分析动力学三个问题的研究进展

摘要

分析力学的发展涉及理论的和应用的诸多方面。本文在分析力学与数学交缘的三个问题上综述分析力学的近代发展。第一是利用Lie群和Lie代数的一些成果来研究分析动力学方程的积分问题。第二是将分析力学的经典和近代积分方法应用于一般微分方程的积分问题。第三是将分析动力学方程在一定条件下化成梯度系统的方程，再用梯度系统的性质来研究力学系统的动力学行为。%The development of analytical mechanics involves many aspects of theory and application.This paper summarizes the recent progress of analytical mechanics in three problems on the interdisciplinarity between analyt-ical mechanics and mathematics.The first is to study the integration problem of equations of analytical dynamics by using some results of Lie groups and Lie algebras.The second is to apply the classical and modern integration methods of analytical mechanics to the integration problem of general differential equations.The third is to trans-form the equations of analytical dynamics into the equations of gradient system under certain conditions and then discuss the dynamical behaviors of the mechanical system by using the properties of gradient system.