微扰力系统一阶近似守恒量与对称性研究

楼智美

doi:10.3969/j.issn.1000-5641.2017.03.011

微扰力系统一阶近似守恒量与对称性研究

doi: 10.3969/j.issn.1000-5641.2017.03.011

楼智美

绍兴文理学院物理系, 浙江绍兴 312000

基金项目:

国家自然科学基金 11472177

详细信息

作者简介:
楼智美, 女, 教授, 主要从事分析力学研究. E-mail: louzhimei@usx.edu.cn

中图分类号: O316
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- 文章访问数: 209
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- 被引次数: 0
出版历程
- 收稿日期: 2016-04-01
- 刊出日期: 2017-05-25

The study of the first order approximate conserved quantities and approximate symmetries of perturbed mechanical system

LOU Zhi-mei

Department of Physics, Shaoxing University, Shaoxing Zhejiang 312000, China

摘要

摘要: 提出了用泊松括号求一阶近似守恒量的方法, 将微扰力学系统的Hamilton函数看成是未受微扰作用系统的Hamilton函数和微扰项两部分组成.先根据未受微扰作用力学系统的特点选择一种合适的方法求得其精确守恒量, 再利用泊松括号和偏微分方程的性质求得守恒量的一阶微扰项, 最后根据Noether对称性、Lie对称性和Mei对称性性质, 求得与一阶近似守恒量相应的一阶近似Noether对称性、近似Lie对称性和近似Mei对称性.研究了受微扰作用的二维各向同性谐振子的一阶近似守恒量和近似对称性, 得到了系统的3个一阶近似守恒量及它们相应的一阶近似对称性.结果表明, 与3个一阶近似守恒量相应的一阶近似对称性既是近似Noether对称性, 又是近似Lie对称性, 也是近似Mei对称性.
- 两自由度微扰力学系统 /
- 一阶近似守恒量 /
- 泊松括号法 /
- 一阶近似对称性
Abstract: A Poisson bracket method to obtain the first order approximate conserved quantities of two-dimensional perturbed mechanical system is proposed. We consider the perturbed Hamiltonian function as the combination of Hamiltonian function of unperturbed system and the perturbed term. First, according to the peculiarity of unperturbed system, we select a suitable method to obtain the exact conserved quantities of unperturbed system. Second, we calculate the first order perturbed terms of conserved quantities by using Poisson bracket and the character of partial differential equations. Finally, according to the characters of Noether symmetries, Lie symmetries and Mei symmetries, the first order approximate Noether symmetries, approximate Lie symmetries and approximate Mei symmetries of the first order approximate conserved quantities can be obtained. A perturbed two-dimensional isotropic harmonic oscillator is studied in this paper, and three first order approximate conserved quantities are obtained by using Poisson bracket method, and the first order approximate symmetries of three first order approximate conserved quantities are either approximate Noether symmetries or approximate Lie symmetries and Mei symmetries.
- two-dimensional perturbed mechanical system /
- first order approximate conserved quantities /
- Poisson bracket method /
- first order approximate symmetries

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参考文献(16)

[1]	LEACH P G L, MOYO S, COTSAKIS S, et al. Symmetry, singularities and integrability in complex dynamics Ⅲ: Approximate symmetries and invariants[J]. Journal of Nonlinear Mathematical Physics, 2001, 8(1): 139-156 doi: 10.2991/jnmp.2001.8.1.11
[2]	GOVINDER K S, HEIL T G, UZER T. Approximate Noether symmetries[J]. Physics Letters A, 1998, 240(3): 127-131 doi: 10.1016/S0375-9601(98)00067-X
[3]	NAEEM I, MAHOMED F M. Approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling [J]. Mathematical and Computational Applications, 2010, 15(4): 720-731
[4]	UNAL G. Approximate generalized symmetries, normal forms and approximate first integrals[J]. Physics Letters A, 2000, 266(2): 106-122
[5]	DOLAPIC I T, PAKDEMIRLI M. Approximate symmetries of creeping flow equations of a second grade fluid[J]. International Journal of Non-linear Mechanics, 2004, 39(10): 1603-1619 doi: 10.1016/j.ijnonlinmec.2004.01.002
[6]	KARA A H, MAHOMED F M, QADIR A. Approximate symmetries and conservation laws of the geodesic equations for the Schwarzschild metric[J]. Nonlinear Dynamics, 2008, 51(1-2): 183-188
[7]	GREBENEV V N, OBERLACK M. Approximate Lie symmetries of the Navier-Stokes equations[J]. Journal of Non-linear Mathematical Physics, 2007, 14(2): 157-163 doi: 10.2991/jnmp.2007.14.2.1
[8]	JOHNPILLAI A G, KARA A H, MAHOMED F M. Approximate Noether-typesymmetries and conservation laws via partial Lagrangians for PDEs with a small parameter[J]. Journal of Computational and Applied Mathematics, 2009, 223(1): 508-518 doi: 10.1016/j.cam.2008.01.020
[9]	ZHANG Z Y, YONG X L, CHEN Y F. A new method to obtain approximate symmetry of nonlinear evolution equation form perturbations[J]. Chinese Physics B, 2009, 18(7): 2629-2633 doi: 10.1088/1674-1056/18/7/002
[10]	楼智美.两自由度弱非线性耦合系统的一阶近似Lie对称性与近似守恒量[J].物理学报, 2013, 62(22): 220202. doi: 10.7498/aps.62.220202
[11]	楼智美, 梅凤翔, 陈子栋.弱非线性耦合二维各向异性谐振子的一阶近似Lie对称性与近似守恒量[J].物理学报, 2012, 61(11): 110204. doi: 10.7498/aps.61.110204
[12]	楼智美.微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量[J].物理学报, 2010, 59(10): 6764-6769. doi: 10.7498/aps.59.6764
[13]	楼智美.含非线性微扰项的二阶动力学系统的一阶近似守恒量的一种新求法[J].物理学报, 2014, 63(6): 060202. http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201406002.htm
[14]	梅凤翔.李群和李代数对约束力学系统的应用[M].北京:科学出版社, 1999: 120-126.
[15]	梅凤翔.约束力学系统的对称性与守恒量[M].北京:北京理工大学出版社, 2004: 10-14.
[16]	楼智美.用Noether定理确定各向同性谐振子的守恒量[J].力学与实践2003, 25(1): 72-73. http://www.cnki.com.cn/Article/CJFDTOTAL-LXYS200302029.htm