基于Frobenius定理的Hamilton-Jacobi方法的几何解释  

A Geometric Explanation of Hamilton-Jacobi Methods Based on the Frobenius Theorem

作　　者：肖静 刘畅[2,3] 王勇 XIAO Jing LIU Chang WANG Yong(School of Information Technology, Guangdong Medical University, Dongguan , Guangdong 523808, P.R. China School of Physics, Liaoning University, Shenyang 110035, P.R. China State Key Laboratory of Structural Analysis for Industrial Equipment （ Dalian University of Technology） , Department of Engineering Mechanics, Dalian University of Technology, Dalian , Liaoning 115024, P.R. China)

机构地区：[1]广东医科大学信息工程学院,广东东莞523808 [2]辽宁大学物理学院,沈阳110036 [3]大连理工大学工程力学系工业装备结构分析国家重点实验室(大连理工大学),辽宁大连116024

出　　处：《应用数学和力学》2017年第6期708-714,共7页Applied Mathematics and Mechanics

基　　金：国家自然科学基金(11572145;11202090);辽宁省教育厅科学技术一般项目(L2013005);广东省自然科学基金(2015A030310127);中国博士后科学基金(2014M560203)~~

摘　　要：给出了一阶偏微分方程特征微分方程组的一种基于Frobenius定理的几何解释,通过研究发现根据Frobenius定理可以从一阶偏微分方程直接得到其特征微分方程组;在此基础上说明如何利用几何方法从Hamilton正则方程出发找到与之对应的Hamilton-Jacobi方程.这种方法可以被用于非保守或非完整Hamilton力学问题的研究中,经典Hamilton-Jacobi方法是这种方法的一个特例.With the differential geometry method, a geometric explanation based on the Frobe- nius theorem for characteristic equations of 1st-order partial differential equations was presen- ted. According to the Frobenius theorem, the characteristic equations can be deduced directly from the 1st-order partial differential equations. Based on this, how to use the geometric meth- od to find the corresponding Hamilton-Jacobi equations from Hamiltonian canonical equations was discussed. This method could be utilized to address the nonconservative or nonholonomic Hamiltonian mechanical problems. The classical Hamilton-Jacobi method is only a special case of this method.

关 键 词：Hamilton-Jacobi理论 一阶偏微分方程 Frobenius定理 

分 类 号：O316[理学—一般力学与力学基础]