Lagrange方程应用于流体动力学
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摘要
如何将Lagrange方程应用于流体动力学的问题是一个理论研究的难题。按照从变分学的基本理论研究做起的思想,本文应用变导的概念和运算法则,通过研究Lagrange方程中求导的性质,逐步地将Lagrange方程应用于理想流体动力学。按照从变分学的基本理论研究做起的思想,本文应用Lagrange-Hamilton体系,即非保守系统的Lagrange方程是非保守系统的Hamilton型拟变分原理的拟驻值条件,由不可压缩黏性流体动力学的Hamilton型拟变分原理推导出不可压缩黏性流体动力学的Lagrange方程,进而应用不可压缩黏性流体动力学的Lagrange方程推导出不可压缩黏性流体动力学的控制方程。探讨将Lagrange方程应用于可压缩黏性流体动力学问题中,推导出可压缩黏性流体动力学的控制方程。本文解决了如何将Lagrange方程应用于流体动力学的问题。
The application of the Lagrange equation to fluid dynamics is difficult in theoretical research. In accordance with basic theory research on the calculus of variations,the concept of the variational derivative and algorithm are applied. The Lagrange equation is applied to ideal fluid dynamics gradually by studying the property of derivation from the Lagrange equation. The Lagrange-Hamilton system,which is the Lagrange equation of a non-conservative system,is a quasi-stationary condition of Hamilton-type quasi-variational principle of a non-conservative system. The Lagrange equations for incompressible viscous fluid dynamics are derived from the Hamiltonian quasi-variational principle of the incompressible viscous fluid dynamics successfully. The governing equations of incompressible viscous fluid dynamics are deduced from the Lagrange equation of incompressible viscous fluid dynamics. Finally,application of the Lagrange equation to the questions of compressible viscous fluid dynamics is discussed. This paper comprehensively describes how to apply the Lagrange equation to fluid dynamics.
The application of the Lagrange equation to fluid dynamics is difficult in theoretical research. In accordance with basic theory research on the calculus of variations,the concept of the variational derivative and algorithm are applied. The Lagrange equation is applied to ideal fluid dynamics gradually by studying the property of derivation from the Lagrange equation. The Lagrange-Hamilton system,which is the Lagrange equation of a non-conservative system,is a quasi-stationary condition of Hamilton-type quasi-variational principle of a non-conservative system. The Lagrange equations for incompressible viscous fluid dynamics are derived from the Hamiltonian quasi-variational principle of the incompressible viscous fluid dynamics successfully. The governing equations of incompressible viscous fluid dynamics are deduced from the Lagrange equation of incompressible viscous fluid dynamics. Finally,application of the Lagrange equation to the questions of compressible viscous fluid dynamics is discussed. This paper comprehensively describes how to apply the Lagrange equation to fluid dynamics.
引文
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[10]赵晓兵,方秦.Lagrange方程在防护工程中的应用及其相关的力学问题[J].防护工程,2000(2):28-32.ZHAO Xiaobing,FANG Qin.Application of Lagrange equation in protection engineering and its related mechanical problems[J].Protection engineering,2000(2):28-32.
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[12]颜振珏.非惯性参照系中的Lagrange方程[J].黔南民族师范学院学报,2004,24(6):8-11.YAN Zhenyu.Lagrange equation in uninertia system[J].Journal of Qiannan Normal University for Nationalities,2004,24(6):8-11.
[13]廖旭.非惯性系中的Lagrange方程及其应用[J].云南大学学报(自然科学版),2004,26(B07):122-124.LIAO Xu.Lagrange equation in noninertial frame and application[J].Journal of Yunnan University(natural sciences edition),2004,26(B07):122-124.
[14]和兴锁,宋明,邓峰岩.非惯性系下考虑剪切变形的柔性梁的动力学建模[J].物理学报,2011,60(4):323-328.HE Xingsuo,SONG Ming,DENG Fengyan.Dynamic modeling of flexible beam with considering shear deformation in non-inertial reference frame[J].Acta physica sinica,2011,60(4):323-328.
[15]沈惠川.弹性力学的Lagrange形式:用Routh方法建立弹性有限变形问题的基本方程[J].数学物理学报,1998,18(1):78-88.SHEN Huichuan.Lagrange formalism of elasticity:building the basic equations on finite-deformation problems by routh's method[J].Acta mathematica scientia,1998,18(1):78-88.
