Geometrization of Lie and Noether symmetries with applications in Cosmology

TL;DR

This paper links Lie and Noether symmetries of dynamical systems to the geometric properties of Riemannian spaces, providing a framework to identify symmetries through space geometry, with applications in Newtonian mechanics and cosmology.

Contribution

It expresses symmetry vectors in terms of space's projective and homothetic vectors, establishing a geometric approach to symmetry analysis in Riemannian spaces.

Findings

Derived theorems for symmetry determination in Riemannian spaces.

Connected symmetries to special geometric vectors of the space.

Applied the framework to Newtonian and cosmological systems.

Abstract

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in a $n-$dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the special projective and the homothetic vectors of the space. Therefore the Lie and the Noether symmetries for these equations are geometric symmetries or, equivalently, the geometry of the space is modulating the motion of dynamical systems in that space. We give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether symmetries of a specific dynamical system in a given Riemannian space. We apply the theorems to various interesting situations covering Newtonian 2d and 3d systems as well as dynamical systems in cosmology.