线性微分系统的一种约化形式及其在哈密顿系统可积性中的应用
A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
摘要：
设具有$A\in \mathrm{M}_n (k)$的$[A]: Y'=AY$是一个微分线性系统。我们说矩阵$R\in {\cal M}_{n}(\bar{k})$是$[A]$的{ \em约化形式}，如果$R\in \mathfrak{g}(\bar{k})$且存在$P\in GL_n (\bar{k})$使得$R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$。这样的形式通常是通过规范变换而不引入新的超越量所能达到的稀疏可能性。在这篇文章中，我们讨论如何计算一些辛微分系统的约化形式，产生的变分方程的哈密顿系统。利用这一点，我们给出了关于哈密顿系统（不可）积性的Morales-Ramis定理的一个有效形式.
英文摘要：
Let $[A]: Y'=AY$ with $A\in \mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $R\in {\cal M}_{n}(\bar{k})$ is a {\em reduced form} of $[A]$ if $R\in \mathfrak{g}(\bar{k})$ and there exists $P\in GL_n (\bar{k})$ such that $R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.
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