GENERALIZED VARIATIONAL PRINCIPLE OF HERGLOTZ TYPE FOR NONCONSERVATIVE SYSTEM IN PHASE SPACE AND NOETHER'S THEOREM
-
摘要: 与经典变分原理相比，基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述，它不仅能够描述所有采用经典变分原理能够描述的动力学过程，而且能够应用于经典变分原理不能适用的非保守或耗散系统．将Herglotz广义变分原理拓展到相空间，研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理．首先，提出相空间中Herglotz广义变分原理，给出相空间中非保守系统的变分描述，导出相应的Hamilton正则方程；其次，基于非等时变分与等时变分之间的关系，导出相空间中Hamilton-Herglotz作用量变分的两个基本公式；再次，给出Noether对称变换的定义和判据，提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理，揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系．在经典条件下，Herglotz广义变分原理退化为经典变分原理，与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理．文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性．
-
关键词:
- Herglotz广义变分原理 /
- Noether定理 /
- 非保守动力学 /
- 相空间
Abstract: Compared with the classical variational principle, the generalized variational principle of Herglotz based upon the action defined by differential equations gives a variational description of nonconservative dynamical system. The principle can describe all dynamical processes and nonconservative or dissipative systems. In the present study, the principle is extended to phase space, and the generalized variational principle of Herglotz type for non-conservative mechanical system in phase space is given and Noether's theorem and its inverse of the system are studied. Firstly, the generalized variational principle of Herglotz type in phase space is presented, a variational description of non-conservative system in phase space is given, and the corresponding Hamilton canonical equations are deduced. Secondly, based upon the relation between non-isochronal variation and isochronal variation, two basic formulae for the variation of Hamilton-Herglotz action in phase space are obtained. Thirdly, the definition and the criterion of Noether symmetry are given, and Noether's theorem and its inverse of nonconservative system for the variational problem of Herglotz type in phase space are proposed and proved, and the inner relation between the Noether symmetry and the conserved quantity for mechanical systems in phase space is revealed. The generalized variational principle of Herglotz type reduces to the classical variational principle under classical conditions, and Noether's theorem for the variational problem of Herglotz type reduces to the classical Noether's theorem of Hamilton system. In the end of the paper, we take the famous Emden equation and damping oscillator with second power as examples to illustrate the application of the results. -
-
1 Noether AE. Invariante variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918, KI Ⅱ:235-257 2 李子平. 经典和量子约束系统及其对称性质. 北京:北京工业大学出版社, 1993(Li Ziping. Classical and Quantal Dynamics of Constrained Systems and their Symmetrical Properties. Beijing:Beijing Polytechnic University Press, 1993(in Chinese)) 3 Djuki? Dj S, Vujanovi? B. Noether's theory in classical nonconservative mechanics. Acta Mechanica, 1975, 23(1-2):17-27 4 李子平. 约束系统的对称变换. 物理学报, 1981, 30(12):1699-1706 (Li Ziping. The transformation properties of constrained system. Acta Physica Sinica, 1981, 30(12):1699-1706(in Chinese)) 5 Bahar LY, Kwatny HG. Extension of Noether's theorem to constrained nonconservative dynamical systems. International Journal of Non-Linear Mechanics, 1987, 22(2):125-138 6 Liu D. Noether's theorem and its inverse of nonholonomic nonconservative dynamical systems. Science in China (Series A), 1991, 34(4):419-429 7 Mei FX. The Noether's theory of Birkhoffan systems. Science in China (Serie A), 1993, 36(12):1456-1467 8 梅凤翔. 广义Birkhoff系统动力学. 北京:科学出版社, 2013(Mei Fengxiang. Dynamics of Generalized Birkhoffan System. Beijing:Science Press, 2013(in Chinese)) 9 Zhang Y, Zhou Y. Symmetries and conserved quantities for fractional action-like Pfa an variational problems. Nonlinear Dynamics, 2013, 73(1-2):783-793 10 Zhang Y, Zhai XH. Noether symmetries and conserved quantities for fractional Birkhoffan systems. Nonlinear Dynamics, 2015, 81(1-2):469-480 11 梅凤翔. 