Symmetries of Differential equations and Applications in Relativistic Physics

TL;DR

This thesis explores the geometric relationship between symmetries of differential equations and manifold collineations, applying it to classify dynamical systems, generalize classical models, analyze quantum symmetries, and identify dark energy models in gravity theories.

Contribution

It introduces a new geometric method linking point symmetries of differential equations with manifold collineations, applicable across various physical systems and theories.

Findings

Classified Newtonian dynamical systems by point symmetries

Generalized Kepler-Ermakov systems in Riemannian spaces

Identified geometric criteria for dark energy models

Abstract

In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. This geometric method is applied in order the two and three dimensional Newtonian dynamical systems to be classified in relation to the point symmetries; to generalize the Newtonian Kepler-Ermakov system in Riemannian spaces; to study the symmetries between classical and quantum systems and to investigate the geometric origin of the Type II hidden symmetries for the homogeneous heat equation and for the Laplace equation in Riemannian spaces. At last but not least, we apply this geometric approach in order to determine the dark energy models by use the Noether symmetries as a geometric criterion in modified theories of gravity.