Autonomous three dimensional Newtonian systems which admit Lie and Noether point symmetries

TL;DR

This paper classifies three-dimensional autonomous Newtonian and Hamiltonian systems based on their Lie and Noether point symmetries, providing geometric methods applicable to higher dimensions.

Contribution

It identifies conditions for symmetries in 3D Newtonian and Hamiltonian systems and applies these to 2D systems in curved spaces for integrability analysis.

Findings

Classified 3D Newtonian systems with Lie symmetries.

Identified 3D Hamiltonian systems with Noether symmetries.

Applied results to 2D curved space systems for integrability.

Abstract

We determine the autonomous three dimensional Newtonian systems which admit Lie point symmetries and the three dimensional autonomous Newtonian Hamiltonian systems, which admit Noether point symmetries. We apply the results in order to determine the two dimensional Hamiltonian dynamical systems which move in a space of constant non-vanishing curvature and are integrable via Noether point symmetries. The derivation of the results is geometric and can be extended naturally to higher dimensions.