刚体碰撞问题的Hamilton原理及其应用  

Hamilton’s principle for rigid body collision problems and its applications

作　　者：黄维璇 李四平[1] HUANG WeiXuan;LI SiPing(School of Naval Architecture,Ocean&Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240,China)

机构地区：[1]上海交通大学船舶海洋与建筑工程学院,上海200240

出　　处：《中国科学：物理学、力学、天文学》2023年第5期73-84,共12页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金(编号:51878407)资助项目。

摘　　要：从d’Alembert-Lagrange原理出发,推导出碰撞的Lagrange方程的新形式——碰撞状态方程;在此基础上,定义了能量形式的碰撞特征函数,提出了碰撞分析的Hamilton原理,并揭示了其物理本质:在任意运动许可的碰撞中,真实的碰撞使系统动能损失在广义速度空间最小(或系统当前动能最大),把碰撞分析理论推进到变分(极值)原理的阶段.进一步的研究显示,以碰撞的Hamilton原理为基础,不仅能推出碰撞的三大基本定理,还能推出碰撞的相对动能定理的新结果.文中还探讨了若干应用的具体问题,厘清了牛顿摆球链散开原因的历史争议,表明碰撞的Hamilton原理能够突破传统方法的局限,兼具理论的普适性和应用的便捷性.Based on the d’Alembert-Lagrange principle,a new type of the Lagrange equations of collision,that is,the collision state equations,are derived.According to these equations and the collision characteristic function defined in energy form,the Hamilton’s principle of collision is proposed.The physical essence of the principle is also revealed:For any collision permitted by motion,the actual collision minimizes the kinetic energy loss,or maximizes the current kinetic energy of the system,in the generalized velocity space.Moreover,the analytical mechanical theory of collision is extended to the field of variational principles(or extremum principles).Results reveal that based on Hamilton’s principle of collision,not only the three fundamental theorems of collision but also the relative kinetic energy theorem of collision can be derived,which has never been done before.In this study,several relevant application problems of rigid body system collision are discussed,and the historical controversy about the cause of ball chain dispersion in Newton’s cradle is clarified.These examples illustrate that Hamilton’s principle of collision not only has the universality of theory and the convenience of application but also can break through the limitations of traditional methods.

关 键 词：碰撞 碰撞Lagrange方程 冲量势 碰撞特征函数 碰撞Hamilton原理 球链碰撞模型 

分 类 号：O313.4[理学—一般力学与力学基础]