Generalizing the autonomous Kepler Ermakov system in a Riemannian space

TL;DR

This paper extends the Kepler Ermakov system to three dimensions and Riemannian spaces, identifying integrable cases and applying these to cosmological models in General Relativity.

Contribution

It generalizes the Kepler Ermakov system to three dimensions and Riemannian spaces, and applies the framework to specific cosmological models in General Relativity.

Findings

Identified all three-dimensional Liouville integrable Kepler Ermakov systems via Noether symmetries.

Derived the Riemannian Ermakov invariant for systems with gradient homothetic vectors.

Reduced gravitational field equations in certain cosmological models to integrable Kepler Ermakov systems.

Abstract

We generalize the two dimensional autonomous Hamiltonian Kepler Ermakov dynamical system to three dimensions using the sl(2,R) invariance of Noether symmetries and determine all three dimensional autonomous Hamiltonian Kepler Ermakov dynamical systems which are Liouville integrable via Noether symmetries. Subsequently we generalize the autonomous Kepler Ermakov system in a Riemannian space which admits a gradient homothetic vector by the requirements (a) that it admits a first integral (the Riemannian Ermakov invariant) and (b) it has sl(2,R) invariance. We consider both the non-Hamiltonian and the Hamiltonian systems. In each case we compute the Riemannian Ermakov invariant and the equations defining the dynamical system. We apply the results in General Relativity and determine the autonomous Hamiltonian Riemannian Kepler Ermakov system in the spatially flat Friedman Robertson Walker spacetime. We consider a locally rotational symmetric (LRS) spacetime of class A and discuss two cosmological models. The first cosmological model consists of a scalar field with exponential potential and a perfect fluid with a stiff equation of state. The second cosmological model is the f(R) modified gravity model of {\Lambda}_{bc}CDM. It is shown that in both applications the gravitational field equations reduce to those of the generalized autonomous Riemannian Kepler Ermakov dynamical system which is Liouville integrable via Noether integrals.