欧拉-拉格朗日方程在一维波动方程中的应用

APPLICATION OF EULER-LAGRANGE EQUATION IN ONE-DIMENSIONAL WAVE EQUATION

本文以一维弦上微元的动能和势能为基础,推导出了一维波动方程。文章首先介绍了通过力学分析得到一维波动方程的方法。然后分析了一维自由运动粒子的动能和势能,引入系统的哈密顿量和拉格朗日函数,由最小作用原理得到了欧拉-拉格朗日方程,也就是粒子的运动方程。将这一方法用于分析一维弦上波动,给出微元的拉格朗日密度函数,得到可以描写无穷多自由度系统的欧拉-拉格朗日方程,从而导出了一维波动方程。最后分析了一维弦上波动的拉格朗日密度与弦理论中Polyakov作用量中的拉格朗日密度的关系。

In this paper,the one-dimensional wave function is studied in frame of the kinetic energy and potential energy.In general,one-dimensional wave equation is obtained through the force analysis of an arbitrary string element and Newton＇s second law.In this paper,we introduce the Lagrangian of a particle,which moves in the potential V.Then Euler-Lagrange equation,Which is also the motion equation of particle,is given based on the principle of the least action.In the frame of this theory,we give the kinetic energy and potential energy of the string element.Then the Lagrange density function of the 1-dimension string element is defined.The Euler-Lagrange equation to describe a system with infinite degrees of freedom is obtained.Based on these,the one-dimensional wave equation is revealed.At last,we give the relations between Lagrange density function in one-dimensional wave and Lagrange density function in Polyakov interaction in string theory.