浅谈变分原理及其一些应用  

A brief discussion on variational principle and some applications

作　　者：郑神州[1] 于海燕[2] ZHENG Shen-zhou;YU Hai-yan(School of Mathematics and Statistics,Beijing Jiaotong University,Beijing 100044,China;School of Mathematics and Physics,Inner Mongolia University for Nationalities,Tongliao,Inner Mongolia 028043,China)

机构地区：[1]北京交通大学数学与统计学院,北京100044 [2]内蒙古民族大学数理学院,内蒙古通辽028043

出　　处：《大学物理》2023年第10期1-5,共5页College Physics

基　　金：国家自然科学基金项目(12071021);北京交通大学研究生课程建设项目(134001103522)资助。

摘　　要：给出了变分原理的基本形式以及变分极值的欧拉-拉格朗日方程的推导,其次以短程线、测地线、最速降线和等周问题为例,分别推导出各具体形式的欧拉-拉格朗日方程表示式,并对相应原问题给出求解过程以及各种相关的推广.The variational principle is introduced,and its Euler-Lagrange equation for the minimum of variaional problems in the setting of one dimension is described.Then,the corresponding specific expression of these Euler-Lagrange equations for taking a geodesic line,geodesic curve,Brachistochrone and isoperimetric problems are deduced as examples,respectively.The solutions of the corresponding original problems are obtained and the relevant generalizations are given.

关 键 词：变分原理 积分泛函 欧拉-拉格朗日方程 极值函数 

分 类 号：O4-1[理学—物理]