A Group Theoretical Identification of Integrable Equations in the Liénard Type Equation $\ddot{x}+f(x)\dot{x}+g(x)=0$ : Part II: Equations having Maximal Lie Point Symmetries

Abstract

In this second of the set of two papers on Lie symmetry analysis of a class of Liénard type equation of the form $\ddot{x}+f(x)\dot{x}+g(x)=0$, where over dot denotes differentiation with respect to time and $f(x)$ and $g(x)$ are smooth functions of their variables, we isolate the equations which possess maximal Lie point symmetries. It is well known that any second order nonlinear ordinary differential equation which admits eight parameter Lie point symmetries is linearizable to free particle equation through point transformation. As a consequence all the identified equations turn out to be linearizable. We also show that one can get maximal Lie point symmetries for the above Liénard equation only when $f_{xx}=0$ (subscript denotes differentiation). In addition, we discuss the linearising transformations and solutions for all the nonlinear equations identified in this paper.

Accepted for publication in Journal of Mathematical Physics

References in corpus (2)

• On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator
• A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

Cited by in corpus (11)

• Classification of Lie point symmetries for quadratic Li$\acute{\text{e}}$nard type equation $\ddot{x}+f(x){\dot{x}}^2+g(x)=0$
• Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
• A class of exact solutions of the Liénard type ordinary non-linear differential equation
• Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second order differential equations
• Isochronicity and limit cycle oscillation in chemical systems
• Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study
• 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
• Classical Noether's theory with application to the linearly damped particle
• When an oscillating center in an open system undergoes power law decay
• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order