A Group Theoretical Identification of Integrable Cases of the Li\'{e}nard Type Equation $\ddot{x}+f(x)\dot{x}+g(x) = 0$ : Part I: Equations having Non-maximal Number of Lie point Symmetries

TL;DR

This paper classifies Lie point symmetries of the Lie9nard equation, identifying integrable and linearizable cases with fewer symmetries, and proves their integrability through solutions or Hamiltonian constructions.

Contribution

It provides a detailed classification of Lie9nard equations with non-maximal symmetries and establishes their integrability and linearizability.

Findings

Identified families of integrable equations with fewer symmetries.

Proved integrability of these equations via solutions or Hamiltonians.

First-time identification of several equations through group analysis.

Abstract

We carry out a detailed Lie point symmetry group classification of the Li\'enard type equation, $\ddot{x}+f(x)\dot{x}+g(x) = 0$, where $f(x)$ and $g(x)$ are arbitrary smooth functions of $x$. We divide our analysis into two parts. In the present first part we isolate equations that admit lesser parameter Lie point symmetries, namely, one, two and three parameter symmetries, and in the second part we identify equations that admit maximal (eight) parameter Lie-point symmetries. In the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations (under point transformations). Further, we prove the integrability of all of the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent Hamiltonians. Several of these equations are being identified for the first time from the group theoretical analysis.