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Chin. Phys. B, 2011, Vol. 20(3): 030202    DOI: 10.1088/1674-1056/20/3/030202
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Mei symmetries and Mei conserved quantities for higher-order nonholonomic constraint systems

Jiang Wen-An(姜文安), Li Zhuang-Jun(李状君), and Luo Shao-Kai(罗绍凯)
Institute of Mathematical Mechanics and Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
Abstract  This paper presents the Mei symmetries and new types of non-Noether conserved quantities for a higher-order nonholonomic constraint mechanical system. On the basis of the form invariance of differential equations of motion for dynamical functions under general infinitesimal transformation, the determining equations, the constraint restriction equations and the additional restriction equations of Mei symmetries of the system are constructed. The criterions of Mei symmetries, weak Mei symmetries and strong Mei symmetries of the system are given. New types of conserved quantities, i.e. the Mei symmetrical conserved quantities, the weak Mei symmetrical conserved quantities and the strong Mei symmetrical conserved quantities of a higher-order nonholonomic system, are obtained. Then, a deduction of the first-order nonholonomic system is discussed. Finally, two examples are given to illustrate the application of the method and then the results.
Keywords:  higher-order nonholonomic system      Mei symmetry      Mei conserved quantity  
Received:  21 September 2010      Revised:  20 October 2010      Accepted manuscript online: 
PACS:  02.20.Sv (Lie algebras of Lie groups)  
  11.30.-j (Symmetry and conservation laws)  
  03.50.Kk (Other special classical field theories)  
  11.10.Ef (Lagrangian and Hamiltonian approach)  
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10372053).

Cite this article: 

Jiang Wen-An(姜文安), Li Zhuang-Jun(李状君), and Luo Shao-Kai(罗绍凯) Mei symmetries and Mei conserved quantities for higher-order nonholonomic constraint systems 2011 Chin. Phys. B 20 030202

