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The Noether symmetries and the conserved quantities of dynamics for non-conservative systems with time delay are proposed and studied. Firstly, the Hamilton principle for non-conservative systems with time delay is established, and the Lagrange equations with time delay are obtained. Secondly, based upon the invariance of the Hamilton action with time delay under a group of infinitesimal transformations which depends on the generalized velocities, the generalized coordinates and the time, the Noether symmetric transformations and the Noether quasi-symmetric transformations of the system are defined and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theory of non-conservative systems with time delay is established At the end of the paper, some examples are given to illustrate the application of the results.
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Keywords:
- system with time delay /
- non-conservative mechanics /
- Noether symmetry /
- conserved quantity
[1] Hu H Y, Wang Z H 1999 Adv. Mech. 29 501 (in Chinese) [胡海岩, 王在华 1999 力学进展 29 501]
[2] Xu J, Pei L J 2006 Adv. Mech. 36 17 (in Chinese) [徐鉴, 裴利军 2006 力学进展 36 17]
[3] Wang Z H, Hu H Y 2013 Adv. Mech. 43 3 (in Chinese) [王在华, 胡海岩 2013 力学进展 43 3]
[4] Djukić Dj S, Vujanović B 1975 Acta Mech. 23 17
[5] Li Z P 1981 Acta Phys. Sin. 30 1699 (in Chinese) [李子平 1981 30 1699]
[6] Bahar L Y, Kwatny H G 1987 Int. J. Non-Linear Mech. 22 125
[7] Mei F X 2001 Int. J. Non-Linear Mech. 36 817
[8] Xu X J, Mei F X 2005 Chin. Phys. 14 449
[9] Luo S K 2007 Chin. Phys. Lett. 24 3017
[10] Fu J L, Chen B Y, Chen L Q 2009 Phys. Lett. A 373 409
[11] Zhang Y, Zhou Y 2013 Nonlinear Dyn. 73 783
[12] Bluman G W, Anco S C 2002 Symmety and Integration Methods for Differential Equations (New York: Springer-Verlag)
[13] Lutzky M 1979 J. Phys. A: Math. Gen. 12 973
[14] Hojman S A 1992 J. Phys. A: Math. Gen. 25 L291
[15] Wang P, Wang X M, Fang J H 2009 Chin. Phys. Lett. 26 034501
[16] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用 (北京: 科学出版社)]
[17] Zhang Y 2002 Acta Phys. Sin. 51 461 (in Chinese) [张毅 2002 51 461]
[18] Long Z X, Zhang Y 2013 Acta Mech. Doi: 10.1007/s00707-013-0956-5
[19] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 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)]
[20] Hojman S 1984 J. Phys. A: Math. Gen. 17 2399
[21] Mei F X, Wu H B 2008 Phys. Lett. A 372 2141
[22] Zhang Y 2011 Chin. Phys. B 20 034502
[23] El’sgol’c L E 1964 Qualitative Methods in Mathematical Analysis (Providence: American Mathematical Society)
[24] Hughes D K 1968 J. Optim. Theory Appl. 2 1
[25] Palm W J, Schmitendorf W E 1974 J. Optim. Theory Appl. 14 599
[26] Rosenblueth J F 1988 IMA J. Math. Control Inform. 5 125
[27] Chan W L, Yung S P 1993 J. Optim. Theory Appl. 76 131
[28] Lee C H, Yung S P 1996 J. Optim. Theory Appl. 88 157
[29] Frederico G S F, Torres D F M 2012 Control Optim. 2 619
[30] Mei F X, Wu H B 2009 Dynamics of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press)
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[1] Hu H Y, Wang Z H 1999 Adv. Mech. 29 501 (in Chinese) [胡海岩, 王在华 1999 力学进展 29 501]
[2] Xu J, Pei L J 2006 Adv. Mech. 36 17 (in Chinese) [徐鉴, 裴利军 2006 力学进展 36 17]
[3] Wang Z H, Hu H Y 2013 Adv. Mech. 43 3 (in Chinese) [王在华, 胡海岩 2013 力学进展 43 3]
[4] Djukić Dj S, Vujanović B 1975 Acta Mech. 23 17
[5] Li Z P 1981 Acta Phys. Sin. 30 1699 (in Chinese) [李子平 1981 30 1699]
[6] Bahar L Y, Kwatny H G 1987 Int. J. Non-Linear Mech. 22 125
[7] Mei F X 2001 Int. J. Non-Linear Mech. 36 817
[8] Xu X J, Mei F X 2005 Chin. Phys. 14 449
[9] Luo S K 2007 Chin. Phys. Lett. 24 3017
[10] Fu J L, Chen B Y, Chen L Q 2009 Phys. Lett. A 373 409
[11] Zhang Y, Zhou Y 2013 Nonlinear Dyn. 73 783
[12] Bluman G W, Anco S C 2002 Symmety and Integration Methods for Differential Equations (New York: Springer-Verlag)
[13] Lutzky M 1979 J. Phys. A: Math. Gen. 12 973
[14] Hojman S A 1992 J. Phys. A: Math. Gen. 25 L291
[15] Wang P, Wang X M, Fang J H 2009 Chin. Phys. Lett. 26 034501
[16] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用 (北京: 科学出版社)]
[17] Zhang Y 2002 Acta Phys. Sin. 51 461 (in Chinese) [张毅 2002 51 461]
[18] Long Z X, Zhang Y 2013 Acta Mech. Doi: 10.1007/s00707-013-0956-5
[19] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 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)]
[20] Hojman S 1984 J. Phys. A: Math. Gen. 17 2399
[21] Mei F X, Wu H B 2008 Phys. Lett. A 372 2141
[22] Zhang Y 2011 Chin. Phys. B 20 034502
[23] El’sgol’c L E 1964 Qualitative Methods in Mathematical Analysis (Providence: American Mathematical Society)
[24] Hughes D K 1968 J. Optim. Theory Appl. 2 1
[25] Palm W J, Schmitendorf W E 1974 J. Optim. Theory Appl. 14 599
[26] Rosenblueth J F 1988 IMA J. Math. Control Inform. 5 125
[27] Chan W L, Yung S P 1993 J. Optim. Theory Appl. 76 131
[28] Lee C H, Yung S P 1996 J. Optim. Theory Appl. 88 157
[29] Frederico G S F, Torres D F M 2012 Control Optim. 2 619
[30] Mei F X, Wu H B 2009 Dynamics of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press)
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