哈密顿系统高阶变分方程的一种降阶方法
A Reduction Method for Higher Order Variational Equations of Hamiltonian Systems
摘要：
设$\mathbf{k}$是一个微分场，$[A]\,:\,Y'=A\,Y$是一个线性微分系统，其中$A\in\mathrm{Mat}(n\,,\,\mathbf{k})$ .我们说$A$是一个约化形式，如果$A\in\mathfrak{g}(\bar{\mathbf{k}})$，其中$\mathfrak{g}$是$[A]$的李代数，$\bar{\mathbf{k}}$表示$\mathbf{k}$的代数闭包.我们欠存在这样的减少形式的一个结果，由于Kolchin和Kovacic \cite{Ko71a}。本文研究了复解析哈密顿系统$X$的（高阶）变分方程沿特解的约化形式。使用以前的结果\cite{ApWea}，我们将假设一阶变分方程有一个阿贝尔李代数，使在一阶，有没有伽罗瓦障碍刘维尔可积。如果$m$阶变分方程已经是约化形式且其李代数是交换的，则给出了$m+1$阶变分方程的（部分）约化策略.我们的程序停止时，我们遇到障碍的亚纯可积的$X$。我们充分利用了变分方程的下块三角结构和线性微分系统的相关李代数的概念（基于魏和诺曼在\cite{WeNo63a}中的工作）。可积性的障碍出现时，在某些步骤中，我们得到一个非平凡的交换子之间的对角元素和幂零（次对角）元素的相关李代数。我们使用我们的方法加上一个推理的多项式给出一个新的和系统的证明的Hénon-Heiles系统的不可积性。我们猜想，我们的方法不仅是一个部分约简过程，但一个完整的约简算法。在复杂的哈密顿系统的背景下，这意味着我们的方法将是莫拉莱斯-拉米斯-西莫定理的有效版本。
英文摘要：
Let $\mathbf{k}$ be a differential field and let $[A]\,:\,Y'=A\,Y$ be a linear differential system where $A\in\mathrm{Mat}(n\,,\,\mathbf{k})$. We say that $A$ is in a reduced form if $A\in\mathfrak{g}(\bar{\mathbf{k}})$ where $\mathfrak{g}$ is the Lie algebra of $[A]$ and $\bar{\mathbf{k}}$ denotes the algebraic closure of $\mathbf{k}$. We owe the existence of such reduced forms to a result due to Kolchin and Kovacic \cite{Ko71a}. This paper is devoted to the study of reduced forms, of (higher order) variational equations along a particular solution of a complex analytical hamiltonian system $X$. Using a previous result \cite{ApWea}, we will assume that the first order variational equation has an abelian Lie algebra so that, at first order, there are no Galoisian obstructions to Liouville integrability. We give a strategy to (partially) reduce the variational equations at order $m+1$ if the variational equations at order $m$ are already in a reduced form and their Lie algebra is abelian. Our procedure stops when we meet obstructions to the meromorphic integrability of $X$. We make strong use both of the lower block triangular structure of the variational equations and of the notion of associated Lie algebra of a linear differential system (based on the works of Wei and Norman in \cite{WeNo63a}). Obstructions to integrability appear when at some step we obtain a non-trivial commutator between a diagonal element and a nilpotent (subdiagonal) element of the associated Lie algebra. We use our method coupled with a reasoning on polylogarithms to give a new and systematic proof of the non-integrability of the Hénon-Heiles system. We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm. In the context of complex Hamiltonian systems, this would mean that our method would be an effective version of the Morales-Ramis-Simó theorem.
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