[1] Hertz H R 1894 Die Prinzipien der Mechanik (Leibzing: Gesammelte Werke) p37
[2] Niu Q P 1964 Acta Mech. Sin. 7 139 (in Chinese)
[3] Mei F X 2000 Appl. Mech. Rev. 53 283
[4] Li Z P 1981 Acta Phys. Sin. 30 1659 (in Chinese)
[5] Ge Z M and Chen Y H 1982 J. Appl. Mech. 49 429
[6] Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)
[7] Zhao G K and Zhao Y Y 1985 Appl. Math. Mech. 6 1101 (in Chinese)
[8] Luo S K 1988 J. Shangqiu Norm. Univ. 4 3 (in Chinese)
[9] Luo S K 1988 J. Shangqiu Norm. Univ. 4 20 (in Chinese)
[10] Mu X W and Guo Z H 1989 Sci. Chin. Ser. A 19 949 (in Chinese)
[11] Zhang J F 1989 Chin. Sci. Bull. 34 1756 (in Chinese)
[12] Chen L Q 1990 Chin. Sci. Bull. 35 1836
[13] Luo S K 1990 Proc. ICDVC (Beijing: Peking University Press) p645
[14] Zhang Y 1991 J. Tsinghua Univ. 31 108 (in Chinese)
[15] Chen B 1991 Acta Mech. Sin. 23 379 (in Chinese)
[16] Li Z P 1993 Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properies (Beijing: Beijing Polytechnic University Press) (in Chinese)
[17] Liang L F 1993 Acta Mech. Sol. Sin. 6 357
[18] Ge W K and Mei F X 1993 J. Beijing Inst. Technol. 14 457 (in Chinese)
[19] Fang J H and Li Y C 1994 J. Xinjiang Norm. Univ. 13 53 (in Chinese)
[20] Luo S K 1994 Acta Math. Sci. 14 168 (in Chinese)
[21] Li Z P and Jiang J H 2002 Symmetries in Constrained Canonical Systems (Beijing: Science Press)
[22] Li A M, Zhang Y and Li Z P 2004 Acta Phys. Sin. 53 2816 (in Chinese)
[23] Liang L F and Hu H C 2007 Sci. Chin. Ser. G 37 76 (in Chinese)
[24] Liu Y Z 1995 Spacecraft Attitude Dynamics (Beijing: National Defense Industry Press) (in Chinese)
[25] Ge X S, Chen L Q and Liu Y Z 2006 Engine. Mech. 23 63 (in Chinese)
[26] Ostrovskaya S and Angels J 1998 Appl. Mech. Rev. 51 415
[27] Gao F Z, Yuan F S and Gao C C 2009 Fuzzy Systems Maths. 23 158 (in Chinese)
[28] Zhu Y, Zhang T and Song J Y 2010 Contr. Theor. Appl. 27 152 (in Chinese)
[29] Fu Y L, Zheng Z W, Zhang F H and Wang S G 2010 Machinery Design & Manufacture 4 153 (in Chinese)
[30] Papastavridis J G 1998 Appl. Mech. Rev. 51 239
[31] Noether A E 1918 Nachr. Akad. Wiss. Gottingen: Math. Phys. 2 235
[32] Lutzky M 1979 J. Phys. A 12 973
[33] Lutzky M 1979 J. Math. Phys. A: Math. Gen. 19 105
[34] Liu D 1990 Sci. Chin. Ser. A 11 1189
[35] Borner H G and Davidson W F 1978 Phys. Rev. Lett. 40 167
[36] Aguirre M and Krause J 1988 J. Math. Phys. 29 9
[37] Fuchs J C 1991 J. Math. Phys. 32 1703
[38] Ostrovsky V N and Prudov N V 1995 J. Phys. B 20 4435
[39] Luo S K 1991 Chin. Sci. Bull. 36 1930
[40] Mei F X 1993 Sci. Chin. Ser. A 36 1456
[41] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[42] Zhao Y Y and Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) (in Chinese)
[43] Mei F X 2004 Symmetry and Conserved Quantity of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)
[44] Luo S K and Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese)
[45] Mei F X 1998 Chin. Sci. Bull. 43 1937
[46] Zhang Y and Xue Y 2001 Acta Phys. Sin. 50 816 (in Chinese)
[47] Zhang H B, Chen L Q, Liu R W and Gu S L 2005 Acta Phys. Sin. 54 2489 (in Chinese)
[48] Luo S K 2003 Chin. Phys. 12 841
[49] Luo S K and Jia L Q 2003 Commun. Theor. Phys. 40 265
[50] Luo S K 2003 Chin. Phys. Lett. 20 597
[51] Fan J H, Peng Y, Liao Y P and Li H 2004 Commun. Theor. Phys. 42 440
[52] Cai J L, Luo S K and Mei F X 2008 Chin. Phys. B 17 3170
[53] Xie J F, Gang T Q and Mei F X 2008 Chin. Phys. B 17 3175
[54] Chen X W, Liu C and Mei F X 2008 Chin. Phys. B 17 3180
[55] Chen X W, Zhao Y H and Li Y M 2009 Chin. Phys. B 18 3139
[56] Xia L L and Cai J L 2010 Chin. Phys. B 19 040302
[57] Mei F X 2000 J. Beijing Inst. Technol. 9 120
[58] Luo S K 2003 Acta Phys. Sin. 52 2941 (in Chinese)
[59] Fan J H 2003 Commun. Theor. Phys. 40 269
[60] Chen X W, Luo S K and Mei F X 2002 Appl. Math. Mech. 23 47 (in Chinese)
[61] Wang S Y and Mei F X 2001 Chin. Phys. 10 373
[62] Luo S K 2002 Chin. Phys. Lett. 19 449
[63] Luo S K 2002 Commun. Theor. Phys. 38 257
[64] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese)
[65] Fang J H, Liao Y P and Peng Y 2005 Acta Phys. Sin. 54 496 (in Chinese)
[66] Ge W K and Zhang Y 2006 Acta Phys. Sin. 55 4985 (in Chinese)
[67] Xia L L, Li Y C, Wang J and Hou Q B 2006 Commun. Theor. Phys. 46 415
[68] Zheng S W and Jia L Q 2007 Acta Phys. Sin. 56 661 (in Chinese)
[69] Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese)
[70] Ding N and Fang J H 2009 Acta Phys. Sin. 58 7740 (in Chinese)
[71] Cui J C, Zhang Y Y and Jia L Q 2010 Chin. Phys. B 19 030304
[72] Cai J L 2010 Int. J. Theor. Phys. 49 201
[73] Zhang Z Y, Yong X L and Chen Y F 2009 Chin. Phys. B 18 2629
[74] Yang X F, Jia L Q, Cui J C and Luo S K 2010 Chin. Phys. B 19 030305
[75] Zhang Y 2010 Chin. Phys. B 19 080301 endfootnotesize
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