[1]刘艳东,张 毅.Caputo导数下分数阶Hamilton系统的Noether准对称性定理[J].南京理工大学学报(自然科学版),2018,42(03):374.[doi:10.14177/j.cnki.32-1397n.2018.42.03.018]
　Liu Yandong,Zhang Yi.Noether quasi-symmetry theorems for fractional Hamilton systemin terms of Caputo derivatives[J].Journal of Nanjing University of Science and Technology,2018,42(03):374.[doi:10.14177/j.cnki.32-1397n.2018.42.03.018]
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Caputo导数下分数阶Hamilton系统的Noether 准对称性定理()
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《南京理工大学学报》（自然科学版）[ISSN:1005-9830/CN:32-1397/N]

卷:
42卷
期数:
2018年03期
页码:
374
栏目:
出版日期:
2018-06-30

文章信息/Info

Title:
Noether quasi-symmetry theorems for fractional Hamilton system in terms of Caputo derivatives
文章编号:
1005-9830(2018)03-0374-06
作者:
刘艳东1; 张 毅2
1.苏州科技大学 数理学院,江苏 苏州 215009; 2.苏州科技大学 土木工程学院,江苏 苏州 215011
Author(s):
Liu Yandong1; Zhang Yi2
1.School of Mathematics and Physics,Suzhou University of Science and Technology,Suzhou 215009,China; 2.School of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,China
关键词:
Caputo导数;  分数阶Hamilton系统;  Noether准对称性;  非保守动力学系统;  分数阶守恒量
Keywords:
Caputo derivatives;  fractional Hamilton system;  Noether quasi-symmetry;  non-conservative dynamical systems;  fractional conserved quantities
分类号:
O316
DOI:
10.14177/j.cnki.32-1397n.2018.42.03.018
摘要:
为了探究分数阶模型下非保守动力学系统的对称性与守恒量之间的内在关系,该文提出并研究Caputo导数下分数阶Hamilton系统的Noether准对称性与分数阶守恒量问题。建立分数阶模型下Hamilton系统的Noether准对称性的定义和判据。基于Frederico-Torres分数阶守恒量概念,利用重新参数化方法导出Noether准对称性定理。以某分数阶Hamilton系统为例,给出该系统的准对称性及其相应的分数阶守恒量。该文研究方法和结果具有普遍性,可进一步推广到非完整非保守动力学系统等。
Abstract:
The fractional conserved quantities and the Noether quasi-symmetry for a fractional Hamilton system in terms of Caputo derivatives are proposed and studied to explore the internal relation between symmetry and conserved quantities for non-conservative dynamical systems under fractional models. The definition and the criterion of Noether quasi-symmetry for the fractional Hamilton system in terms of Caputo derivatives are established. The Noether quasi-symmetry theorem is deduced by using the time-reparameterization method based on the concept of Frederico-Torres fractional conserved quantity. A fractional Hamilton system is taken as an example,and the quasi-symmetries of the system and corresponding fractional conserved quantities are given. The methods and results of this study are universal and can be extended to non-holonomic non-conservative dynamic systems,etc.

参考文献/References:

[1] Podlubny I. Fractional differential equations[M]. San Diego,USA:Academic Press,1999. [2]孙文,孙洪广,李西成. 力学与工程问题的分数阶导数建模[M]. 北京:科学出版社,2010. [3]Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E,1996,53(2):1890-1899. [4]Riewe F. Mechanics with fractional derivatives[J]. Physical Review E,1997,55(3):3581-3592. [5]Agrawal O P. Formulation of Euler-Lagrange equations for fractional variational problems[J]. Journal of Mathematical Analysis and Applications,2002,272(1):368-379 [6]AtanackoviAc’ T M,Konjik S,PilipoviAc’ S. Variational problems with fractional derivatives:Euler-Lagrange equations[J]. Journal of Physics A:Mathematical and Theoretical,2008,41(9):095201. [7]Baleanu D,Trujillo J I. A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives[J]. Communications in Nonlinear Science and Numerical Simulation,2010,15(5):1111-1115. [8]Frederico G S F,Torres D F M. A formulation of Noether’s theorem for fractional problems of the calculus of variations[J]. Journal of Mathematical Analysis and Applications,2007,334(2):834-846. [9]Frederico G S F,Torres D F M. Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem[J]. International Mathematical Forum,2008,3(10):479-493. [10]Zhou Sha,Fu Hao,Fu Jingli. Symmetry theories of Hamiltonian systems with fractional derivatives[J]. Science China(Physics,Mechanics & Astronomy),2011,54(10):1847-1853. [11]Zhou Yan,Zhang Yi. Noether’s theorems of a fractional Birkhoffian system within Riemann-Liouville deriva-tives[J]. Chinese Physics B,2014,23(12):124502. [12]Zhang Yi,Zhai Xianghua. Noether symmetries and conserved quantities for fractional Birkhoffian systems[J]. Nonlinear Dynamics,2015,81(1-2):469-480. [13]Zhai Xianghua,Zhang Yi. Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay[J]. Communications in Nonlinear Science and Numerical Simulation,2016,36:81-97. [14]张毅,周燕. 基于Riesz导数的分数阶Birkhoff系统的Noether对称性与守恒量[J]. 北京大学学报(自然科学版),2016,52(4):658-668. Zhang Yi,Zhou Yan. Noether symmetry and conserved quantity for fractional Birkhoffian systems in terms of Riesz derivatives[J]. Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):658-668. [15]张毅,丁金凤. 基于El-Nabulsi分数阶模型的广义Birkhoff系统Noether对称性研究[J]. 南京理工大学学报,2014,38(3):409-413. Zhang Yi,Ding Jinfeng. Noether symmetries of generalized Birkhoff systems based on El-Nabulsi fractional model[J]. Journal of Nanjing University of Science and Technology,2014,38(3):409-413. [16]张毅. Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理[J]. 力学学报,2017,49(3):693-702. Zhang Yi. Quasi-symmetry and Noether’s theorem for fractional Birkhoffian systems in terms of Caputo derivatives[J]. Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):693-702. [17]Malinowska A B,Torres D F M. Introduction to the fractional calculus of variations[M]. London,UK:Imperial College Press,2012.

备注/Memo

备注/Memo:
收稿日期:2017-07-18 修回日期:2018-01-09 基金项目:国家自然科学基金(11572212; 11272227); 苏州科技大学研究生科研创新计划(SKYCX16_004) 作者简介:刘艳东(1988-),男,硕士生,主要研究方向:力学中的数学方法,E-mail:996364538@qq.com; 通讯作者:张毅(1964-),男,教授,博士生导师,主要研究方向:分析力学,E-mail:zhy@mail.usts.edu.cn。 引文格式:刘艳东,张毅. Caputo导数下分数阶Hamilton系统的Noether准对称性定理[J]. 南京理工大学学报,2018,42(3):374-379. 投稿网址:http://zrxuebao.njust.edu.cn
更新日期/Last Update: 2018-06-30