标题： 微分伽罗瓦理论与可积性 显示英文标题

标题： Differential Galois theory and Integrability

摘要： 本文是对我们相关工作的综述，这些工作涉及自然哈密顿系统具有齐次势能以及牛顿方程具有与速度无关的齐次力的可积性研究。 这两种类型的可积性障碍被提出。 第一种是局部障碍，与沿非平衡特解的变分方程的微分伽罗瓦群的分析有关。 第二种是全局障碍，通过同时分析属于某一类的所有特解相关的变分方程得到。 这两种可积性障碍的结合使得能够实现所有可积齐次系统的分类计划。 展示了具有两个或更多自由度的系统的可积性分析的主要步骤以及新的可积系统。 显示英文摘要

摘要： This paper is an overview of our works which are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realise the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.