2011 # 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…

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(114 citation statements)

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“…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.…”

confidence: 58%

“…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.…”

confidence: 58%

“…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.…”

confidence: 94%

“…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.…”

confidence: 99%

“…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.…”

confidence: 99%