讲授力学原理统一性——以弹簧振子为例  

UNIFICATION OF FIVE MECHANICS PRINCIPLES BASED ON THE SPRING-MASS SYSTEM

作　　者：刘建林[1] LIU Jianlin(College of Pipeline and Civil Engineering,China University of Petroleum(East China),Qingdao,Shandong 266580)

机构地区：[1]中国石油大学(华东)储运与建筑工程学院,山东青岛266580

出　　处：《物理与工程》2020年第4期69-72,83,共5页Physics and Engineering

基　　金：国家自然科学基金(11672335)资助。

摘　　要：对一个系统进行力学分析可以运用不同的力学原理而得到相同结果。本文以弹簧振子为例,详细阐述了牛顿第二定律、机械能守恒定理、拉格朗日方程、哈密顿正则方程、哈密顿变分原理这五种力学原理之间的相互关系。其中牛顿第二定律是从受力角度进行分析,而机械能守恒定理、拉格朗日方程、哈密顿正则方程则是从能量角度出发。哈密顿变分原理则为更为普遍的力学原理,通过对其变分可以推导出拉格朗日方程、哈密顿正则方程以及运动微分方程。弄清楚这五种力学原理的统一性,有助于我们更加合理地选择运用它们去解决工程问题。To analyze a system,one can get the same result according to different mechanical principles.We take the example of one spring-mass system,in order to the relations among the Newton's second law,conservation law of mechanical energy,Lagrange theorem,Hamilton canonical equation,Hamilton variational principle.The Newton's second law is based on the force analysis,but the conservation law of mechanical energy,Lagrange equation,and Hamilton canonical equation all start from the energy viewpoint.The Hamilton principle is more general,and by variation it can lead to the Lagrange equation,the Hamilton canonical equation and the differential equation of motion.Understanding the unification of these five mechanics principles are helpful to select them properly to solve engineering problems.

关 键 词：牛顿第二定律 机械能守恒定律 拉格朗日定理 哈密顿正则方程 哈密顿原理 

分 类 号：O3[理学—力学]