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Chin. Phys. B, 2017, Vol. 26(8): 084501    DOI: 10.1088/1674-1056/26/8/084501
ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS Prev   Next  

Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales

Jing Song(宋静)1, Yi Zhang(张毅)2
1 College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
2 College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
Abstract  

This paper focuses on studying the Noether symmetry and the conserved quantity with non-standard Lagrangians, namely exponential Lagrangians and power-law Lagrangians on time scales. Firstly, for each case, the Hamilton principle based on the action with non-standard Lagrangians on time scales is established, with which the corresponding Euler-Lagrange equation is given. Secondly, according to the invariance of the Hamilton action under the infinitesimal transformation, the Noether theorem for the dynamical system with non-standard Lagrangians on time scales is established. The proof of the theorem consists of two steps. First, it is proved under the infinitesimal transformations of a special one-parameter group without transforming time. Second, utilizing the technique of time-re-parameterization, the Noether theorem in a general form is obtained. The Noether-type conserved quantities with non-standard Lagrangians in both classical and discrete cases are given. Finally, an example in Friedmann-Robertson-Walker spacetime and an example about second order Duffing equation are given to illustrate the application of the results.

Keywords:  time scale      non-standard Lagrangian      Noether symmetry      conserved quantity     
Received:  28 February 2017      Published:  05 August 2017
PACS:  45.20.Jj (Lagrangian and Hamiltonian mechanics)  
  11.30.Na (Nonlinear and dynamical symmetries (spectrum-generating symmetries))  
  45.10.Db (Variational and optimization methods)  
Fund: 

Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227) and the Innovation Program of Suzhou University of Science and Technology, China (Grant No. SKYCX16_012).

Corresponding Authors:  Yi Zhang     E-mail:  zhy@mail.usts.edu.cn
About author:  0.1088/1674-1056/26/8/

Cite this article: 

Jing Song(宋静), Yi Zhang(张毅) Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales 2017 Chin. Phys. B 26 084501

[1] Noether A E 1918 Nachr kgl Ges Wiss Göttingen. Math. Phys. KI II 235
[2] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[3] Luo S K, Guo Y X and Mei F X 2004 Acta Phys. Sin. 53 1271 (in Chinese)
[4] Shi S Y and Huang X H 2008 Chin. Phys. B 17 1554
[5] Zhang Y 2009 Chin. Phys. B 18 4636
[6] Jin S X and Zhang Y 2014 Chin. Phys. B 23 054501
[7] Zhou Y and Zhang Y 2015 J. Dyn. Control 13 410
[8] Arnold V I 1978 Mathematical methods of classical mechanics (New York: Springer)
[9] Alekseev A I and Arbuzov B A 1984 Theor. Math. Phys. 59 372
[10] Dimitrijevic D D and Milosevic M 2012 AIP Conf. Proc. 1472 41
[11] El-Nabulsi R A 2013 Can. J. Phys. 56 216
[12] El-Nabulsi R A 2013 J. Theor. Math. Phys. 7 1
[13] El-Nabulsi R A 2014 J. At. Mol. Sci. 5 268
[14] El-Nabulsi R A 2015 Math. 3 727
[15] El-Nabulsi R A 2015 Appl. Math. Lett. 43 120
[16] El-Nabulsi R A 2014 Nonlinear Dyn. 79 2055
[17] Musielak Z 2008 J. Phys. A: Math. Theor. 41 295
[18] Musielak Z 2009 Chaos, Solitons and Fractals 42 2645
[19] Cieslinski J and Nikiciuk T 2009 J. Phys. A: Math. Theor. 43 1489
[20] El-Nabulsi R A 2013 Qual. Theory Dyn. Syst. 12 273
[21] El-Nabulsi R A 2013 Nonlinear Dyn. 74 381
[22] El-Nabulsi R A, Soulati T and Rezazadeh H 2013 J. Adv. Res. Dyn. Control Theory 5 50
[23] El-Nabulsi R A 2014 Proc. Natl. Acad. Sci. India Sect. A: Phys. Sci. 84 563
[24] El-Nabulsi R A 2014 Comp. Appl. Math. 33 163
[25] Zhang Y and Zhou X S 2016 Nonlinear Dyn. 84 1867
[26] Zhou X S and Zhang Y 2016 Chin. Quart. Mech. 37 15
[27] Hilger S 1988 Ein makettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. thesis, (Universität Würzburg)
[28] Bohner M and Peterson A 2001 Dynamic equations on time scales: An introduction with applications (Boston: Birkhäuser)
[29] Bohner M and Peterson A 2003 Advances in dynamic equations on time scales (Boston: Birkhäuser)
[30] Ahlbrandt C, Bohner M and Ridenhour J 2000 J. Math. Anal. Appl. 250 561
[31] Bartosiewicz Z, Martins N and Torres D F M 2011 Eur. J. Control 17 9
[32] Malinowska A B and Ammi M R S 2014 Int. J. Differ. Equ. ISSN 0973
[33] Bartosiewicz Z and Torres D F M 2008 J. Math. Anal. Appl. 342 1220
[34] Malinowska A B and Martins N 2013 Abst. Appl. Anal. 2013 1728
[35] Zhang Y 2016 Chin. Q. Mech. 37 214
[36] Song C J and Zhang Y 2015 J. Math. Phys. 56 102701
[37] Zu Q H and Zhu J Q 2016 J. Math. Phys. 57 082701
[38] Jin S X and Zhang Y 2017 Chin. Phys. B 26 014501
[39] Atici F M, Biles D C and Lebedinsky A 2006 Math. Comp. Mod. 43 718
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