[16]方刚,张斌.弹性介质的Lagrange动力学与地震波方程[J].物理学报,2013,(15):248-253.FANG Gang,ZHANG Bin.Lagrangian dynamics and seismic wave align of elastic medium[J].Acta physica sinica,2013,(15):248-253.
[17]薛纭,翁德玮,陈立群.精确Cosserat弹性杆动力学的分析力学方法[J].物理学报,2013,(4):312-318.XUE Yun,WENG Dewei,CHEN Liqun.Methods of analytical mechanics for exact Cosserat elastic rod dynamics[J].Acta physica sinica,2013,(4):312-318.
[18]马善钧,徐学翔,黄沛天,等.完整系统三阶Lagrange方程的一种推导与讨论[J].物理学报,2004,53(11):3648-3651.MA Shanjun,XU Xuexiang,HUANG Peitian,et al.The discussion on Lagrange equation containing third order derivatives[J].Acta physica sinica,2004,53(11):3648-3651.
[19]丁光涛.状态空间Lagrange函数和运动方程[J].中国科学(G辑),2009(6):813-820.DING Guangtao.The state space Lagrange function and the corresponding equation of motion[J].Science in China(series G),2009(6):813-820.
[20]梅凤翔.广义Birkhoff系统动力学[M].北京:北京理工大学出版社,1996.MEI Fengxiang.Generalized Birkhoff system dynamics[M].Beijing:Beijing Institute of Technology Press,1996.
[21]LIANG Lifu,SHI Zhifei.On the inverse problem in calculus of variations[J].Applied mathematics and mechanics,1994,15(9):815-830.
[22]LIANG Lifu,HU Haichang.Generalized variational principle of three kinds of variables in general mechanics[J].Science in China(A),2001,44(6):770-776.
[23]梁立孚.变分原理及其应用[M].哈尔滨:哈尔滨工程大学出版社,2005.LIANG Lifu.Variational principles and their applications[M].Harbin:Harbin Engineering University Press,2005.
[24]陈滨.分析动力学[M].2版.北京:北京大学出版社,2010.CHEN Bin.Analytical mechanics[M].2 ed.Beijing:Peking University Press,2010.
[25]梅凤翔.分析力学[M].北京:北京理工大学出版社,2013.MEI Fengxiang.Analytical mechanics[M].Beijing:Beijing Institute of Technology Press,2013.
[2]HAMILTON W R.On a general method in dynamics[M].[S.l.]:Philosophical Transaction of the Royal Society,Part I,1834:247-308.
[3]汪家訸.分析动力学[M].北京:高等教育出版社,1958.
[4]GOLDSTEIN H.Classical Mechanics[M].2nd ed.[S.l.]:Addison-Wesley Publishing Co.,1980.
[5]赵俊伟,李雪锋,陈国强.基于Lagrange方法的3-PRS并联机构动力学分析[J].机械设计与研究,2015,31(2):1-5.ZHAO Junwei,LI Xuefeng,CHEN Guoqiang.The dynamics equation of A 3-PRS parallel manipulator based on lagrange method[J].Machine design&research,2015,31(2):1-5.
[6]林良明,吴俊.用Lagrange方程描述假手机构动力学的研究[J].中国生物医学工程学报,1989,8(1):1-8.LIN Liangming,WU Jun.Description of mechanism dynamics of prosthetichand using lagrangian equation[J].Chinese journal of biomedical engineering,1989,8(1):1-8.
[7]王启明,汪劲松,刘辛军,等.二移动自由度并联操作臂的动力学建模[J].清华大学学报(自然科学版),2002,42(11):1469-1472.WANG Qiming,WANG Jinsong,LIU Xinjun,et al.Dynamic modeling of a parallel manipulator with two translational degrees of freedom[J].Journal of Tsinghua University(science and technology),2002,42(11):1469-1472.
[8]孙伟,汪博,鲁明,等.基于拉格朗日方程的直线滚动导轨系统解析建模[J].计算机集成制造系统,2012,18(4):781-786.SUN Wei,WANG Bo,LU Ming,et al.Analytical modeling of linear rolling guide system based on lagrange equation[J].Computer integrated manufacturing systems,2012,18(4):781-786.
[9]卢长福,傅鹏,黄诚,等.Lagrange方程在振动系统中的应用[J].江西科学,2013,3(2):148-150.LU Changfu,FU Peng,HUANG Cheng,et al.The application of lagrange equation in vibration systen[J].Jiangxi science,2013,3(2):148-150.
