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

doi:10.1088/1674-1056/26/8/084501

中国物理B ›› 2017, Vol. 26 ›› Issue (8): 84501-084501.doi: 10.1088/1674-1056/26/8/084501

• ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS • 上一篇下一篇

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

Jing Song(宋静), Yi Zhang(张毅)

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

收稿日期:2017-02-28 修回日期:2017-04-10 出版日期:2017-08-05 发布日期:2017-08-05
通讯作者: Yi Zhang E-mail:zhy@mail.usts.edu.cn
基金资助:
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).

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

Received:2017-02-28 Revised:2017-04-10 Online:2017-08-05 Published:2017-08-05
Contact: Yi Zhang E-mail:zhy@mail.usts.edu.cn
About author:0.1088/1674-1056/26/8/
Supported by:
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).

摘要/Abstract

摘要：

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.

关键词: time scale, non-standard Lagrangian, Noether symmetry, conserved quantity

Abstract:

Key words: time scale, non-standard Lagrangian, Noether symmetry, conserved quantity

中图分类号: (Lagrangian and Hamiltonian mechanics)

45.20.Jj

11.30.Na (Nonlinear and dynamical symmetries (spectrum-generating symmetries)) 45.10.Db (Variational and optimization methods)

宋静, 张毅. Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales[J]. 中国物理B, 2017, 26(8): 84501-084501.

Jing Song(宋静), Yi Zhang(张毅). Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales[J]. Chin. Phys. B, 2017, 26(8): 84501-084501.

参考文献 39

[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

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

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

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 39

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Jian Zhang(张健), Yiming Liu(刘一鸣), and Zhanchun Tu(涂展春). Exploring fundamental laws of classical mechanics via predicting the orbits of planets based on neural networks[J]. 中国物理B, 2022, 31(9): 94502-094502.
[2]	金世欣, 张毅. Generalized Chaplygin equations for nonholonomic systems on time scales[J]. 中国物理B, 2018, 27(2): 20502-020502.
[3]	张冉, 彭淼, 张正娣, 毕勤胜. Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model[J]. 中国物理B, 2018, 27(11): 110501-110501.
[4]	韩青爽, 陈帝伊, 张浩. Bursting oscillations in a hydro-turbine governing system with two time scales[J]. 中国物理B, 2017, 26(12): 128202-128202.
[5]	金世欣, 张毅. Methods of reduction for Lagrange systems on time scaleswith nabla derivatives[J]. 中国物理B, 2017, 26(1): 14501-014501.
[6]	王琳莉, 傅景礼. Non-Noether symmetries of Hamiltonian systems withconformable fractional derivatives[J]. 中国物理B, 2016, 25(1): 14501-014501.
[7]	夏丽莉, 陈立群, 傅景礼, 吴旌贺. Symmetries and variational calculationof discrete Hamiltonian systems[J]. , 2014, 23(7): 70201-070201.
[8]	金世欣, 张毅. Noether symmetry and conserved quantity for a Hamilton system with time delay[J]. 中国物理B, 2014, 23(5): 54501-054501.
[9]	周燕, 张毅. Noether's theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives[J]. 中国物理B, 2014, 23(12): 124502-124502.
[10]	孙毅, 陈本永, 傅景礼. Lie symmetry theorem of fractional nonholonomic systems[J]. , 2014, 23(11): 110201-110201.
[11]	赵纲领, 陈立群, 傅景礼, 洪方昱. Mei symmetry and conservation laws of discrete nonholonomic dynamical systems with regular and irregular lattices[J]. 中国物理B, 2013, 22(3): 30201-030201.
[12]	王鹏, 薛纭, 刘宇陆. Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod[J]. 中国物理B, 2013, 22(10): 104503-104503.
[13]	陈蓉, 许学军. Conformal invariance, Noether symmetry, Lie symmetry and conserved quantities of Hamilton systems[J]. 中国物理B, 2012, 21(9): 94501-094501.
[14]	崔金超, 韩月林, 贾利群 . A type of conserved quantity of Mei symmetry of Nielsen equations for a holonomic system[J]. 中国物理B, 2012, 21(8): 80201-080201.
[15]	张斌, 方建会, 张伟伟 . Symmetry of Lagrangians of holonomic nonconservative system in event space[J]. 中国物理B, 2012, 21(7): 70208-070208.