedMechanical 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)]
[4] Zhao Y Y, Mei F X 1999 Symmetries and Invariants of MechanicalSystems (Beijing: Science Press)(in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与不变量(北京:科学出版社)]
[5] Mei F X, Zheng G H 2002 Acta Mech. Sin. 18 414
[6] Ge W K 2002 Acta Phys. Sin. 51 1156 (in Chinese) [葛伟宽 2002 51 1156]
[7] Fang J H 2003 Commun. Theor. Phys. 40 269
[8] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of TechnologyPress) (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] Jiang W A, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075
[10] Li Z J, Jiang W A, Luo S K 2012 Nonlinear Dyn. 67 445
[11] Luo S K 2007 Chin. Phys. Lett. 24 2463
[12] Jia L Q, Zhang Y Y, Zheng S W 2007 Acta Phys. Sin. 56 649 (in Chinese) [贾利群,张耀宇,郑世旺 2007 56 649]
[13] Luo S K 2007 Chin. Phys. Lett. 24 3017
[14] Luo S K , Zhang Y F 2008 Advances in the Study of Dynamics ofConstrained Systems (Beijing: Science Press) (in Chinese) [罗绍凯,张永发 2008 约束系统动力学研究进展(北京:科学出版社)]
[15] Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems(Moscow: UFN) (in Russian)
[16] Cai J L 2008 Chin. Phys. Lett. 25 1523
[17] Mei F X, Xie J F, Gang T Q 2008 Acta Mech. Sin. 24 583
[18] He G, Mei F X 2008 Chin. Phys. B 17 2764
[19] Cai J L, Mei F X 2008 Acta Phys. Sin. 57 5369 (in Chinese) [蔡建乐,梅凤翔 2008 57 5369]
[20] Cai J L, Luo S K, Mei F X 2008 Chin. Phys. B 17 3170
[21] Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[22] Cai J L 2009 Acta Phys. Pol. A 115 854
[23] Xia L L, Cai J L, Li Y C 2009 Chin. Phys. B 18 3158
[24] Luo Y P 2009 Int. J. Theor. Phys. 48 2665
[25] Li Y C, Xia L L, Wang X M 2009 Chin. Phys. B 18 4643
[26] Zhang M J, Fang J H, Lin P, Lu K, Pang T 2009 Commun. Theor.Phys. 52 561
[27] Chen X W, Li Y M, Zhao Y H 2009 Chin. Phys. B 18 3139
[28] Zhang Y 2009 Chin. Phys. B 18 4636
[29] Cai J L 2010 Acta Phys. Pol. A 117 445
[30] Xia L L, Cai J L 2010 Chin. Phys. B 19 040302
[31] Zhang Y 2010 Commun. Theor. Phys. 53 166
[32] Luo Y P, Fu J L 2010 Chin. Phys. B 19 090303
[33] Luo Y P, Fu J L 2010 Chin. Phys. B 19 090304
[34] Luo Y P, Fu J L 2011 Chin. Phys. B 20 021102
[35] Li Y, Fang J H, Zhang K J 2011 Chin. Phys. B 20 030201
[36] Cai J L 2010 Int. J. Theor. Phys. 49 201
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