[10]赵晓兵,方秦.Lagrange方程在防护工程中的应用及其相关的力学问题[J].防护工程,2000(2):28-32.ZHAO Xiaobing,FANG Qin.Application of Lagrange equation in protection engineering and its related mechanical problems[J].Protection engineering,2000(2):28-32.
[11]靳希,鲁炜.Lagrange方程应用于电系统和机电系统运动分析[J].上海电力学院学报,2003,19(4):1-4.JIN Xi,LU Wei.The application of lagrange equation in the analysis of electric and electromechanical system[J].Journal of Shanghai University of Electric Power,2003,19(4):1-4.
[12]颜振珏.非惯性参照系中的Lagrange方程[J].黔南民族师范学院学报,2004,24(6):8-11.YAN Zhenyu.Lagrange equation in uninertia system[J].Journal of Qiannan Normal University for Nationalities,2004,24(6):8-11.
[13]廖旭.非惯性系中的Lagrange方程及其应用[J].云南大学学报(自然科学版),2004,26(B07):122-124.LIAO Xu.Lagrange equation in noninertial frame and application[J].Journal of Yunnan University(natural sciences edition),2004,26(B07):122-124.
[14]和兴锁,宋明,邓峰岩.非惯性系下考虑剪切变形的柔性梁的动力学建模[J].物理学报,2011,60(4):323-328.HE Xingsuo,SONG Ming,DENG Fengyan.Dynamic modeling of flexible beam with considering shear deformation in non-inertial reference frame[J].Acta physica sinica,2011,60(4):323-328.
[15]沈惠川.弹性力学的Lagrange形式:用Routh方法建立弹性有限变形问题的基本方程[J].数学物理学报,1998,18(1):78-88.SHEN Huichuan.Lagrange formalism of elasticity:building the basic equations on finite-deformation problems by routh's method[J].Acta mathematica scientia,1998,18(1):78-88.
[16]方刚,张斌.弹性介质的Lagrange动力学与地震波方程[J].物理学报,2013,(15):248-253.FANG Gang,ZHANG Bin.Lagrangian dynamics and seismic wave align of elastic medium[J].Acta physica sinica,2013,(15):248-253.
[17]薛纭,翁德玮,陈立群.精确Cosserat弹性杆动力学的分析力学方法[J].物理学报,2013,(4):312-318.XUE Yun,WENG Dewei,CHEN Liqun.Methods of analytical mechanics for exact Cosserat elastic rod dynamics[J].Acta physica sinica,2013,(4):312-318.
[18]马善钧,徐学翔,黄沛天,等.完整系统三阶Lagrange方程的一种推导与讨论[J].物理学报,2004,53(11):3648-3651.MA Shanjun,XU Xuexiang,HUANG Peitian,et al.The discussion on Lagrange equation containing third order derivatives[J].Acta physica sinica,2004,53(11):3648-3651.
[19]丁光涛.状态空间Lagrange函数和运动方程[J].中国科学(G辑),2009(6):813-820.DING Guangtao.The state space Lagrange function and the corresponding equation of motion[J].Science in China(series G),2009(6):813-820.
[20]梅凤翔.广义Birkhoff系统动力学[M].北京:北京理工大学出版社,1996.MEI Fengxiang.Generalized Birkhoff system dynamics[M].Beijing:Beijing Institute of Technology Press,1996.
[21]LIANG Lifu,SHI Zhifei.On the inverse problem in calculus of variations[J].Applied mathematics and mechanics,1994,15(9):815-830.
[22]LIANG Lifu,HU Haichang.Generalized variational principle of three kinds of variables in general mechanics[J].Science in China(A),2001,44(6):770-776.
[23]梁立孚.变分原理及其应用[M].哈尔滨:哈尔滨工程大学出版社,2005.LIANG Lifu.Variational principles and their applications[M].Harbin:Harbin Engineering University Press,2005.
[24]陈滨.分析动力学[M].2版.北京:北京大学出版社,2010.CHEN Bin.Analytical mechanics[M].2 ed.Beijing:Peking University Press,2010.
[25]梅凤翔.分析力学[M].北京:北京理工大学出版社,2013.MEI Fengxiang.Analytical mechanics[M].Beijing:Beijing Institute of Technology Press,2013.