Generalizing the autonomous Kepler Ermakov system in a Riemannian space

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 Λ_{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.

Reference [25] update, 21 pages

References in corpus (7)

• f(R) Theories Of Gravity
• Phantom crossing, equation-of-state singularities, and local gravity constraints in f(R) models
• Power-laws f(R) theories are cosmologically unacceptable
• Two dimensional dynamical systems which admit Lie and Noether symmetries
• Dynamical symmetries and the Ermakov invariant
• On the Hamiltonian structure of Ermakov systems
• On the Lie symmetries of a class of generalized Ermakov systems

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• Quadratic first integrals of autonomous conservative dynamical systems
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• Quadratic first integrals of time-dependent dynamical systems of the form ${\ddot{q}}^a=-{\Gamma }_{bc}^a{\dot{q}}^b{\dot{q}}^c-\omega (t)Q^a(q)$
• Symmetries and conservation laws for the generalized $n$-dimensional Ermakov system