Classification of Lie point symmetries for quadratic Li$\acute{\textbf{e}}$nard type equation $\ddot{x}+f(x)\dot{x}^2+g(x)=0$

TL;DR

This paper classifies all Lie point symmetries of the quadratic Liénard equation, identifying conditions for maximal symmetry, linearizability, and integrability, with applications to physical models.

Contribution

It provides a complete classification of Lie point symmetries for the quadratic Liénard equation, including conditions for maximal symmetry and integrability, and explores physical applications.

Findings

Maximal symmetry group leads to linearizable and isochronic equations.

Non-maximal symmetry cases are all integrable.

Includes physically relevant examples like the Mathews-Lakshmanan oscillator.

Abstract

In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li$\acute{e}$nard type equation, $\ddot {x} + f(x){\dot {x}}^{2} + g(x)= 0$, where $f(x)$ and $g(x)$ are arbitrary functions of $x$. The symmetry analysis gets divided into two cases, $(i)$ the maximal (eight parameter) symmetry group and $(ii)$ non-maximal (three, two and one parameter) symmetry groups. We identify the most general form of the quadratic Li$\acute{e}$nard equation in each of these cases. In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic. We present specific examples of physical interest. For the nonmaximal cases, the identified equations are all integrable and include several physically interesting examples such as the Mathews-Lakshmanan oscillator, particle on a rotating parabolic well, etc. We also analyse the underlying equivalence transformations.