Noether's theory of Lagrange systems in discrete case

doi:10.1088/1674-1056/20/1/011101

中国物理B ›› 2011, Vol. 20 ›› Issue (1): 11101-011101.doi: 10.1088/1674-1056/20/1/011101

• THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS • 上一篇下一篇

Noether's theory of Lagrange systems in discrete case

吕洪升¹, 张宏彬², 顾书龙²

(1)Department of Mathematics, Chaohu University, Chaohu 238000, China; (2)Department of Physics , Chaohu University, Chaohu 238000, China

收稿日期:2010-02-28 修回日期:2010-07-01 出版日期:2011-01-15 发布日期:2011-01-15
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 10872037) and the Natural Science Foundation of Anhui Province, China (Grant No. 070416226).

Noether's theory of Lagrange systems in discrete case

Lü Hong-Sheng(吕洪升)^a), Zhang Hong-Bin(张宏彬)^b)†, and Gu Shu-Long(顾书龙)^b)

^a Department of Mathematics, Chaohu University, Chaohu 238000, China; ^b Department of Physics , Chaohu University, Chaohu 238000, China

Received:2010-02-28 Revised:2010-07-01 Online:2011-01-15 Published:2011-01-15
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 10872037) and the Natural Science Foundation of Anhui Province, China (Grant No. 070416226).

摘要/Abstract

摘要： In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the well-known Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether's theorem is presented. An example was discussed to illustrate these results.

Abstract: In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the well-known Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether's theorem is presented. An example was discussed to illustrate these results.

Key words: discrete Lagrange system, difference functional, Noether-type identity, Noether-type theorem

中图分类号: (Symmetry and conservation laws)

11.30.-j

45.20.Jj (Lagrangian and Hamiltonian mechanics)

吕洪升, 张宏彬, 顾书龙. Noether's theory of Lagrange systems in discrete case[J]. 中国物理B, 2011, 20(1): 11101-011101.

Lü Hong-Sheng(吕洪升), Zhang Hong-Bin(张宏彬), and Gu Shu-Long(顾书龙) . Noether's theory of Lagrange systems in discrete case[J]. Chin. Phys. B, 2011, 20(1): 11101-011101.

参考文献 40

[1]	Noether A E 1918 Nachr . Akad. Wiss. Gottingen Math. Phys. KI II 235
[2]	Li Z P 1981 Acta Phys. Sin. 30 1659 (in Chinese)
[3]	Mei F X 1999 Applications of Lie Group and Lie Algebra to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[4]	Zhao Y Y and Mei F X 1999 The Symmetry and Invariant and of Mechanical Systems (Beijing: Science Press) (in Chinese)
[5]	Fu J L and Chen L Q 2003 Phys. Lett. A 317 255
[6]	Fu J L and Chen L Q 2004 Mech. Res. Commun. 31 9
[7]	Chen X W, Li Y M and Zhao Y H 2005 Phys. Lett. A 337 274
[8]	Liu R W and Chen L Q 2004 Chin. Phys. 13 1615
[9]	Zhang H B and Chen L Q 2005 J. Phys. Soc. Jap. 74 905
[10]	Zhang Y and Mei F X 2000 Chin. Sci. Bull. 45 1354
[11]	Luo S K 2002 Chin. Phys. Lett. 19 449
[12]	Guo Y X, Wang Y, Chen G Y and Mei F X 2005 J. Math. Phys. 46 062902
[13]	Guo Y X, Liu S X, Liu C, Luo S K and Wang Y 2007 J. Math. Phys. 48 082901
[14]	Wang S Y and Mei F X 2002 Chin. Phys. 11 5
[15]	Wu H B 2005 Chin. Phys. 14 452
[16]	Xu X J, Qin M C and Mei F X 2004 Acta Mech. Sin. 20 668
[17]	Fang J H, Liao Y P, Ding N and Wang P 2006 Chin. Phys. 15 2792
[18]	Li Y C, Jing H X, Xia L L, Wang J and Hou Q B 2007 Chin. Phys. 16 2154
[19]	Ge W K and Mei F X 2007 Acta Phys. Sin. 56 2479 (in Chinese)
[20]	Lou Z M 2007 Acta Phys. Sin. 56 2475 (in Chinese)
[21]	Ge W K and Mei F X 2009 Acta Phys. Sin. 58 699 (in Chinese)
[22]	Liu C , Zhao Y H and Chen X W 2010 Acta Phys. Sin. 59 11 (in Chinese)
[23]	Gu S L and Zhang H B 2010 Acta Phys. Sin. 59 716 (in Chinese)
[24]	Levi D and Winternitz P 1991 Phys. Lett. A 152 335
[25]	Levi D and Winternitz P 1993 J. Math. Phys. 34 3713
[26]	Levi D and Winternitz P 1996 J. Math. Phys. 37 5551
[27]	Levi D and Rodriguez M A 1999 J. Phys. A: Math. Gen. 32 8303
[28]	Levi D, Tremblay S and Winternitz P 2000 J. Phys. A: Math. Gen. 33 8507
[29]	Hydon P E 2000 Proc. R. Soc. A 456 2835
[30]	Levi D and Winternitz P 2006 J. Phys. A: Math.Gen. 36 R1
[31]	Dorodnitsyn V A 1991 J. Sov. Math. 55 1490
[32]	Dorodnitsyn V A 1998 Int. J. Mod. Phys. 5 723
[33]	Dorodnitsyn V A and Winternitz P 2000 Nonlinear Dynamics 22 49
[34]	Dorodnitsyn V A, Kozlov R and Winternitz P 2000 J. Math. Phys. 41 480
[35]	Dorodnitsyn V A and Kozlov R 2003 J. Nonl. Math. Phys. 10 16
[36]	Dorodnitsyn V A, Kozlov R and Winternitz P 2004 J. Math. Phys. 45 336
[37]	Zhang H B, Chen L Q and Liu R W 2005 Chin. Phys. 14 1063
[38]	Fu J L, Dai G D, Jim'enez S and Tang Y F 2007 Chin. Phys. 16 570
[39]	Dorodnitsyn V A 2001 Appl. Numer. Math. 307 321
[40]	Fu J L, Chen L Q and Chen B Y 2009 Sin. China Ser. G 39 1320 endfootnotesize

