The purpose of this paper is to illustrate the theory and methods of analytical mechanics that can be effectively applied to the research of some nonlinear nonconservative systems through the case study of two-dimensionally coupled Mathews-Lakshmanan oscillator (abbreviated as M-L oscillator). (1) According to the inverse problem method of Lagrangian mechanics, the Lagrangian and Hamiltonian function in the form of rectangular coordinates of the two-dimensional M-L oscillator is directly constructed from an integral of the two-dimensional M-L oscillators. (2) The Lagrange and Hamiltonian function in the form of polar coordinate was rewritten by using coordinate transformation. (3) By introducing the vector form variables, the two-dimensional M-L oscillator motion differential equation, the first integral, and the Lagrange function are written. Therefore, the two-dimensional M-L oscillator is directly extended to the three-dimensional case, and it is proved that the three-dimensional M-L oscillator can be reduced to the two-dimensional case. (4) The two direct integration methods were provided to solve the two-dimensional M-L oscillator by using polar coordinate Lagrangian and pointed out that the one-dimensional M-L oscillator is a special case of the two-dimensional M-L oscillator.

中文翻译：

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二维Mathews-Lakshmanan振荡器的拉格朗日和哈密顿量

本文的目的是通过二维耦合Mathews-Lakshmanan振荡器（简称为ML振荡器）的案例研究，来说明可以有效地应用于某些非线性非保守系统研究的分析力学的理论和方法。（1）根据拉格朗日力学的反问题方法，二维ML振荡器的直角坐标形式的拉格朗日函数和汉密尔顿函数直接由二维ML振荡器的积分构造。（2）通过坐标变换重写了极坐标形式的拉格朗日函数和哈密顿函数。（3）通过引入向量形式变量，编写了二维ML振荡器运动微分方程，第一积分和Lagrange函数。因此，将二维ML振荡器直接扩展到三维情况，并且证明了可以将三维ML振荡器简化为二维情况。（4）利用极坐标拉格朗日方程提供了两种直接积分方法来求解二维ML振荡器，并指出一维ML振荡器是二维ML振荡器的特例。