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Chin. Phys. B, 2013, Vol. 22(3): 030201    DOI: 10.1088/1674-1056/22/3/030201
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Mei symmetry and conservation laws of discrete nonholonomic dynamical systems with regular and irregular lattices

Zhao Gang-Linga b d, Chen Li-Quna b, Fu Jing-Lic, Hong Fang-Yuc
a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China;
b Department of Mechanics, Shanghai University, Shanghai 200444, China;
c Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China;
d Department of Physics and Information Engineering, Shangqiu Normal University, Shangqiu 476000, China
Abstract  In this paper, Noether symmetry and Mei symmetry of discrete nonholonomic dynamical systems with regular and the irregular lattices are investigated. Firstly, the equations of motion of discrete nonholonomic systems are introduced on the regular and rregular lattices. Secondly, for cases of the two lattices, based on the invariance of the Hamiltomian functional under the infinitesimal transformation of time and generalized coordinates, we present the quasi-extremal equation, the discrete analogues of Noether identity, Noether theorems, and the Noether conservation laws of the systems. Thirdly, in cases of the two lattices, we study the Mei symmetry in which we give the discrete analogues of the criterion, the theorem, and the conservative laws of Mei symmetry for the systems. Finally, an example is discussed for applications of the results.
Keywords:  regular and irregular lattice      nonholonomic system      Noether symmetry      Mei symmetry     
Received:  17 June 2012      Published:  01 February 2013
PACS:  02.20.-a (Group theory)  
  02.30.lk  
  11.30.-j (Symmetry and conservation laws)  
  45.05.+x (General theory of classical mechanics of discrete systems)  
Fund: Project supported by the National Outstanding Young Scientist Fund of China (Grant No. 10725209), the National Natural Science Foundation of China (Grant No. 11072218), and the Natural Science Foundation of Zhejiang Province, China (Grant No. Y6110314).
Corresponding Authors:  Fu Jing-Li     E-mail:  sqfujingli@163.com

Cite this article: 

Zhao Gang-Ling, Chen Li-Qun, Fu Jing-Li, Hong Fang-Yu Mei symmetry and conservation laws of discrete nonholonomic dynamical systems with regular and irregular lattices 2013 Chin. Phys. B 22 030201

[1] Lutzky M 1979 J. Phys. A: Math. Gen. 12 973
[2] Ibragimov N H 1985 Transformation Groups Applied to Mathematical Physics (Boston: Reidel)
[3] Hydon P 1999 Symmetry Methods for Ordinary Differential Equations (Berlin: Springer)
[4] Bluman G W and Kumei S 1989 Symmetries of Differential Equations (Berlin: Springer)
[5] Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer)
[6] Mei F X 1999 Applications of Lie Group and Lie Algebra to Constraint Mechanical Systems (Beijing: Science Press) (in Chinese)
[7] Zhao Y Y and Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) (in Chinese)
[8] Guo Y X, Jiang L Y and Yu Y 2001 Chin. Phys. 10 181
[9] Chen X W and Li Y M 2003 Chin. Phys. 12 1349
[10] Zhang H B, Chen L Q and Liu R W 2005 Chin. Phys. 14 1063
[11] Wu H B and Mei F X 2006 Acta Phys. Sin. 55 3825 (in Chinese)
[12] Lou Z M 2007 Acta Phys. Sin. 56 2475 (in Chinese)
[13] Lutzky M 1982 J. Phys. A: Math. Gen. 15 87
[14] Fu J L and Chen L Q 2003 Phys. Lett. A 317 255
[15] Ovisiannikov L V 1982 Group Analysis of Difference Equations (New York: Academic)
[16] Levi D, Vinet L and Winternitz P 1997 J. Phys. A: Math. Gen. 30 633
[17] Levi D and Martina L 2001 J. Phys. A: Math. Gen. 34 10357
[18] Levi D, Tremblay S and Winternitz P 2000 J. Phys. A: Math. Gen. 33 8507
[19] Levi D, Tremblay S and Winternitz P 2001 J. Phys. A: Math. Gen. 34 9507
[20] Dorodnitsyn V 1991 J. Sov. Math. 55 1490
[21] Dorodnitsyn V, Kozlov R and Winternitz P 2000 J. Math. Phys. 41 480
[22] Dorodnitsyn V 2001 Appl. Numer. Math. 39 307
[23] Dorodnitsyn V, Kozlov R and Winternitz P 2004 J. Math. Phys. 45 336
[24] Dorodnitsyn V 2011 Applications of Lie Groups to Difference Equations (New York: CRC Press)
[25] Guo H Y and Wu K 2003 J. Math. Phys. 44 5978
[26] Fu J L, Chen L Q and Chen B Y 2010 Sci. Chin. A: Phys. Mech. Astron. 53 545
[27] Fu J L, Chen B Y, Fu H, Zhao G L, Liu R W and Zhu Z Y 2011 Sci. Chin. A: Phys. Mech. Astron. 54 288
[28] Wang X Z, Fu H and Fu J L 2012 Chin. Phys. B 21 040201
[29] Mei F X 2000 J. Beijing Inst. Technol. 9 120
[30] Mei F X 2001 Chin. Phys. 10 177
[31] Mei F X and Chen X W 2001 J. Beijing Inst. Technol. 2 138
[32] Mei F X and Xu X J 2005 Chin. Phys. 14 449
[33] Fang J H, Ding N and Wang P 2007 Chin. Phys. 16 887
[34] Chen X W, Zhao Y H and Liu C 2009 Acta Phys. Sin. 58 5150 (in Chinese)
[35] Ge W K 2002 Acta Phys. Sin. 51 939 (in Chinese)
[36] Luo S K 2002 Acta Phys. Sin. 51 712 (in Chinese)
[37] Fang J H 2003 Commun. Theor. Phys. 40 269
[38] Zhang Y and Mei F X 2003 Chin. Phys. 12 1058
[39] Zhang Y 2004 Commun. Theor. Phys. 42 899
[40] Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese)
[41] Luo Y P and Fu J L 2011 Chin. Phys. B 20 021102
[42] Zheng S W, Jia L Q and Yu H S 2006 Chin. Phys. 15 1399
[43] Xia L L, Li Y C, Wang J and Hou Q B 2006 Commun. Theor. Phys. 46 415
[44] Wang X X, Sun X T, Zhang M L, Han Y L and Jia L Q 2012 Chin. Phys. B 21 050201
[45] Shi S Y, Chen L Q and Fu J L 2008 Commun. Theor. Phys. 50 607
[46] Shi S Y and Fu J L 2011 Chin. Phys. B 20 021101
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