弱非线性动力学方程的Noether准对称性与近似Noether守恒量  

NOETHER QUASI-SYMMETRY AND APPROXIMATE NOETHER CONSERVATION LAWS FOR WEAKLY NONLINEAR DYNAMICAL EQUATIONS

作　　者：张毅[1] Zhang Yi(College of Civil Engineerings Suzhou University of Science and Technology,Suzhou 215011,Jiangsu,China)

机构地区：[1]苏州科技大学土木工程学院,江苏苏州215011

出　　处：《力学学报》2020年第6期1765-1773,共9页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金(11972241,11572212);江苏省自然科学基金(BK20191454)资助项目。

摘　　要：自然界和工程技术领域存在大量的非线性问题,它们通常需要用非线性微分方程来描述.守恒量在微分方程的求解、约化和定性分析方面发挥重要作用.因此,研究非线性动力学方程的近似守恒量具有重要意义.文章利用Noether对称性方法研究弱非线性动力学方程的近似守恒量.首先,将弱非线性动力学方程化为一般完整系统的Lagrange方程,在Lagrange框架下建立Noether准对称性的定义和广义Noether等式,给出近似Noether守恒量.其次,将弱非线性动力学方程化为相空间中一般完整系统的Hamilton方程,在Hamilton框架下建立Noether准对称性的定义和广义Noether等式,给出近似Noether守恒量.再次,将弱非线性动力学方程化为广义Birkhoff方程,在Birkhoff框架下建立Noether准对称性的定义和广义Noether等式,给出近似Noether守恒量.最后,以著名的van der Pol方程,Duffing方程以及弱非线性耦合振子为例,分析三个不同框架下弱非线性系统的Noether准对称性与近似Noether守恒量的计算.结果表明:同一弱非线性动力学方程可以化为不同的一般完整系统或不同的广义Birkhoff系统;Hamilton框架下的结果是Birkhoff框架的特例,而Lagrange框架下的结果与Hamilton框架的等价.利用Noether对称性方法寻找弱非线性动力学方程的近似守恒量不仅方便有效,而且具有较大的灵活性.There are a lot of nonlinear problems in nature and engineering technology,which need to be described by nonlinear differential equations.Conservation laws play an important role in solving,reducing and qualitative analysis of differential equations.Therefore,it is of great significance to study the approximate conservation laws of nonlinear dynamical equations.In this paper,we apply the Noether symmetry method to the study of approximate conservation laws of weakly nonlinear dynamical equations.Firstly,the weakly nonlinear dynamical equations are transformed into the Lagrange equations of general holonomic system.Under the Lagrangian framework,the definition of Noether quasisymmetry and the generalized Noether identities are established,and the approximate Noether conservation laws are obtained.Secondly,the weakly nonlinear dynamical equations are transformed into the Hamilton equations of general holonomic system in phase space.Under the Hamiltonian framework,the definition of Noether quasi-symmetry and the generalized Noether identities are established,and the approximate Noether conservation laws are obtained.Thirdly,the weakly nonlinear dynamical equations are transformed into the generalized Birkhoff’s equations.Under the Birkhoffian framework,the definition of Noether quasi-symmetry and the generalized Noether identities are established,and the approximate Noether conservation laws are obtained.Finally,taking the famous Van der Pol equation,the Duffing equation and the weakly nonlinear coupled oscillators as examples,the computation of Noether quasi-symmetries and approximate conservation laws for weakly nonlinear systems under three different frameworks is analyzed.The results show that the same weakly nonlinear dynamical equation can be reduced to different general holonomic systems or different generalized Birkhoff systems.The result under the Hamiltonian framework is a special case of the Birkhoffian framework,while the result under the Lagrangian framework is equivalent to that under the Hamiltonian

关 键 词：弱非线性动力学 近似Noether守恒量 Noether准对称性 广义Noether等式 

分 类 号：O316[理学—一般力学与力学基础]