The inverse problem of a mixed Li\'enard type nonlinear oscillator equation from symmetry perspective

TL;DR

This paper investigates the inverse problem of a mixed Lie9nard nonlinear oscillator using symmetry methods, deriving integrals, Lagrangian, Hamiltonian, and analyzing its dynamics including isochronous oscillations.

Contribution

It introduces a novel approach connecting Lie symmetries, Jacobi last multiplier, and Prelle-Singer method to solve the inverse problem for this oscillator.

Findings

Constructed a time-independent integral for the oscillator.

Identified non-standard Lagrangian and Hamiltonian functions.

Discussed classical dynamics and isochronous oscillations.

Abstract

In this paper, we discuss the inverse problem for a mixed Li\'enard type nonlinear oscillator equation $\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0$, where $f(x),\,g(x)$ and $h(x)$ are arbitrary functions of $x$. Very recently, we have reported the Lie point symmetries of this equation. By exploiting the interconnection between Jacobi last multiplier, Lie point symmetries and Prelle-Singer procedure we construct a time independent integral for the case exhibiting maximal symmetry from which we identify the associated conservative non-standard Lagrangian and Hamiltonian functions. The classical dynamics of the nonlinear oscillator is also discussed and certain special properties including isochronous oscillations are brought out.