Conservation laws for Birkhoffian systems of Herglotz type

doi:10.1088/1674-1056/27/9/090502

中国物理B ›› 2018, Vol. 27 ›› Issue (9): 90502-090502.doi: 10.1088/1674-1056/27/9/090502

• SPECIAL TOPIC—Recent advances in thermoelectric materials and devices • 上一篇下一篇

Conservation laws for Birkhoffian systems of Herglotz type

Yi Zhang(张毅), Xue Tian(田雪)

1 College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China;
2 College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China

收稿日期:2018-03-20 修回日期:2018-05-29 出版日期:2018-09-05 发布日期:2018-09-05
通讯作者: Yi Zhang E-mail:zhy@mail.usts.edu.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province, China (Grant No. KYZZ16_0479), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology, China (Grant No. SKCX16_058).

Conservation laws for Birkhoffian systems of Herglotz type

Yi Zhang(张毅)¹, Xue Tian(田雪)²

1 College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China;
2 College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China

Received:2018-03-20 Revised:2018-05-29 Online:2018-09-05 Published:2018-09-05
Contact: Yi Zhang E-mail:zhy@mail.usts.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227), the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province, China (Grant No. KYZZ16_0479), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology, China (Grant No. SKCX16_058).

摘要/Abstract

摘要：

Conservation laws for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are studied. We propose a new differential variational principle, called the Pfaff-Birkhoff-d'Alembert principle of Herglotz type. Birkhoff's equations for both the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are obtained. According to the relationship between the isochronal variation and the nonisochronal variation, the conditions of the invariance for the Pfaff-Birkhoff-d'Alembert principle of Herglotz type are given. Then, the conserved quantities for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are deduced. Furthermore, the inverse theorems of the conservation theorems are also established.

关键词: Birkhoffian system, conversation law, differential variational principle, variational problem of Herglotz type

Abstract:

Key words: Birkhoffian system, conversation law, differential variational principle, variational problem of Herglotz type

中图分类号: (Nonlinear dynamics and chaos)

05.45.-a

11.30.-j (Symmetry and conservation laws) 45.10.Db (Variational and optimization methods)

张毅, 田雪. Conservation laws for Birkhoffian systems of Herglotz type[J]. 中国物理B, 2018, 27(9): 90502-090502.

Yi Zhang(张毅), Xue Tian(田雪). Conservation laws for Birkhoffian systems of Herglotz type[J]. Chin. Phys. B, 2018, 27(9): 90502-090502.

参考文献 42

[1]	Mei F X 2013 Analytical Mechanics (Ⅱ) (Beijing:Beijing Institute of Technology Press) p. 440 (in Chinese)
[2]	Zhang Y and Zhou Y 2013 Nonlinear Dynam. 73 783 doi: 10.1007/s11071-013-0831-x
[3]	Zhou Y and Zhang Y 2014 Chin. Phys. B 23 124502 doi: 10.1088/1674-1056/23/12/124502
[4]	Jin S X and Zhang Y 2014 Chin. Phys. B 23 054501 doi: 10.1088/1674-1056/23/5/054501
[5]	Yan B and Zhang Y 2016 Acta Mech. 227 2439 doi: 10.1007/s00707-016-1622-5
[6]	Song J and Zhang Y 2017 Chin. Phys. B 26 084501 doi: 10.1088/1674-1056/26/8/084501
[7]	Fu J L, Chen L Q and Xie F P 2004 Chin. Phys. B 13 1611 doi: 10.1088/1009-1963/13/10/005
[8]	Luo S K, Guo Y X and Mei F X 2004 Acta Phys. Sin. 53 1271 (in Chinese)
[9]	Fang J H, Miao Y P, Ding N, et al. 2006 Chin. Phys. B 15 2792 doi: 10.1088/1009-1963/15/12/005
[10]	Shi S Y and Huang X H 2008 Chin. Phys. B 17 01554 doi: 10.1088/1674-1056/17/5/004
[11]	Fang J H, Wang P and Ding N 2008 Commun. Theor. Phys. 48 977
[12]	Fang J H, Liu Y K and Zhang X N 2008 Chin. Phys. B 17 1962 doi: 10.1088/1674-1056/17/6/005
[13]	Li Y C, Xia L L, Wang X M, et al. 2010 Acta Phys. Sin. 59 3639 (in Chinses)
[14]	Zhao G L, Chen L Q, Fu J L, et al. 2013 Chin. Phys. B 22 030201 doi: 10.1088/1674-1056/22/3/030201
[15]	Vujanović B 1978 Int. J. Nonlinear Mech. 13 185 doi: 10.1016/0020-7462(78)90007-0
[16]	Liu D 1989 Acta Mech. Sin. 5 167 doi: 10.1007/BF02489142
[17]	Mei F X 1998 J. Beijing Inst. Technol. 18 17 (in Chinese)
[18]	Fang J H 2000 Appl. Math. Mech. Engl. 21 836 doi: 10.1007/BF02428382
[19]	Vujanović B 1987 Acta Mech. 65 63 doi: 10.1007/BF01176873
[20]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing:Science Press) p. 288 (in Chinese)
[21]	Wang C D, Liu S X and Mei F X 2010 Acta. Phys. Sin. 59 8322 (in Chinese)
[22]	Mei F X, Liu D and Luo Y 1991 Advanced Analytical Mechanics (Beijing:Beijing Institute of Technology Press) (in Chinese)
[23]	Herglotz G 1979 Gesammelte Schriften (Göttingen:Vandenhoeck & Ruprecht) doi: melte Schriften ???G
[24]	Mrugała R 1980 Acta Phys. Polon. A 58 19
[25]	Furta K, Sano A and Atherton D 1988 State Variable Methods in Automatic Control (New York:John Wiley)
[26]	Georgieva B and Guenther R 2002 Topol. Method. Nonlinear Anal. 20 261 doi: 10.12775/TMNA.2002.036
[27]	Georgieva B, Guenther R and Bodurov T 2003 J. Math. Phys. 44 3911 doi: 10.1063/1.1597419
[28]	Georgieva B and Bodurov T 2013 J. Math. Phys. 54 062901 doi: 10.1063/1.4807728
[29]	Santos S P S, Martins N and Torres D F M 2015 Discr. Contin. Dyn. Syst. 35 4593 doi: 10.3934/dcds.2015.35.4593
[30]	Santos S P S, Martins N and Torres D F M 2018 Discr. Contin. Dyn. Syst. (Ser. S) 11 91
[31]	Donchev V 2014 J. Math. Phys. 55 032901 doi: 10.1063/1.4867626
[32]	Zhang Y 2016 Chin. J. Theor. Appl. Mech. 48 1382 (in Chinese)
[33]	Lazo M J, Paiva J, Amaral J T S, et al. 2017 Phys. Rev. D 95 101501 doi: 10.1103/PhysRevD.95.101501
[34]	Lazo M J, Paiva J, Amaral J T S, et al. 2018 J. Math. Phys. 59 032902 doi: 10.1063/1.5019936
[35]	Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York:Springer)
[36]	Mei F X, Wu H B, Li Y M, et al. 2016 Chin. J. Theor. Appl. Mech. 48 263 (in Chinese)
[37]	Galiullan A S 1989 Analytical Dynamics (Moscow:Nauka) (in Russian)
[38]	Mei F X, Shi R C, Zhang Y F and Wu H B 1996 Dynamics of Birkhoffian System (Beijing:Beijing Institute of Technology Press) (in Chinese)
[39]	Galiullin A S, Gafarov G G, Malaishka R P and Khwan A M 1997 Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems (Moscow:UFN) (in Russian)
[40]	Mei F X 2013 Dynamics of Generalized Birkhoffian System (Beijing:Science Press) (in Chinese)
[41]	Zhang Y 2017 Acta Mech. 228 1481 doi: 10.1007/s00707-016-1758-3
[42]	Tian X and Zhang Y 2018 Int. J. Theor. Phys. 57 887 doi: 10.1007/s10773-017-3621-2

Conservation laws for Birkhoffian systems of Herglotz type

Conservation laws for Birkhoffian systems of Herglotz type

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