Symmetries and variational calculationof discrete Hamiltonian systems

doi:10.1088/1674-1056/23/7/070201

Abstract We present a numerical simulation method of Noether and Lie symmetries for discrete Hamiltonian systems. The Noether and Lie symmetries for the systems are proposed by investigating the invariance properties of discrete Lagrangian in phase space. The numerical calculations of a two-degree-of-freedom nonlinear harmonic oscillator show that the difference discrete variational method preserves the exactness and the invariant quantity.

Keywords: discrete Hamiltonian systems discrete variational integrators symmetry conserved quantity

Received: 17 October 2013
Revised: 26 January 2014
Accepted manuscript online:

PACS:	02.20.Sv	(Lie algebras of Lie groups)
	02.20.Qs	(General properties, structure, and representation of Lie groups)
	11.30.-j	(Symmetry and conservation laws)
	45.20.Jj	(Lagrangian and Hamiltonian mechanics)

Fund: Project supported by the Key Program of National Natural Science Foundation of China (Grant No. 11232009), the National Natural Science Foundation of China (Grant Nos. 11072218, 11272287, and 11102060), the Shanghai Leading Academic Discipline Project, China (Grant No. S30106), the Natural Science Foundation of Henan Province, China (Grant No. 132300410051), and the Educational Commission of Henan Province, China (Grant No. 13A140224).

Corresponding Authors: Chen Li-Qun
E-mail: lqchen@straff.shu.edu.cn

About author: 02.20.Sv; 02.20.Qs; 11.30.-j; 45.20.Jj

Cite this article:

Xia Li-Li (夏丽莉), Chen Li-Qun (陈立群), Fu Jing-Li (傅景礼), Wu Jing-He (吴旌贺) Symmetries and variational calculationof discrete Hamiltonian systems 2014 Chin. Phys. B 23 070201

[1]	Noether A E 1918 Math. Phys. KI. 2 235
[2]	Djukic D D S and Vujanovic B D 1975 Acta Mech. 23 17
[3]	Sarlet W and Cantrijn F 1981 SIAM Rev. 23 467
[4]	Lutzky M 1979 J. Phys. A: Math. Gen. 12 973
[5]	Lutzky M 1979 Phys. Lett. A 72 86
[6]	Mei F X 2000 J. Beijing Inst. Technol. 9 120
[7]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[8]	Mei F X, Xu X J and Zhang Y F 2004 Acta Mech. Sin. 20 668
[9]	Dorodnitsyn V 2011 Applications of Lie Groups to Difference Equations (Boca Raton, FL: Chapman & Hall/CRC)
[10]	Logan J D 1973 Aequat. Math. 9 210
[11]	Dorodnitsyn V 2001 Appl. Numer. Math. 39 307
[12]	Dorodnitsyn V and Kozlov R 2009 J. Phys. A: Math. Theor. 42 454007
[13]	Fu J L, Dai G D, Salvador J and Tang Y F 2007 Chin. Phys. 16 570
[14]	Levi D, Tremblay S and Winternitz P 2000 J. Phys. A: Math. Gen. 33 8507
[15]	Xia L L and Chen L Q 2012 Nonlinear Dyn. 70 1223
[16]	Grinspun E, Desbrun M, Polthier K, Schröder P and Stern A 2006 Discrete Differential Geometry: An Applied Introduction – The 33rd International Conference and Exhibition on Computer Graphics and Interactive Techniques, July 30-August 3, 2006 Boston, USA, (ACM SIGGRAPH 2006 Course 1)
[17]	Cadzow J A 1970 Int. J. Control 11 393
[18]	Wendlandt J M and Marsden J E 1997 Physica D: Nonlinear Phenomena 106 223
[19]	Marsden J E, Patrick G W and Shkoller S 1998 Commun. Math. Phys. 199 351
[20]	Kane C, Marsden J E and Ortiz M 1999 J. Math. Phys. 40 3353
[21]	Cortés J and Martínez S 2001 Nonlinearity 14 1365
[22]	Guo H Y, Wu K, Wang S H, Wang S K and Wei J M 2000 Commun. Theor. Phys. 34 307
[23]	Guo H Y, Li Y Q and Wu K 2001 Commun. Theor. Phys. 35 703
[24]	Chen J B, Guo H Y and Wu K 2003 J. Math. Phys. 44 1688
[25]	McLachlan R and Perlmutter M 2006 J. Nonlinear Sci. 16 283
[26]	Guo H Y and Wu K 2003 J. Math. Phys. 44 5978
[27]	Liu S X, Liu C and Guo Y X 2011 Chin. Phys. B 20 034501
[28]	Zhang H B, Chen L Q and Liu R W 2005 Chin. Phys. 14 1063
[29]	Kane C, Marsden J E, Ortiz M and West M 1999 Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems (Ph.D. dissertation) (California: Caltech)
[30]	Maeda S 1980 Math. Japonica 25 405
[31]	Levi D and Winternitz P 1991 Phys. Lett. A 152 335
[32]	Dorodnitsyn V and Winternitz P 2000 Nonlinear Dyn. 22 49
[33]	Bahar L Y and Kwatny H G 1987 Int. J. Nonlinear Mech. 22 125
[34]	Miller K S 1968 The American Mathematical Monthly 75 630
[35]	Marsden J E and West M 2001 Acta Numerica 10 357
[36]	Hairer E, Lubich C and Wanner G 2003 Acta Numerica 12 399

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