Please wait a minute...
Chin. Phys. B, 2014, Vol. 23(5): 054501    DOI: 10.1088/1674-1056/23/5/054501
ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS Prev   Next  

Noether symmetry and conserved quantity for a Hamilton system with time delay

Jin Shi-Xin (金世欣)a, Zhang Yi (张毅)b
a College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
b College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
Abstract  In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are discussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed, the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.
Keywords:  time delay      Hamilton system      Noether symmetry      conserved quantity  
Received:  20 September 2013      Revised:  19 October 2013      Accepted manuscript online: 
PACS:  45.20.Jj (Lagrangian and Hamiltonian mechanics)  
  11.30.Na (Nonlinear and dynamical symmetries (spectrum-generating symmetries))  
  02.30.Ks (Delay and functional equations)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 10972151 and 11272227), the Innovation Program for Scientific Research in Higher Education Institution of Jiangsu Province, China (Grant No. CXLX11_0961), and the Innovation Program for Scientific Research of Suzhou University of Science and Technology, China (Grant No. SKCX12S_039).
Corresponding Authors:  Zhang Yi     E-mail:  weidiezh@gmail.com
About author:  45.20.Jj; 11.30.Na; 02.30.Ks

Cite this article: 

Jin Shi-Xin (金世欣), Zhang Yi (张毅) Noether symmetry and conserved quantity for a Hamilton system with time delay 2014 Chin. Phys. B 23 054501

