Classification of Lie point symmetries for quadratic Liénard type equation $\ddot{x}+f(x)\dot{x}^2+g(x)=0$ẍ+f(x)ẋ2+g(x)=0

Abstract

In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li\documentclass[12pt]{minimal}\begin{document}$\acute{e}$\end{document}énard type equation, \documentclass[12pt]{minimal}\begin{document}$\ddot{x} + f(x){\dot{x}}^{2} + g(x)\break = 0$\end{document}ẍ+f(x)ẋ2+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\documentclass[12pt]{minimal}\begin{document}$\acute{e}$\end{document}énard 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 non-maximal 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.