On the Symmetries of a Liénard Type Nonlinear Oscillator Equation

Abstract

In the contemporary nonlinear dynamics literature, the nonlinear oscillator equation $\ddot{x}+kx \dot{x}+\frac{k^2}{9}x^3+\tilde{\lambda } x=0$ is being analyzed in various contexts both classically and quantum mechanically. Classically this nonlinear oscillator equation has been shown to admit three different types of dynamics depending upon the sign and magnitude of the parameter $\tilde{\lambda }$, namely (i) $\tilde{\lambda }=0$, (ii) $\tilde{\lambda }>0$ and (iii) $\tilde{\lambda }<0$. By considering its importance, in this paper, we present the symmetries of its Lagrangian and underlying equation of motion for all the three cases. In particular, we present Lie point symmetries, $\lambda$-symmetries, Noether symmetries and telescopic symmetries of this equation. The utility of the symmetries for all the three cases is demonstrated explicitly.