Lie group analysis of Poisson's equation and optimal system of subalgebras for Lie algebra of $3-$dimensional rigid motions

TL;DR

This paper applies Lie symmetry analysis to Poisson's equation, identifying its symmetry group related to rigid motions in 3D, and classifies its subalgebras to aid in solving the equation.

Contribution

It determines the Lie point symmetries of Poisson's equation with a specific symmetry group and classifies its subalgebras for solution analysis.

Findings

Identified the Lie symmetry group of Poisson's equation as related to 3D rigid motions.

Classified the optimal system of subalgebras of the symmetry group.

Provided properties of solutions based on the subalgebra classification.

Abstract

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.