Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics

doi:10.1088/1674-1056/20/3/034502

Abstract This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics. The differential equations of motion of the system are established. The definition and the criterion of the symmetry of Hamiltonian of the system are given. A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given. Since a Hamilton system is a special case of the generalized classical mechanics, the results above are equally applicable to the Hamilton system. The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian. Finally, two examples are given to illustrate the application of the results.

Keywords: symmetry of Hamiltonian generalized classical mechanics conserved quantity

Received: 11 September 2010
Revised: 10 October 2010
Accepted manuscript online:

PACS:	45.20.Jj	(Lagrangian and Hamiltonian mechanics)
	11.30.Na	(Nonlinear and dynamical symmetries (spectrum-generating symmetries))
	02.30.Hq	(Ordinary differential equations)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10972151).

Cite this article:

Zhang Yi(张毅) Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics 2011 Chin. Phys. B 20 034502

[1]	Podolsky B 1942 Phys. Rev. 65 228
[2]	De Le'on M and Rodrigues P R 1985 Generalized Classical Mechanics and Field Theory (Amsterdam, North-Holland: Elservier Science Publishers B V)
[3]	Li Z P 1993 Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properties (Beijing: Beijing Polytechnic University Press)
[4]	Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)
[5]	Li Z P and Jiang J H 2002 Symmetries in Constrained Canonical Systems (Beijing: Science Press)
[6]	Mei F X 1990 Appl. Math. Mech. 11 569
[7]	Qiao Y F, Yue Q W and Dong Y A 1994 Appl. Math. Mech. 15 877
[8]	Zhang Y, Shang M and Mei F X 2000 Chin. Phys. 9 401
[9]	Qiao Y F, Li R J and Zhao S H 2001 Acta Phys. Sin. 50 811 (in Chinese)
[10]	Qiao Y F, Zhang Y L and Han G C 2002 Chin. Phys. 11 988
[11]	Luo S K 2002 Acta Phys. Sin. 51 1416 (in Chinese)
[12]	Zhang Y 2006 Chin. Phys. 15 1935
[13]	Zhang P Y, Fang J H, Wang P and Ding N 2006 Commun. Theor. Phys. (Beijing, China) 45 961
[14]	Currie D G and Saletan E F 1966 J. Math. Phys. 7 967
[15]	Hojman S and Harleston H 1981 J. Math. Phys. 22 1414
[16]	Zhao Y Y and Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press)
[17]	Hojman S 1984 J. Phys. A: Math. Gen. 17 2399
[18]	Mei F X and Wu H B 2008 Phys. Lett. A 372 2141
[19]	Wu H B and Mei F X 2009 Chin. Phys. B 18 3145
[20]	Mei F X and Wu H B 2009 Acta Phys. Sin. 58 5919 (in Chinese)

[1]	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.
[2]	Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales Jing Song(宋静), Yi Zhang(张毅). Chin. Phys. B, 2017, 26(8): 084501.
[3]	Non-Noether symmetries of Hamiltonian systems withconformable fractional derivatives Lin-Li Wang (王琳莉) and Jing-Li Fu(傅景礼). Chin. Phys. B, 2016, 25(1): 014501.
[4]	Symmetries and variational calculationof discrete Hamiltonian systems Xia Li-Li (夏丽莉), Chen Li-Qun (陈立群), Fu Jing-Li (傅景礼), Wu Jing-He (吴旌贺). Chin. Phys. B, 2014, 23(7): 070201.
[5]	Noether symmetry and conserved quantity for a Hamilton system with time delay Jin Shi-Xin (金世欣), Zhang Yi (张毅). Chin. Phys. B, 2014, 23(5): 054501.
[6]	Noether's theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives Zhou Yan (周燕), Zhang Yi (张毅). Chin. Phys. B, 2014, 23(12): 124502.
[7]	Lie symmetry theorem of fractional nonholonomic systems Sun Yi (孙毅), Chen Ben-Yong (陈本永), Fu Jing-Li (傅景礼). Chin. Phys. B, 2014, 23(11): 110201.
[8]	Conformal invariance, Noether symmetry, Lie symmetry and conserved quantities of Hamilton systems Chen Rong (陈蓉), Xu Xue-Jun (许学军). Chin. Phys. B, 2012, 21(9): 094501.
[9]	A type of conserved quantity of Mei symmetry of Nielsen equations for a holonomic system Cui Jin-Chao (崔金超), Han Yue-Lin (韩月林), Jia Li-Qun (贾利群 ). Chin. Phys. B, 2012, 21(8): 080201.
[10]	Symmetry of Lagrangians of holonomic nonconservative system in event space Zhang Bin(张斌), Fang Jian-Hui(方建会), and Zhang Wei-Wei(张伟伟) . Chin. Phys. B, 2012, 21(7): 070208.
[11]	Noether conserved quantities and Lie point symmetries for difference nonholonomic Hamiltonian systems in irregular lattices Xia Li-Li(夏丽莉) and Chen Li-Qun(陈立群) . Chin. Phys. B, 2012, 21(7): 070202.
[12]	Form invariance and approximate conserved quantity of Appell equations for a weakly nonholonomic system Jia Li-Qun(贾利群), Zhang Mei-Ling(张美玲), Wang Xiao-Xiao(王肖肖), and Han Yue-Lin(韩月林) . Chin. Phys. B, 2012, 21(7): 070204.
[13]	Symmetry of Lagrangians of a holonomic variable mass system Wu Hui-Bin(吴惠彬) and Mei Feng-Xiang(梅凤翔) . Chin. Phys. B, 2012, 21(6): 064501.
[14]	Mei symmetry and Mei conserved quantity of the Appell equation in a dynamical system of relative motion with non-Chetaev nonholonomic constraints Wang Xiao-Xiao(王肖肖), Sun Xian-Ting(孙现亭), Zhang Mei-Ling(张美玲), Han Yue-Lin(韩月林), and Jia Li-Qun(贾利群) . Chin. Phys. B, 2012, 21(5): 050201.
[15]	Mei conserved quantity directly induced by Lie symmetry in a nonconservative Hamilton system Fang Jian-Hui(方建会), Zhang Bin(张斌), Zhang Wei-Wei(张伟伟), and Xu Rui-Li(徐瑞莉) . Chin. Phys. B, 2012, 21(5): 050202.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics

Cite this article:

Online attention