Dynamical symmetries and the Ermakov invariant

TL;DR

This paper explores the connection between dynamical symmetries and the Ermakov invariant, showing how certain symmetries lead to conserved quantities that simplify the integration of Ermakov systems.

Contribution

It identifies Ermakov systems with Noether point symmetry within a Lagrangian framework and links the Ermakov invariant to a dynamical symmetry, providing a new perspective on their integrability.

Findings

Ermakov systems with Noether point symmetry are characterized.

The Ermakov invariant arises from a dynamical symmetry.

These invariants enable reduction of systems to quadratures.

Abstract

Ermakov systems possessing Noether point symmetry are identified among the Ermakov systems that derive from a Lagrangian formalism and, the Ermakov invariant is shown to result from an associated symmetry of dynamical character. The Ermakov invariant and the associated Noether invariant, are sufficient to reduce these systems to quadratures.