论文标题
Poisson Quasi-Nijenhuis歧管和Toda系统
Poisson quasi-Nijenhuis manifolds and the Toda system
论文作者
论文摘要
Poisson Quasi-Nijenhuis的概念概括了Poisson-Nijenhuis歧管的概念。自双汉米尔顿（Bi-Hamiltonian）对整合性方法的诞生以来，后者在完全集成系统的理论中的相关性已经建立。在本说明中，我们讨论了在有限维的整合系统的背景下，Poisson Quasi-Nijenhuis歧管概念的相关性。一般而言（正如我们以$ 3 $的自由度为例所示的那样）Poisson Quasi-nijenhuis结构在很大程度上太笼统了，无法确保系统的liouville集成性。但是，我们证明可以在这种几何结构中构建封闭的（或周期性的）$ n $ - 零件toda晶格，并且其运动的众所周知积分可以作为“ Quasi-nijenhuis recursion ocurs ocurs of type $ n $ n $ n $ n $ n $ n $ n ote ote（1,1）$ 1,1）的“ Quasi-Nijenhuis recursion ocurs”的光谱不变性。这个示例及其某些概括用于理解是否可以在合理意义上定义{\ em涉及\}泊松Quasi-nijenhuis歧管的概念。开放式（或非周期性）与封闭的TODA系统之间的几何联系也是在连接Poisson Quasi-Nijenhuis和Poisson-Nijenhuis歧管的一般方案的背景下进行的。
The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with $3$ degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) $n$-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field $N$ of type $(1,1)$ defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of {\em involutive\} Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.
