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

Abstract

We carry out a detailed Lie point symmetry group classification of the Liénard 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.

Accepted for publication in Journal of Mathematical Physics

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• A class of exact solutions of the Liénard type ordinary non-linear differential equation
• On connections of the Liénard equation with some equations of Painlevé--Gambier type
• Exact solutions of the Liénard and generalized Liénard type ordinary non-linear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator
• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order
• On the integrability of the Abel and of the extended Liénard equations
• Notes on Lie symmetry group methods for differential equations