场方法的改进及其在积分Riemann-Cartan空间运动方程中的应用  

Improvement of field method and its application to integrating motion equation in Riemann-Cartan space

作　　者：王勇[1,2] 梅凤翔 曹会英[2] 郭永新 Wang Yong;Mei Feng-Xiang;Cao Hui-Ying;Guo Yong-Xin(School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China;School of Information Engineering, Guangdong Medical University, Dongguan 523808, China;College of Physics, Liaoning University, Shenyang 110036, China;Physics of Medical Imaging Department, Eastern Liaoning University, Dandong 118001, China)

机构地区：[1]北京理工大学宇航学院 [2]广东医科大学信息工程学院 [3]辽宁大学物理学院 [4]辽东学院影像物理教研室

出　　处：《物理学报》2018年第3期133-139,共7页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:11772144;11572145;11272050;11572034;11202090;11472124);广东省自然科学基金(批准号:2015AO30310178)资助的课题~~

摘　　要：和Hamilton-Jacobi方法类似,Vujanovi?场方法把求解常微分方程组特解的问题转化为寻找一个一阶拟线性偏微分方程(基本偏微分方程)完全解的问题,但Vujanovi?场方法依赖于求出基本偏微分方程的完全解,而这通常是困难的,这就极大地限制了场方法的应用.本文将求解常微分方程组特解的Vujanovi?场方法改进为寻找动力学系统运动方程第一积分的场方法,并将这种方法应用于一阶线性非完整约束系统Riemann-Cartan位形空间运动方程的积分问题中.改进后的场方法指出,只要找到基本偏微分方程的包含m(m≤ n,n为基本偏微分方程中自变量的数目)个任意常数的解,就可以由此找到系统m个第一积分.特殊情况下,如果能够求出基本偏微分方程的完全解(完全解是m=n时的特例),那么就可以由此找到≤系统全部第一积分,从而完全确定系统的运动.Vujanovi?场方法等价于这种特殊情况.Like the Hamilton-Jacobi method, the Vujanovi？ field method transforms the problem of seeking the particular solution of an ordinary differential equations into the problem of finding the complete solution of a first order quasilinear partial differential equation, which is usually called the basic partial differential equation. Due to no need of the strong restrictive conditions required in the classic Hamilton-Jacobi method, the Vujanovi？ field method may be used in many fields, such as non-conservative systems, nonholonomic systems, Birkhoff systems, controllable mechanical systems, etc. Even so, there is still a fundamental difficulty in the Vujanovi？ field method. That is, for most of dynamical systems, it is hard to find the complete solution of the basic partial differential equation. In this paper, the Vujanovic field method is improved into a new field method. The purpose of the improved field method is to find the first integrals of the motion equations, but not the particular solutions of the motion equations. The improved field method points out that for a basic partial differential equation with n independent variables, m （m ≤ n） first integrals of a dynamical system can be found as long as a solution with m arbitrary constants of the basic partial differential equation is found. In particular, if the complete solution （the complete solution is a special case of m=n） of the basic partial differential equation is found, all first integrals of the dynamical system can be found. That means that the motion of the dynamical system is completely determined. The Vujanovic field method is just equivalent to this particular case. The improved field method expands the applicability of the field method, and is simpler than the Vujanovic field method. Two examples are given to illustrate the effectiveness of the method. In addition, the improved field method is used to integrate the motion equations in Riemann-Cartan space. For a first-order linear homogenous scleronomous nonholonomic system subj

关 键 词：场方法 第一积分 Riemann-Cartan空间 非完整约束系统 

分 类 号：O175[理学—数学]