低维泊松流形上Casimir函数的符号计算与可积性分析

Symbolic Computation of Casimir Functions and Integrability Analysis on Low-Dimensional Poisson Manifolds

摘要: 本文研究低维泊松流形上Casimir函数的符号计算及其在完全可积性分析中的应用。对给定的泊松双向量$\pi =\frac{1}{2}{\pi }^{ij}\left(x\right){\partial }_i\wedge {\partial }_j$ ，Casimir函数$C$ 满足线性偏微分方程组${\pi }^{ij}{\partial }_{j}C=0$ 。利用计算机代数系统符号求解该方程组，获得函数独立的Casimir函数，从而确定泊松流形的辛叶结构。在此基础上，给出低维情形($dimM\le 4$ )下基于Casimir函数的Liouville可积性判据：若$rank\pi =2r$ ，则最大独立对合首次积分数为$dimM-r$ 。分别讨论二维、三维及四维泊松流形的典型例子，展示符号计算推导过程与可积性分析细节。

Abstract: This paper investigates the symbolic computation of Casimir functions on low-dimensional Poisson manifolds and its application to complete integrability analysis. For a given Poisson bivector $\pi =\frac{1}{2}{\pi }^{ij}\left(x\right){\partial }_i\wedge {\partial }_j$ , the Casimir functions $C$ are determined by the linear PDE system ${\pi }^{ij}{\partial }_{j}C=0$ . By symbolically solving this system using computer algebra, functionally independent Casimir functions are obtained, revealing the symplectic foliation of the manifold. A Liouville integrability criterion in low dimensions ($dimM\le 4$ ) is established based on the Casimir functions: if $rank\pi =2r$ , then the maximal number of independent involutive first integrals is $dimM-r$ . Typical examples in two, three, and four dimensions are discussed in great detail, with explicit symbolic computations and integrability analysis.