分析力学(下卷). 北京:北京理工大学出版社, 2013(Mei Fengxiang. Analytical Mechanics (Ⅱ). Beijing:Beijing Institute of Technology Press, 2013(in Chinese)) 12 梅凤翔. 李群和李代数对约束力学系统的应用. 北京:科学出版社, 1999(Mei Fengxiang. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems. Beijing:Science Press, 1999 (in Chinese)) 13 梅凤翔. 约束力学系统的对称性与守恒量. 北京:北京理工大学出版社, 2004(Mei Fengxiang. Symmetries and Conserved Quantities of Constrained Mechanical Systems. Beijing:Beijing Institute of Technology Press, 2004(in Chinese)) 14 梅凤翔. 经典约束力学系统对称性与守恒量研究进展. 力学进展, 2009, 39(1):37-43(Mei Fengxiang. Advances in the symmetries and conserved quantities of classical constrained systems. Advances in Mechanics, 2009, 39(1):37-43(in Chinese)) 15 Ferreira RAC, Malinowska AB. A counterexample to a Frederico-Torres fractional Noether-type theorem. Journal of Mathematical Analysis and Applications, 2015, 429(2):1370-1373 16 Atanackovi? TM, Konjik S, Pilipovi? S, et al. Variational problems with fractional derivatives:invariance conditions and Noether's theorem. Nonlinear Analysis, 2009, 71(5-6):1504-1517 17 Malinowska AB, Torres DFM. Introduction to the Fractional Calculus of Variations. London:Imperial College Press, 2012 18 Mei FX, Xie JF, Gang TQ. Weakly Noether symmetry for nonholonomic systems of Chetaev's type. Commun Theor Phys (Beijing, China), 2008, 49(6):1413-1416 19 Fu JL, Chen LQ, Chen BY. Noether-type theory for discrete mechanico-electrical dynamical systems with nonregular lattices. Science China:Physics, Mechanics, Astronomy, 2010, 53(9):1687-1698 20 Long ZX, Zhang Y. Noether's theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space. Acta Mechanica, 2014, 225(1):77-90 21 Zhai XH, Zhang Y. Noether symmetries and conserved quantities for fractional Birkhoffan systems with time delay. Communications in Nonlinear Science and Numerical Simulation, 2016, 36:81-97 22 Song CJ, Zhang Y. Noether theorem for Birkhoffan systems on time scales. Journal of Mathematical Physics, 2015, 56(10):102701 23 Zhang Y, Zhou XS. Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians. Nonlinear Dynamics, 2016, 84(4):1867-1876 24 Herglotz G. Berührungstransformationen. Lectures at the University of Göttingen, Göttingen, 1930 25 Georgieva B. Symmetries of the Herglotz variational principle in the case of one independent variable. Annual of Sofia University, the Faculty of Mathematics and Informatics, 2010, 100:113-122 26 Santos SPS, Martins N, Torres DFM. Higher-order variational problems of Herglotz type. Vietnam Journal of Mathematics, 2014, 42(4):409-419 27 Georgieva B, Guenther R. First Noether-type theorem for the generalized variational principle of Herglotz. Topological Methods in Nonlinear Analysis, 2002, 20(2):261-273 28 Georgieva B, Guenther R, Bodurov T. Generalized variational principle of Herglotz for several independent variables. First Noethertype theorem. Journal of Mathematical Physics, 2003, 44(9):3911-3927 29 Santos SPS, Martins N, Torres DFM. Variational problems of Herglotz type with time delay:Dubois-Reymond condition and Noether's first theorem. Discrete and Continuous Dynamical Systems, 2015, 35(9):4593-4610 30 Donchev V. Variational symmetries, conserved quantities and identities for several equations of mathematical physics. Journal of Mathematical Physics, 2014, 55(3):032901 31 梅凤翔. 经典力学从牛顿到伯克霍夫. 力学与实践, 1996, 18(4):1-8(Mei Fengxiang. Classical mechanics from Newton to Birkhoffian. Mechanics in Engineering, 1996, 18(4):1-8(in Chinese)) 32 Goldstein H, Poole C, Safko J. Classical Mechanics (Third Edition). Beijing:Higher Education Press, 2005 33 梅凤翔,刘端,罗勇. 高等分析力学. 北京:北京理工大学出版社, 1991 (Mei Fengxiang, Liu Duan, Luo Yong. Advanced Analytical Mechanics. Beijing:Beijing Institute of Technology Press, 1991(in Chinese)) -
期刊类型引用(21)
1. 王文静，张毅. Herglotz型Birkhoff系统的积分因子与守恒律（英文）. Wuhan University Journal of Natural Sciences. 2025(01): 49-56 . 