[1] Hu H Y and Wang Z H 1999 Adv. Mech. 29 501 (in Chinese)
[2] Xu J and Pei L J 2006 Adv. Mech. 36 17 (in Chinese)
[3] Wang Z H and Hu H Y 2013 Adv. Mech. 43 3 (in Chinese)
[4] El'sgol'c L E, Brown A A and Danskin J M 1964 Qualitative Methods of Mathematical Analysis (Providence: American Mathematical Society)
[5] Hughes D K 1968 J. Optim. Theory Appl. 2 1
[6] Palm W J and Schmitendorf W E 1974 J. Optim. Theory Appl. 14 599
[7] Rosenblueth J F 1988 IMA J. Math. Control Inform. 5 125
[8] Rosenblueth J F 1988 IMA J. Math. Control. Inform. 5 285
[9] Chan W L and Yung S P 1993 J. Optim. Theory Appl. 76 131
[10] Lee C H and Yung S P 1996 J. Optim. Theory. Appl. 88 157
[11] Frederico G S F and Torres D F M 2012 Numer. Algebra Control Optim. 2 619
[12] Djukic Dj S and Vujanovic B 1975 Acta Mech. 23 17
[13] Li Z P 1981 Acta Phys. Sin. 30 1699 (in Chinese)
[14] Bahar L Y and Kwatny H G 1987 Int. J. Non-Linear Mech. 22 125
[15] Liu D 1991 Sci. China Ser. A 34 419
[16] Luo S K 1991 Appl. Math. Mech. 12 927
[17] Mei F X 1993 Sci. China Ser. A 36 1456
[18] Zhang Y, Shang M and Mei F X 2000 Chin. Phys. 9 401
[19] Fu J L, Chen B Y and Chen L Q 2009 Phys. Lett. A 373 409
[20] Bluman G W and Anco S C 2002 Symmety and Integration Methods for Differential Equations (New York: Springer-Verlag)
[21] Lutzky M 1979 J. Phys. A: Math. Gen. 12 973
[22] Hojman S A 1992 J. Phys. A: Math. Gen. 25 L291
[23] Zhao Y Y 1994 Acta Mech. Sin. 26 380 (in Chinese)
[24] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[25] Zhang Y 2002 Acta Phys. Sin. 51 461 (in Chinese)
[26] Luo S K, Cai J L and Jia L Q 2005 Commun. Theor. Phys. 43 193
[27] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)
[28] Hojman S 1984 J. Phys. A: Math. Gen. 17 2399
[29] Mei F X and Wu H B 2008 Phys. Lett. A 372 2141
[30] Zhang Y 2011 Chin. Phys. B 20 034502
[1] Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh-Rose and FitzHugh-Nagumo neurons with two time delays
Zhan-Hong Guo(郭展宏), Zhi-Jun Li(李志军), Meng-Jiao Wang(王梦蛟), and Ming-Lin Ma(马铭磷). Chin. Phys. B, 2023, 32(3): 038701.
[2] Effect of autaptic delay signal on spike-timing precision of single neuron
Xuan Ma(马璇), Yaya Zhao(赵鸭鸭), Yafeng Wang(王亚峰), Yueling Chen(陈月玲), and Hengtong Wang(王恒通). Chin. Phys. B, 2023, 32(3): 038703.
[3] Exploring fundamental laws of classical mechanics via predicting the orbits of planets based on neural networks
Jian Zhang(张健), Yiming Liu(刘一鸣), and Zhanchun Tu(涂展春). Chin. Phys. B, 2022, 31(9): 094502.
[4] Inferring interactions of time-delayed dynamic networks by random state variable resetting
Changbao Deng(邓长宝), Weinuo Jiang(蒋未诺), and Shihong Wang(王世红). Chin. Phys. B, 2022, 31(3): 030502.
[5] Review on typical applications and computational optimizations based on semiclassical methods in strong-field physics
Xun-Qin Huo(火勋琴), Wei-Feng Yang(杨玮枫), Wen-Hui Dong(董文卉), Fa-Cheng Jin(金发成), Xi-Wang Liu(刘希望), Hong-Dan Zhang(张宏丹), and Xiao-Hong Song(宋晓红). Chin. Phys. B, 2022, 31(3): 033101.
[6] Bifurcation and dynamics in double-delayed Chua circuits with periodic perturbation
Wenjie Yang(杨文杰). Chin. Phys. B, 2022, 31(2): 020201.
[7] Finite-time Mittag—Leffler synchronization of fractional-order complex-valued memristive neural networks with time delay
Guan Wang(王冠), Zhixia Ding(丁芝侠), Sai Li(李赛), Le Yang(杨乐), and Rui Jiao(焦睿). Chin. Phys. B, 2022, 31(10): 100201.
[8] Delayed excitatory self-feedback-induced negative responses of complex neuronal bursting patterns
Ben Cao(曹奔), Huaguang Gu(古华光), and Yuye Li(李玉叶). Chin. Phys. B, 2021, 30(5): 050502.
[9] Modeling and dynamics of double Hindmarsh-Rose neuron with memristor-based magnetic coupling and time delay
Guoyuan Qi(齐国元) and Zimou Wang(王子谋). Chin. Phys. B, 2021, 30(12): 120516.
[10] Stabilization strategy of a car-following model with multiple time delays of the drivers
Weilin Ren(任卫林), Rongjun Cheng(程荣军), and Hongxia Ge(葛红霞). Chin. Phys. B, 2021, 30(12): 120506.
[11] Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen-Grossberg neural networks
Yu-Jiao Huang(黄玉娇), Xiao-Yan Yuan(袁孝焰), Xu-Hua Yang(杨旭华), Hai-Xia Long(龙海霞), Jie Xiao(肖杰). Chin. Phys. B, 2020, 29(2): 020703.
[12] Enhanced vibrational resonance in a single neuron with chemical autapse for signal detection
Zhiwei He(何志威), Chenggui Yao(姚成贵), Jianwei Shuai(帅建伟), and Tadashi Nakano. Chin. Phys. B, 2020, 29(12): 128702.
[13] Design of passive filters for time-delay neural networks with quantized output
Jing Han(韩静), Zhi Zhang(章枝), Xuefeng Zhang(张学锋), and Jianping Zhou(周建平). Chin. Phys. B, 2020, 29(11): 110201.
[14] Validity of extracting photoionization time delay from the first moment of streaking spectrogram
Chang-Li Wei(魏长立), Xi Zhao(赵曦). Chin. Phys. B, 2019, 28(1): 013201.
[15] Synchronization performance in time-delayed random networks induced by diversity in system parameter
Yu Qian(钱郁), Hongyan Gao(高红艳), Chenggui Yao(姚成贵), Xiaohua Cui(崔晓华), Jun Ma(马军). Chin. Phys. B, 2018, 27(10): 108902.
No Suggested Reading articles found!