Noether's theory of Lagrange systems in discrete case

Noether's theory of Lagrange systems in discrete case

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 40

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Xilin Lu, Zi-Min Li, Vladimir V Mangazeev, and Murray T Batchelor. Hidden symmetry operators for asymmetric generalized quantum Rabi models[J]. 中国物理B, 2022, 31(1): 14210-014210.
[2]	. [J]. 中国物理B, 2020, 29(12): 124501-.
[3]	夏丽莉, 戈新生, 陈立群. Discrete symmetrical perturbation and variational algorithm of disturbed Lagrangian systems[J]. 中国物理B, 2019, 28(3): 30201-030201.
[4]	张毅, 田雪. Conservation laws for Birkhoffian systems of Herglotz type[J]. 中国物理B, 2018, 27(9): 90502-090502.
[5]	陈向炜, 张晔, 梅凤翔. An application of a combined gradient system to stabilize a mechanical system[J]. 中国物理B, 2016, 25(10): 100201-100201.
[6]	梅凤翔, 吴惠彬. Skew-gradient representation of generalized Birkhoffian system[J]. 中国物理B, 2015, 24(10): 104502-104502.
[7]	Worasak Prarokijjak, Bumned Soodchomshom. Electron with arbitrary pseudo-spins in multilayer graphene[J]. , 2015, 24(4): 48101-048101.
[8]	张智勇, 陈玉福. Conservation laws of the generalized short pulse equation[J]. , 2015, 24(2): 20201-020201.
[9]	夏丽莉, 陈立群, 傅景礼, 吴旌贺. Symmetries and variational calculationof discrete Hamiltonian systems[J]. , 2014, 23(7): 70201-070201.
[10]	赵纲领, 陈立群, 傅景礼, 洪方昱. Mei symmetry and conservation laws of discrete nonholonomic dynamical systems with regular and irregular lattices[J]. 中国物理B, 2013, 22(3): 30201-030201.
[11]	王肖肖, 韩月林, 张美玲, 贾利群. Lie symmetry and its generation of conserved quantity of Appell equation in a dynamical system of the relative motion with Chetaev-type nonholonomic constraints[J]. 中国物理B, 2013, 22(2): 20201-020201.
[12]	荆建春, 李彪. Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients[J]. Chin. Phys. B, 2013, 22(1): 10303-010303.
[13]	韩月林, 孙现亭, 王肖肖, 张美玲, 贾利群. A type of new conserved quantity deduced from Mei symmetry for Nielsen equations in a holonomic system with unilateral constraints[J]. Chin. Phys. B, 2012, 21(12): 120201-120201.
[14]	张世华, 陈本永, 傅景礼. Hamilton formalism and Noether symmetry for mechanico–electrical systems with fractional derivatives[J]. 中国物理B, 2012, 21(10): 100202-100202.
[15]	张美玲, 王肖肖, 韩月林, 贾利群. Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system of relative motion[J]. 中国物理B, 2012, 21(10): 100203-100203.