弱非线性动力学方程的 Noether 准对称性与近似 Noether 守恒量

张毅

doi:10.6052/0459-1879-20-242

弱非线性动力学方程的 Noether 准对称性与近似 Noether 守恒量

doi: 10.6052/0459-1879-20-242

张毅^2), ,

苏州科技大学土木工程学院,江苏苏州 215011

基金项目: 1) 国家自然科学基金(11972241);国家自然科学基金(11572212);江苏省自然科学基金(BK20191454)

详细信息

作者简介:
2) 张毅, 教授, 主要研究方向: 分析力学. E-mail: zhy@mail.usts.edu.cn

通讯作者:
张毅

中图分类号: O316
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出版历程
- 收稿日期: 2020-07-06
- 刊出日期: 2020-12-10

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

Zhang Yi^{2)
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College of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,Jiangsu,China

摘要

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.
- 弱非线性动力学 /
- 近似 Noether 守恒量 /
- Noether 准对称性 /
- 广义 Noether 等式
Abstract: 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 quasi-symmetry 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 framework. Using Noether symmetry method to find approximate conservation laws of weakly nonlinear dynamical equations is not only convenient and effective, but also has great flexibility.
- weakly nonlinear dynamics /
- approximate Noether conservation laws /
- Noether quasi-symmetry /
- generalized Noether identity

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参考文献(1)

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