Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

doi:10.1088/1674-1056/22/10/104503

Abstract In this paper, we investigate the Noether symmetry and Noether conservation law of elastic rod dynamics with two independent variables: time t and arc coordinate s. Starting from the Lagrange equations of Cosserat rod dynamics, the criterion of Noether symmetry with Lagrange style for rod dynamics is given and the Noether conserved quantity is obtained. Not only are the conservations of generalized moment and generalized energy obtained, but also some other integrals.

Keywords: analytical mechanics Noether symmetry conservation laws Cosserat elastic rod dynamics

Received: 15 January 2013
Revised: 09 April 2013
Accepted manuscript online:

PACS:	45.20.Jj	(Lagrangian and Hamiltonian mechanics)
	02.20.Sv	(Lie algebras of Lie groups)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11262019 and 10972143).

Corresponding Authors: Liu Yu-Lu
E-mail: ylliu@staff.shu.edu.cn

Cite this article:

Wang Peng (王鹏), Xue Yun (薛纭), Liu Yu-Lu (刘宇陆) Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod 2013 Chin. Phys. B 22 104503

[1]	Benham C J 1977 Proc. Natl. Acad. Sci. USA 74 2397
[2]	Shi Y M and Hearst J E 1994 J. Chem. Phys. 101 5186
[3]	Zhou H J and Ouyang Z C 1999 J. Chem. Phys. 110 1247
[4]	Goriely A and Tabor M 2000 Nonlinear Dyn. 21 101
[5]	Liu Y Z 2006 Nonlinear Mechanics of Thin Elastic Rod – Theoretical Basis of Mechanical Model of DNA (Beijing: Tsinghua University Press and Springer) pp. 124-131 (in Chinese)
[6]	Cao D Q and Tucker R W 2008 Int. J. Solids Structures 45 460
[7]	Xue Y and Liu Y Z 2009 Acta Phys. Sin. 58 6737 (in Chinese)
[8]	Wang W, Zhang Q C and Jin G 2012 Acta Phys. Sin. 61 064602 (in Chinese)
[9]	Noether E 1918 Kgl. Ges. Wiss. Nachr. Göttingen Math. Phys. 235; Chinese translation: Mei F X 2004 Advances in Mechanics 34 116
[10]	Mei F X 2000 Appl. Mech. Rev. 53 283
[11]	Hydon P E 2001 J. Phys. A: Math. Gen. 34 10347
[12]	Fu J L, Chen B Y and Chen L Q 2009 Phys. Lett. A 373 409
[13]	Ge W K, Zhang Y and Xue Y 2010 Acta Phys. Sin. 59 4434 (in Chinese)
[14]	Lou Z M and Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese)
[15]	Zhang S H, Chen B Y and Fu J L 2012 Chin. Phys. B 21 100202
[16]	Xia L L and Chen L Q 2012 Chin. Phys. B 21 070202
[17]	Wang P 2012 Nonlinear Dyn. 68 53
[18]	Liu S X, Song D, Jia L, Liu C and Guo Y X 2013 Acta Phys. Sin. 62 034501 (in Chinese)
[19]	Lutzky M 1979 J. Phys. A: Math. Gen. 12 973
[20]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[21]	Chen X W, Li Y M and Zhao Y H 2005 Phys. Lett. A 337 274
[22]	Zhang H B, Lü H S and Gu S L 2010 Acta Phys. Sin. 59 5213 (in Chinese)
[23]	Fang J H 2010 Chin. Phys. B 19 040301
[24]	Li Z J, Jiang W A and Luo S K 2012 Nonlinear Dyn. 67 445
[25]	Jiang W A and Luo S K 2012 Nonlinear Dyn. 67 475
[26]	Zheng S W, Wang J B, Chen X W, Li Y M and Xie J F 2012 Acta Phys. Sin. 61 111101 (in Chinese)
[27]	Wang X X, Han Y L, Zhang M L and Jia L Q 2013 Chin. Phys. B 22 020201
[28]	Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)
[29]	Fang J H, Zhang M J and Zhang W W 2010 Phys. Lett. A 374 1806
[30]	Jia L Q, Xie Y L and Luo S K 2011 Acta Phys. Sin 60 040201 (in Chinese)
[31]	Wang P, Xue Y and Liu Y L 2012 Chin. Phys. B 21 070203
[32]	Mei F X, Xu X J and Qin M C 2004 Acta Mech. Sin. 20 668
[33]	Liu X W and Li Y C 2011 Chin. Phys. B 20 070204
[34]	Liu C, Mei F X and Guo Y X 2008 Acta Phys. Sin. 57 6704 (in Chinese)
[35]	Zhang Y 2009 Chin. Phys. B 18 4636
[36]	Cai J L, Shi S S, Fang H J and Xu J 2012 Meccanica 47 63
[37]	Chen R and Xu X J 2012 Chin. Phys. B 21 094501
[38]	Guo Y X, Jiang L Y and Yu Y 2001 Chin. Phys. 10 181
[39]	Zhang Y 2011 Chin. Phys. B 20 034502
[40]	Bluman G W and Anco S C 2002 Symmetry and Integration Methods for Differential Equation (Berlin: Springer-Verlag)
[41]	Maddocks J H and Dichmann D J 1994 J. Elasticity 34 83
[42]	Coleman B D, Dill E H and Seigond D A 1995 Arch. Rational Mech. Anal. 129 147
[43]	Jung P, Leyendecker S, Linn J and Ortiz M 2011 Int. J. Numer. Meth. Engng. 85 31
[44]	Xue Y and Wang P 2011 Acta Phys. Sin. 60 114501 (in Chinese)
[45]	Langer J and Singer D A 1996 SIAM Rev. 38 605
[46]	Xue Y, Liu Y Z and Chen L Q 2005 Acta Mech. Sin. 37 485 (in Chinese)
[47]	Xue Y, Liu Y Z and Chen L Q 2006 Acta Phys. Sin. 55 3845 (in Chinese)
[48]	Xue Y, Weng D W and Chen L Q 2013 Acta Phys. Sin. 62 044601 (in Chinese)
[49]	Liu Y Z and Xue Y 2011 Appl. Math. Mech. Engl. Ed. 32 603

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Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod

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