Non-Noether symmetries of Hamiltonian systems withconformable fractional derivatives

doi:10.1088/1674-1056/25/1/014501

Abstract In this paper, we present the fractional Hamilton's canonical equations and the fractional non-Noether symmetry of Hamilton systems by the conformable fractional derivative. Firstly, the exchanging relationship between isochronous variation and fractional derivatives, and the fractional Hamilton principle of the system under this fractional derivative are proposed. Secondly, the fractional Hamilton's canonical equations of Hamilton systems based on the Hamilton principle are established. Thirdly, the fractional non-Noether symmetries, non-Noether theorem and non-Noether conserved quantities for the Hamilton systems with the conformable fractional derivatives are obtained. Finally, an example is given to illustrate the results.

Keywords: conformable fractional derivative Hamilton's canonical equation non-Noether conserved quantity

Received: 26 June 2015
Revised: 28 August 2015
Accepted manuscript online:

PACS:	45.10.Hj	(Perturbation and fractional calculus methods)
	02.30.Xx	(Calculus of variations)
	11.10.Ef	(Lagrangian and Hamiltonian approach)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272287 and 11472247), the Program for Changjiang Scholars and Innovative Research Team in University, China (Grant No. IRT13097), and the Key Science and Technology Innovation Team Project of Zhejiang Province, China (Grant No. 2013TD18).

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

Cite this article:

Lin-Li Wang (王琳莉) and Jing-Li Fu(傅景礼) Non-Noether symmetries of Hamiltonian systems withconformable fractional derivatives 2016 Chin. Phys. B 25 014501

[1]	Noether A E 1918 Math. Phys. KI II 235
[2]	Li Z P 1981 Acta Phys. Sin. 30 1659 (in Chinese)
[3]	Mei F X 2000 J. Beijing Inst. Technol. 9 120
[4]	Mei F X 2001 Chin. Phys. 10 177
[5]	Hojman S A 1992 J. Phys. A: Math. Gen. 25 L291
[6]	Gonzaler-Cascon F 1994 J. Phys. A: Math. Gen. 27 L59
[7]	Lutzky M 1995 J. Phys. A: Math. Gen. 28 L637
[8]	Mei F X 2002 Chin. Sci. Bull. 47 1544 (in Chinese)
[9]	Mei F X 2003 Acta Phys. Sin. 52 1048 (in Chinese)
[10]	Fu J L, Chen L Q and Xie F P 2004 Chin. Phys. Soc. 13 1611
[11]	Fang J H, Liao Y P and Peng Y 2004 Chin. Phys. Soc. 13 1620
[12]	Zhou S, Fu H and Fu J L 2011 Sci. Chin. Phys. Mech. Astron. 54 1847
[13]	Zhang S H, Chen B Y and Fu J L 2012 Chin. Phys. B 21 100202
[14]	Zhou Y and Zhang Y 2014 Chin. Phys. B 23 124502
[15]	Sun Y, Chen B Y and Fu J L 2014 Chin. Phys. B 23 110201
[16]	Zaslavsky G M 2005 Hamiltonian Chaos and Fractional Dynamics (Oxford: Oxford University Press)
[17]	Tarasov V E 2010 Fractional Dynamics (Beijing: Higher Education Press)
[18]	Oldham K B and Spanier J 1974 The Fractional Calculus (New York: Academic Press)
[19]	Podlubny I 1999 Fractional Differential Equations (San Diego CA: Academic Press)
[20]	Miller K S and Ross B 1993 An Introduction to the Fractional Calculus and Fractional Differential Equations (New York: A Wiley-Interscience Publication)
[21]	Hilf R 2000 Applications of Fractional Calculus in Physics (Singapore: World Scientific Publishing Company)
[22]	Samko S G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives-Theory and Applications (Linghorne: Gordon and Breach Science Publishers)
[23]	Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations (Amsterdam: Elsevier B V)
[24]	Zaslavsky G M 2002 Phys. Rep. 371 461
[25]	Zaslavsky G M and Edelman M A 2004 Physica D 193 128
[26]	Yang J H 2012 Chin. Phys. Lett. 29 104501
[27]	Tan C, Liang Z S and Zhang J Q 2014 Acta Phys. Sin. 63 200502 (in Chinese)
[28]	Wang F, Xie T T, Deng C and Luo M K 2014 Acta Phys. Sin. 63 160502 (in Chinese)
[29]	Zhou X W, Lin L F, Ma H and Luo M K 2014 Acta Phys. Sin. 63 160503 (in Chinese)
[30]	Tu Z, Lai L and Luo M K 2014 Acta Phys. Sin. 63 120503 (in Chinese)
[31]	Wang F Q and Ma X K 2013 Chin. Phys. B 22 030506
[32]	Riewe F 1996 Phys. Rev. E 53 1890
[33]	Riewe F 1997 Phys. Rev. E 55 3581
[34]	Agrawal O P 2002 J. Math. Anal. Appl. 272 368
[35]	Agrawal O P 2004 Nonlinear Dyn. 38 323
[36]	Zhang Y 2012 Chin. Phys. B 21 084502
[37]	Wang L L and Fu J L 2014 Chin. Phys. B 23 124501
[38]	Khalil R, Al Horani M, Yousef A and Sababheh M 2014 J. Comput. Appl. Math. 264 65
[39]	Abdeljawad T 2015 J. Comput. Appl. Math. 279 57

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Non-Noether symmetries of Hamiltonian systems withconformable fractional derivatives

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