The geometric nature of Lie and Noether symmetries

Abstract

It is shown that the Lie and the Noether symmetries of the equations of motion of a dynamical system whose equations of motion in a Riemannian space are of the form \({\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x} ^{k}+f(x^{i})=0}\) where f(xi) is an arbitrary function of its argument, are generated from the Lie algebra of special projective collineations and the homothetic algebra of the space respectively. Therefore the computation of Lie and Noether symmetries of a given dynamical system in these cases is reduced to the problem of computation of the special projective algebra of the space. It is noted that the Lie and Noether symmetry vectors are common to all dynamical systems moving in the same background space. The selection of the vectors which are Lie/Noether symmetries for a given dynamical system is done by means of a set of differential conditions involving the vectors and the potential function defining the dynamical system. The general results are applied to a number of different applications concerning (a) The motion in Euclidean space under the action of a general central potential (b) The motion in a space of constant curvature (c) The determination of the Lie and the Noether symmetries of class A Bianchi type hypersurface orthogonal spacetimes filled with a scalar field minimally coupled to gravity (d) The analytic computation of the Bianchi I metric when the scalar field has an exponential potential.