Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Li\'enard equation

TL;DR

This paper revisits Chiellini's integrability criterion for the Lie9nard equation, deriving Hamiltonian structures, identifying the Kukles equation as unique in satisfying specific conditions, and exploring various formulations including metriplectic and complex Hamiltonian approaches.

Contribution

It introduces a novel transformation involving the Jacobi Last Multiplier to derive Hamiltonian structures and characterizes the Kukles equation within the Lie9nard family.

Findings

Derived Hamiltonian and Lagrangian for Lie9nard equation

Identified Kukles equation as unique in satisfying Chiellini and Sabatini conditions

Mapped Lie9nard equation to harmonic oscillator using Chiellini's condition

Abstract

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Li\'enard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Li\'enard equation is derived. We also show that the Kukles equation is the only equation in the Li\'enard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions. In addition we examine this result by mapping the Li\'enard equation to a harmonic oscillator equation using tacitly Chiellini's condition. Finally we provide a metriplectic and complex Hamiltonian formulation of the Li\'enard equation through the use of Chiellini condition for integrability.