New non-standard Lagrangians for the Li\'enard-type equations

TL;DR

This paper introduces a new family of Lie9nard-type equations with non-standard Lagrangians and derives autonomous first integrals, linking their existence to linearizability via nonlocal transformations.

Contribution

It presents a novel family of Lie9nard equations with non-standard Lagrangians and connects their properties to linearizability conditions.

Findings

New family of Lie9nard equations with non-standard Lagrangians

Derivation of autonomous first integrals for these equations

Link between Lagrangian existence conditions and linearizability

Abstract

Li\'enard-type equations are used for the description of various phenomena in physics and other fields of science. Here we find a new family of the Li\'enard-type equations which admits a non-standard autonomous Lagrangian. As a by-product we obtain autonomous first integrals for each member of this family of equations. We also show that some of the previously known conditions for the existence of a non-standard Lagrangian for the Li\'enard-type equations follow from the linearizability of the corresponding equation via nonlocal transformations.