百度学术
2. 张毅. 非完整系统的约束Herglotz方程的梯度化及其解的稳定性. 力学学报. 2025(02): 516-522 . 本站查看
3. 张毅. 非保守广义Chaplygin系统的Herglotz型Noether定理. 力学学报. 2024(09): 2695-2702 . 本站查看
4. 蔡铭俣，张毅. 变质量完整系统的Herglotz型微分变分原理与守恒律. 苏州科技大学学报(自然科学版). 2023(04): 31-38 . 百度学术
5. 蔡锦祥，张毅. 事件空间中Herglotz型非保守Lagrange系统的Noether定理. 力学季刊. 2022(01): 122-131 . 百度学术
6. 张毅，蔡锦祥. 事件空间中非完整力学系统的Herglotz-d′Alembert原理与守恒律. 动力学与控制学报. 2022(02): 15-21 . 百度学术
7. 蔡铭俣，张毅. 变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量. 动力学与控制学报. 2022(06): 106-113 . 百度学术
8. 田雪，张毅. CaputoΔ型分数阶时间尺度Noether定理. 力学学报. 2021(07): 2010-2022 . 本站查看
9. ZHANG Yi，CAI Jinxiang. Noether Theorem of Herglotz-Type for Nonconservative Hamilton Systems in Event Space. Wuhan University Journal of Natural Sciences. 2021(05): 376-382 . 必应学术
10. 张毅，田雪，翟相华，宋传静. 时间尺度上Lagrange系统的Hojman守恒量. 力学学报. 2021(10): 2814-2822 . 本站查看
11. 徐鑫鑫，张毅. 相空间中Herglotz型微分变分原理与一类新型绝热不变量. 中山大学学报(自然科学版). 2020(01): 35-42 . 百度学术
12. 张毅. 非保守动力学的Herglotz广义变分原理的研究进展（英文）. Transactions of Nanjing University of Aeronautics and Astronautics. 2020(01): 13-26 . 百度学术
13. 张毅. 弱非线性动力学方程的Noether准对称性与近似Noether守恒量. 力学学报. 2020(06): 1765-1773 . 本站查看
14. 徐鑫鑫，张毅. 分数阶非保守Lagrange系统的一类新型绝热不变量. 物理学报. 2020(22): 291-298 . 百度学术
15. 夏丽莉，国忠金，张伟. 基于离散变分算子的非保守Hamilton系统的Lie对称性与守恒量. 河北大学学报(自然科学版). 2019(01): 6-10 . 百度学术
16. 张毅. 非保守动力学系统的Herglotz型微分变分原理与守恒律. 南京理工大学学报. 2019(06): 759-764 . 百度学术
17. 田雪，张毅. 非保守Lagrange系统的Herglotz型广义变分原理及其Noether理论. 南京理工大学学报. 2019(06): 765-770+799 . 百度学术
18. 田雪，张毅. 时间尺度上Herglotz变分原理及其Noether定理. 力学季刊. 2018(02): 237-248 . 百度学术
19. 孙晨，朱建青. 时间尺度上相空间中力学系统的Mei对称性及守恒量. 苏州科技大学学报(自然科学版). 2018(04): 18-22+43 . 百度学术
20. 张毅. 基于Herglotz型微分变分原理研究相空间中非保守系统的守恒律. 力学季刊. 2018(04): 681-688 . 百度学术
21. 张毅. Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理. 力学学报. 2017(03): 693-702 . 本站查看
其他类型引用(6)
计量
- 文章访问数: 1351
- HTML全文浏览量: 123
- PDF下载量: 540
- 被引次数: 27
下载:
