Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations

TL;DR

This paper analyzes the Lie and Noether point symmetries of a broad class of second-order differential systems, revealing their geometric structure and applying findings to specific equations like Laplace and Klein-Gordon.

Contribution

It provides a geometric method to determine symmetries and conservation laws for second-order systems, including explicit results for coupled Laplace and Klein-Gordon equations.

Findings

Symmetries generated by collineations of (pseudo)metrics.

Complete invariant group for Klein-Gordon equation.

Invariant solutions for wave functions in hyperbolic space.

Abstract

We study the Lie and Noether point symmetries of a class of systems of second-order differential equations with $n$ independent and $m$ dependent variables ($n\times m$ systems). We solve the symmetry conditions in a geometric way and determine the general form of the symmetry vector and of the Noetherian conservation laws. We prove that the point symmetries are generated by the collineations of two (pseudo)metrics, which are defined in the spaces of independent and dependent variables. We demonstrate the general results in two special cases (a) a system of $m$ coupled Laplace equations and (b) the Klein-Gordon equation of a particle in the context of Generalized Uncertainty Principle. In the second case we determine the complete invariant group of point transformations, and we apply the Lie invariants in order to find invariant solutions of the wave function for a spin-$0$ particle in the two dimensional hyperbolic space.