Two dimensional dynamical systems which admit Lie and Noether symmetries

TL;DR

This paper establishes the relationship between Lie and Noether symmetries of two-dimensional dynamical systems in Riemannian spaces, classifies systems with these symmetries, and applies results to specific physical models including cosmology.

Contribution

It proves new theorems linking symmetries with geometric groups and classifies 2D Newtonian systems with symmetries, applying findings to physical models like the Kepler-Ermakov system and scalar cosmology.

Findings

Classified 2D Newtonian systems with symmetries

Identified integrable scalar cosmological models in FRW backgrounds

Demonstrated symmetry generators can be read from classification tables

Abstract

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler - Ermakov system, which in general is non-conservative and for potentials similar to the H\`enon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.