On the Lie symmetries of Kepler-Ermakov systems

TL;DR

This paper investigates the Lie-point symmetries of Kepler-Ermakov systems, identifying conditions for SL(2,R) symmetry and Lagrangian formulation, revealing these systems as variable-frequency Ermakov systems.

Contribution

It determines the specific forms of the arbitrary function H(x,y) that admit SL(2,R) symmetry and possess a Lagrangian, clarifying the symmetry structure of Kepler-Ermakov systems.

Findings

Identified forms of H(x,y) with SL(2,R) symmetry.

Established that these systems are variable-frequency Ermakov systems.

Connected symmetry properties with the existence of a Lagrangian.

Abstract

In this work, we study the Lie-point symmetries of Kepler--Ermakov systems presented by C. Athorne in J. Phys. A24 (1991), L1385--L1389. We determine the forms of arbitrary function H(x,y) in order to find the members of this class possessing the sl(2,R) symmetry and a Lagrangian. We show that these systems are usual Ermakov systems with the frequency function depending on the dynamical variables.