Poisson方程的李群分析及$3-$维刚体运动李代数的最优子代数系
Lie group analysis of Poisson's equation and optimal system of subalgebras for Lie algebra of $3-$dimensional rigid motions
摘要：
利用基本Lie对称方法，找到了$\nabla u=f(u)$ Poisson方程的最一般Lie点对称群，它有一个与$3-$维特殊欧氏群${\rm SE}(3)$或${\Bbb R}^3$的刚体运动群同构的子代数.通过研究${\rm SE}(3)$在其李代数$\goth{se}(3)$上的伴随表示，可以得到其子代数的完备最优系.后者提供了一些性质的解决方案的泊松方程。
英文摘要：
Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the $\nabla u=f(u)$ Poisson's equation, which has a subalgebra isomorphic to the $3-$dimensional special Euclidean group ${\rm SE}(3)$ or group of rigid motions of ${\Bbb R}^3$. Looking the adjoint representation of ${\rm SE}(3)$ on its Lie algebra $\goth{se}(3)$, we will find the complete optimal system of its subalgebras. This latter provides some properties of solutions for the Poisson's equation.
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