Lie symmetry theorem of fractional nonholonomic systems

doi:10.1088/1674-1056/23/11/110201

Abstract The Lie symmetry theorem of fractional nonholonomic systems in terms of combined fractional derivatives is established, and the fractional Lagrange equations are obtained by virtue of the d'Alembert-Lagrange principle with fractional derivatives. As the Lie symmetry theorem is based on the invariance of differential equations under infinitesimal transformations, by introducing the differential operator of infinitesimal generators, the determining equations are obtained. Furthermore, the limit equations, the additional restriction equations, the structural equations, and the conserved quantity of Lie symmetry are acquired. An example is presented to illustrate the application of results.

Keywords: Lie symmetry conserved quantity fractional nonholonomic systems

Received: 11 March 2014
Revised: 11 July 2014
Accepted manuscript online:

PACS:	02.20.-a	(Group theory)
	45.20.Jj	(Lagrangian and Hamiltonian mechanics)
	45.10.Hj	(Perturbation and fractional calculus methods)
	02.30.Xx	(Calculus of variations)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272287 and 11472247) and the Program for Changjiang Scholars and Innovative Research Team in University of China (Grant No. IRT13097).

Corresponding Authors: Fu Jing-Li
E-mail: sqfujingli@163.com

Cite this article:

Sun Yi (孙毅), Chen Ben-Yong (陈本永), Fu Jing-Li (傅景礼) Lie symmetry theorem of fractional nonholonomic systems 2014 Chin. Phys. B 23 110201

[1]	Mei F X 1999 Applications of Lei Group and Lie Algebra to Constrained Mechanics Systems (Beijing: Science Press) (in Chinese)
[2]	Camci U and Kucukakca Y 2007 Phys. Rev. D 76 084023
[3]	Belmonte-Beitia J, Pérez-García V M and Vekslerchik V 2007 Phys. Rev. Lett. 98 064102
[4]	Mei F X 2000 Acta Phys. Sin. 49 1207 (in Chinese)
[5]	Zhang R C, Chen X W and Mei F X 2000 Chin. Phys. 9 561
[6]	Mei F X and Wu H B 2009 Dynamics of Constrained Mechanical Systems (Beijing: Institute of Technology Press) (in Chinese)
[7]	Mei F X and Shang M 2000 Acta Phys. Sin. 49 1901 (in Chinese)
[8]	Zhang H B 2001 Acta Phys. Sin. 50 1837 (in Chinese)
[9]	Zhang R C, Chen X W and Mei F X 2001 Chin. Phys. 10 12
[10]	Zhang Y and Xue Y 2001 Acta Phys. Sin. 50 816 (in Chinese)
[11]	Mei F X 2001 Chin. Phys. 10 177
[12]	Zhao L, Fu J L and Chen B Y 2011 Chin. Phys. B 20 040201
[13]	Zhang Y 2011 Chin. Phys. B 20 034502
[14]	Fu J L and Chen B Y 2009 Mod. Phys. Lett. B 23 1315
[15]	Fu J L and Chen L Q 2004 Phys. Lett. A 331 138
[16]	Zhang R C, Chen X W and Mei F X 2000 Chin. Phys. 9 801
[17]	Olver P J 1993 Applications of Lie Groups to Differential Equations (New York: Springer)
[18]	Chen X W, Zhao Y H and Li Y M 2009 Chin. Phys. B 18 3139
[19]	Fu J L, Chen B Y, Fu H, Zhao G L, Liu R W and Zhu Z Y 2011 Sci. Chin. Phys. Mech. Astron. 54 288
[20]	Dorodnitsyn V, Kozlov R and Winternitz P 2000 J. Math. Phys. 41 480
[21]	Fu J L, Chen L Q and Bai J H 2003 Chin. Phys. 12 695
[22]	Martins N and Torres D F M 2010 Appl. Math. Lett. 23 1432
[23]	Fu J L, Li X W, Li C R, Zhao W J and Chen B Y 2010 Sci. Chin. Phys. Mech. Astron. 53 1699
[24]	Fu J L, Chen L Q and Chen B Y 2010 Sci. Chin. Phys. Mech. Astron. 53 545
[25]	Fu J L, Chen B Y and Chen L Q 2009 Phys. Lett. A 373 409
[26]	Zhou S, Fu H and Fu J L 2011 Sci. Chin. Phys. Mech. Astron. 54 1847
[27]	Zhang S H, Chen B Y and Fu J L 2012 Chin. Phys. B 21 100202
[28]	Atanacković T M, Konjik S, Pilipović S and Simić S 2009 Nonlinear Anal. 71 1504
[29]	Frederico G S F and Torres D F M 2007 J. Math. Anal. Appl. 334 834
[30]	Rabei E M, Nawafleh K I, Hijjawi R S, Muslih S I and Baleanu D 2007 J. Math. Anal. Appl. 327 891
[31]	Stanislavsky A A 2006 Eur. Phys. J. B 49 93
[32]	Zhang Y 2012 Chin. Phys. B 21 084502
[33]	Baleanu D, Muslih S I and Rabei E M 2008 Nonlinear Dyn. 53 67
[34]	Zhou S, Fu J L and Liu Y S 2010 Chin. Phys. B 19 120301
[35]	Agrawal O P 2002 J. Math. Anal. Appl. 272 368
[36]	Riewe F 1996 Phys. Rev. E 53 1890
[37]	Riewe F 1997 Phys. Rev. E 55 3581
[38]	Agrawal O P 2007 J. Phys. A: Math. Theor. 40 6287
[39]	Cresson J 2007 J. Math. Phys. 48 033504
[40]	Jefferson G F and Carminati J 2014 Comput. Phys. Commun. 185 430

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