Geometrization of Lie and Noether symmetries with applications in Cosmology

Abstract

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in a $n-$dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the special projective and the homothetic vectors of the space. Therefore the Lie and the Noether symmetries for these equations are geometric symmetries or, equivalently, the geometry of the space is modulating the motion of dynamical systems in that space. We give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether symmetries of a specific dynamical system in a given Riemannian space. We apply the theorems to various interesting situations covering Newtonian 2d and 3d systems as well as dynamical systems in cosmology.

15 pages, no figures, 11 tables, Talk given at the 15th Conference on Recent Developments in Gravity (NEB XV), 20-23 June 2012, Chania, Greece

References in corpus (5)

• Using the Noether symmetry approach to probe the nature of dark energy
• Lie and Noether symmetries of geodesic equations and collineations
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• Two dimensional dynamical systems which admit Lie and Noether symmetries
• Lie point symmetries of a general class of PDEs: The heat equation