[1]田 雪,张 毅.非保守Lagrange系统的Herglotz型广义变分原理及其Noether理论[J].南京理工大学学报(自然科学版),2019,43(06):765-770.[doi:10.14177/j.cnki.32-1397n.2019.43.06.014]
 Tian Xue,Zhang Yi.Generalized variational principle of Herglotz type fornon-conservative Lagrangian systems and its Noether’s theory[J].Journal of Nanjing University of Science and Technology,2019,43(06):765-770.[doi:10.14177/j.cnki.32-1397n.2019.43.06.014]
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非保守Lagrange系统的Herglotz型广义变分原理及其Noether理论()
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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:
43卷
期数:
2019年06期
页码:
765-770
栏目:
出版日期:
2019-12-31

文章信息/Info

Title:
Generalized variational principle of Herglotz type fornon-conservative Lagrangian systems and its Noether’s theory
文章编号:
1005-9830(2019)06-0765-06
作者:
田 雪1张 毅2
1.南京理工大学 理学院,江苏 南京 210094; 2.苏州科技大学 土木工程学院,江苏 苏州 215011
Author(s):
Tian Xue1Zhang Yi2
1.School of Sciences,Nanjing University of Science and Technology,Nanjing 210094,China; 2.School of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,China
关键词:
Herglotz型广义变分原理 非保守Lagrange系统 Noether定理 Noether逆定理
Keywords:
generalized variational principle of Herglotz type non-conservative Lagrangian system Noether’s theorem Noether’s inverse theorem
分类号:
O316
DOI:
10.14177/j.cnki.32-1397n.2019.43.06.014
摘要:
为了研究非保守动力学系统,该文利用Herglotz型广义变分原理研究非保守Lagrange系统的Noether定理及其逆定理。根据非保守Lagrange系统的Herglotz型广义变分原理及其动力学方程,给出Herglotz型Noether对称性的定义与判据,并导出Herglotz型Killing方程。建立了Herglotz型Noether定理及其逆定理,揭示了系统的Herglotz型Noether对称性与守恒量之间的内在联系。以Emden方程和二自由度系统为例,表明利用Herglotz型Noether对称性可以系统地研究保守和非保守问题的Noether理论。
Abstract:
In order to study non-conservative dynamic systems,Noether’s theorem and inverse theorem for non-conservative Lagrangian systems are studied by using the generalized variational principle of Herglotz type. According to the generalized variational principle of Herglotz type for the non-conservative Lagrangian system and its dynamic equation,the definition and criteria of Noether symmetry of Herglotz type are given,and the Killing equations of Herglotz type are also derived. Noether’s theorem of Herglotz type and its inverse theorem are obtained,which reveal the inner relation between the Noether symmetry and conserved quantity of the system. Taking the Emden equation and a two-freedom system as examples,the results show the Noether symmetry of Herglotz type can be used to study the Noether theory of conservative and non-conservative problems systematically.

参考文献/References:

[1] Lazo M J,Paiva J,Amaral J T S,et al. An action principle for action-dependent Lagrangians:Toward an action principle to non-conservative systems[J]. Journal of Mathematical Physics,2018,59(3):032902.
[2]Noether E. Invariante variations probleme[J]. Nachrichten von der Gesellschaft der Wissenschaften zu G?ttingen,Mathematisch-Physikalische Klasse,1918,KI(II):235-257.
[3]梅凤翔. 约束力学系统的对称性与守恒量[M]. 北京:北京理工大学出版社,2004.
[4]梅凤翔. 李群和李代数对约束力学系统的应用[M]. 北京:科学出版社,1999.
[5]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.
[6]金世欣,张毅. 相空间中含时滞的非保守力学系统的Noether定理[J]. 中山大学学报(自然科学版),2014,53(4):56-61.
Jin Shixin,Zhang Yi. Noether theorem for nonconservative mechanical system with time delay in phase space[J]. Acta Scientiarum Naturalium Universitatis Sunyateni,2014,53(4):56-61.
[7]傅景礼,付丽萍. 分数阶非完整系统的Noether对称性及其逆问题[J]. 北京大学学报(自然科学版),2016,52(4):643-652.
Fu Jingli,Fu Liping. Noether symmetries and their inverse problems of nonholonomic systems with fractional derivatives[J]. Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):643-652.
[8]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):120-125.
Liu Yandong,Zhang Yi. Noether quasi-symmetry theorems for fractional Hamilton system in terms of Caputo derivatives[J]. Journal of Nanjing University of Science and Technology,2018,42(3):120-125.
[9]宋传静,张毅. 时间尺度上Lagrange系统对称性摄动与绝热不变量[J]. 南京理工大学学报,2017,41(2):181-185.
Song Chuanjing,Zhang Yi. Perturbation to symmetry and adiabatic invariant for Lagrangian system on time scale[J]. Journal of Nanjing University of Science and Technology,2017,41(2):181-185.
[10]Georgieva B,Guenther R. First Noether-type theorem for the generalized variational principle of Herglotz[J]. Topological Methods in Nonlinear Analysis,2002,20(1):261-273.
[11]Georgieva B,Guenther R,Bodurov T. Generalized variational principle of Herglotz for several independent variables. First Noether-type theorem[J]. Journal of Mathematical Physics,2003,44(9):3911-3927.
[12]Georgieva B,Guenther R. Second Noether-type theorem for the generalized variational principle of Herglotz[J]. Topological Methods in Nonlinear Analysis,2005,26(2):307-314.
[13]张毅. 相空间中非保守系统的Herglotz广义变分原理及其Noether定理[J]. 力学学报,2016,48(6):1382-1389.
Zhang Yi. Generalized variational problem of Herglotz type for non-conservative system in phase space and Noether’s theorems[J]. Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1382-1389.
[14]Zhang Yi. Variational problem of Herglotz type for Birkhoffian system and its Noether’s theorems[J]. Acta Mechanica,2017,228(4):1-12.
[15]Zhang Yi,Tian Xue. Conservation laws of nonconservative nonholonomic system based on Herglotz variational problem[J]. Physics Letters A,2019,383(8):691-696.
[16]Georgieva B,Bodurov T. Identities from infinite-dimensional symmetries of Herglotz variational functional[J]. Journal of Mathematical Physics,2013,54(6):062901.
[17]Santos S P S,Martins N,Torres D F M. Noether currents for higher-order variational problems of Herglotz type with time delay[J]. Discrete and Continuous Dynamical Systems(Series S),2018,11(1):91-102.
[18]Donchev V. Variational symmetries,conserved quantities and identities for several equations of mathematical physics[J]. Journal of Mathematical Physics,2014,55(3):032901.
[19]Lazo M J,Paiva J,Amaral J T S,et al. Action principle for action-dependent Lagrangians toward nonconserva-tive gravity:Accelerating universe without dark energy[J]. Physical Review D,2017,95(10):101501.
[20] Almeida R,Malinowska A B. Fractional variational principle of Herglotz[J]. Discrete and Continuous Dynamical Systems(Series B),2014,19(8):2367-2381.
[21]Tian Xue,Zhang Yi. Noether symmetry and conserved quantities of fractional Birkhoffian system in terms of Herglotz variational problem[J]. Communications in Theoretical Physics,2018,70(3):280-288.
[22]Santos S P S,Martins N,Torres D F M. Variational problems of Herglotz type with time delay:Dubois-Reymond condition and Noether’s first theorem[J]. Discrete and Continuous Dynamical Systems,2015,35(9):4593-4610.
[23]Zhang Yi. Noether’s theorem for a time-delayed Birkhoffian system of Herglotz type[J]. International Journal of Non-Linear Mechanics,2018,101(4):36-43.
[24]田雪,张毅. 时间尺度上Herglotz广义变分原理及其Noether定理[J]. 力学季刊,2018,39(2):237-248.
Tian Xue,Zhang Yi. Variational principle of Herglotz type and its Noether’s theorem on time scales[J]. Chinese Quarterly of Mechanics,2018,39(2):237-248.
[25]Tian Xue,Zhang Yi. Noether symmetry and conserved quantity for Hamiltonian system of Herglotz type on time scales[J]. Acta Mechanica,2018,229(9):3601-3611.
[26]梅凤翔,刘端,罗勇. 高等分析力学[M]. 北京:北京理工大学出版社,1991.

备注/Memo

备注/Memo:
收稿日期:2019-03-13 修回日期:2019-04-25
基金项目:国家自然科学基金(11572212); 江苏省普通高校研究生科研创新计划(KYZZ16_0479)
作者简介:田雪(1993-),女,博士生,主要研究方向:分析力学,E-mail:crystaltianxue@njust.edu.cn; 通讯作者:张毅(1964-),男,博士,教授,主要研究方向:分析力学,E-mail:zhy@mail.usts.edu.cn。
引文格式:田雪,张毅. 非保守Lagrange系统的Herglotz型广义变分原理及其Noether理论[J]. 南京理工大学学报,2019,43(6):765-770.
投稿网址:http://zrxuebao.njust.edu.cn
更新日期/Last Update: 2019-12-31