关于分析力学的基础与展望

THE FOUNDATION AND PROSPECT OF ANALYTICAL MECHANICS

本文从分析约束力学系统的“欠定”问题开始,介绍分析力学的基本变分原理和三类运动微分方程,并分析了分析力学具有普适性之缘由.对非完整约束力学系统,着重分析其动力学建模问题、几何结构和重点发展方向,同时又简要介绍了Birkhoff系统所具有的一般辛结构特征和研究意义,以及需要重点解决的问题.文中对力学系统的Noether对称性和运动微分方程的对称性作了较为详细的论述,并列举了相应实例说明两种对称性与守恒量之间的关系.在几何力学部分,重点介绍了分析力学的辛几何结构和对称性约化理论,包括辛流形的Darboux Moser Weinstein局部正则结构、整体拓扑结构及其对量子力学的影响、Lie群与Lie代数的伴随表示和余伴随表示、动量映射、Cartan辛约化、Marsden Weinstein约化等.文中最后论述了完整与非完整力学系统可积性问题的研究方法和成果,指出了非完整力学系统现有可积性方法的局限性.

Starting from the analysis of the under determined problem of constrained mechanical systems,the fundamental variational principles of analytical mechanics and three kinds of differential equations of motion are introduced and the universality of analytical mechanics is analyzed in this paper.For nonholonomic constrained mechanical systems,the dynamic modeling,geometric structure and key development directions are emphatically analyzed.At the same time,the general symplectic structure and research significance of Birkhoffian systems are briefly introduced,as well as the key problems to be solved.The Noether symmetry of mechanical systems and the symmetry of differential equations of motion are discussed in detail,and corresponding examples are given to illustrate the relationship between the two symmetries and conserved quantities.In the part of geometric mechan ics,the symplectic geometric structure and symmetry reduction theory of analytical mechanics are mainly described,including local canonical structure of symplectic manifolds by Darboux Moser Weinstein theorem,global topological structure and its influence on quantum mechanics,adjoint and co adjoint representation of Lie group and Lie algebra,momentum mapping,Cartan symplectic reduction,Marsden Weinstein reduction and so on.At the end of the paper,the research methods and results of the integrability of holonomic and nonholonomic mechanical systems are discussed,and the limitations of the existing integrability methods of nonholonomic mechanical systems are pointed out.