Geometrization of Lie and Noether symmetries and applications

Abstract

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in an n-dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the collineation vectors of the space. More specifically we give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether point symmetries of a specific dynamical system evolving in a given Riemannian space in terms of the projective and the homothetic collineations of space. We apply these theorems to various interesting situations in Newtonian Physics.