动力系统与Poisson结构
Dynamical Systems and Poisson Structures
摘要：
我们首先考虑一般$n=3$系统的哈密顿公式，并证明了${\mathbb R}^3$中的所有动力系统都是双哈密顿的。介绍了一种获得给定动力系统的Poisson结构的算法。其次，我们证明了${\mathbb R}^n$中的所有动力系统都是$(n-1)$ -Hamilton系统。我们也给出了一个类似于${\mathbb R}^3$中的算法，在${\mathbb R}^n$中构造动力系统的秩二Poisson结构.我们给出了动力系统关于向量场$\vec{X}$的不变函数的分类，并证明了${\mathbb R}^n$中的所有自治动力系统都是超可积的。
英文摘要：
We first consider the Hamiltonian formulation of $n=3$ systems in general and show that all dynamical systems in ${\mathbb R}^3$ are bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. We find the Poisson structures of a dynamical system recently given by Bender et al. Secondly, we show that all dynamical systems in ${\mathbb R}^n$ are $(n-1)$-Hamiltonian. We give also an algorithm, similar to the case in ${\mathbb R}^3$, to construct a rank two Poisson structure of dynamical systems in ${\mathbb R}^n$. We give a classification of the dynamical systems with respect to the invariant functions of the vector field $\vec{X}$ and show that all autonomous dynamical systems in ${\mathbb R}^n$ are super-integrable.
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