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 formẍ i + i jkẋis 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.

A study of positive energy condition in Bianchi V spacetimes via Noether symmetries

“…The algebra of the Lie point symmetries of the equations of motion is of dimension 5-8, 10, 12 or 13. The connection between Lie and Noether symmetries with symmetries of the spacetimes like HVs, projective collineations (PCs), and CCs has already been discussed in [16,18]. We highlight the important features of new cosmological solutions in the light of the above results.…”

“…That the simplest Bianchi V model does not attain a maximal algebra of dimension 13 does not come as a surprise because the underlying Riemannian manifold is not flat but contains flat sections. The algebra of Lie symmetries of the equations of motion in a flat space is unique and corresponds to sl(n+2, R) [16]. In our analysis, Bianchi V spacetimes are not flat in general (except when α = 0); the maximum dimension of the Lie algebra of Lie point symmetries is 13.…”

“…In addition Noether symmetry analysis reduces the problem by specifying the unknown functions that appear in the Lagrangian. This line of approach has been followed by several authors, notably in the pioneering works of Capozziello [12,13], Prince [14,15], and Tsamparlis [16,17]. The connection between symmetries of the underlying manifold and those of the differential equations was discussed in [18].…”

Noether symmetries of a modified model in teleparallel gravity and a new approach for exact solutions

“…3.1, we study the general form of the Noether symmetry, which is perceived as a Noether gauge symmetry. However, this terminology is wrong because there is no gauge [48][49][50]. In Sect.…”

Geometrical/Physical Interpretation of the Conserved Quantities Corresponding to Noether Symmetries of Plane Symmetric Space-Times

“…Noether symmetries of the geodetic Lagrangian constitute subalgebra of the algebra of Lie symmetries of the geodesic equations. It was shown in [30] that this subalgebra is generated (in the space of variables) by the coordinate vector associated with the affine parameter of the geodesics and the homothetic Lie algebra of the underlying space-time.…”