Geometrization of Lie and Noether symmetries with applications in Cosmology

NEB 15 – Recent Developments in Gravity

Tsamparlis, Michael^1

Department of Physics, Section of Astronomy, Astrophysics and Mechanics, University of Athens, Panepistemiopolis, Athens 157 83, Greece^1

关键词: 3d systems;    Geometric symmetry;    Newtonians;    Noether condition;    Noether symmetry;    Riemannian spaces;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/453/1/012020/pdf
DOI  :  10.1088/1742-6596/453/1/012020

来源: IOP

【 摘 要 】

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.