^{1,a)}, P. S. Bindu

^{2}, M. Senthilvelan

^{3}and M. Lakshmanan

^{3,b)}

### Abstract

In this second of the set of two papers on Lie symmetry analysis of a class of Liénard-type equation of the form , where overdot denotes differentiation with respect to time and and 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 (subscript denotes differentiation). In addition, we discuss the linearizing transformations and solutions for all the nonlinear equations identified in this paper.

One of us (S.N.P.) is grateful to the Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli, for warm hospitality. The work of S.N.P. forms part of a Department of Science and Technology, Government of India sponsored research project. The work of M.S. forms part of a research project sponsored by the National Board for Higher Mathematics, Government of India. The work of M.L. forms part of a Department of Science and Technology (DST), Ramanna Fellowship and is also supported by a DST-IRHPA research project.

I. INTRODUCTION

II. SYMMETRY DETERMINING EQUATION OF (1) WITH

III. MAXIMAL LIE POINT SYMMETRIES OF LIÉNARD-TYPE SYSTEMS: LINEAR ODEs

A. Linear undamped systems:

1. Free particle motion

2. Free falling particle

3. Free linear harmonic oscillator

4. Displaced linear harmonic oscillator

B. Linear damped systems:

1. Free particle in a viscous medium

2. Damped linear harmonic oscillator

3. Falling particle in a viscous medium

4. Displaced damped harmonic oscillator

IV. MAXIMAL LIE POINT SYMMETRIES OF LIÉNARD-TYPE SYSTEMS: NONLINEAR ODES

A. Modified Emden equations:

1. Modified Emden equation:

2. Modified Emden equation with linear term

3. Modified Emden equation with constant external forcing

4. Modified Emden equation with linear term and constant external forcing

B. Generalized Modified Emden equations:

V. NONMAXIMAL LIE POINT SYMMETRIES FOR

A. General forms of

B. Relationship between the symmetries and the forms of and

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Algebras
- 13.0
- Oscillators
- 13.0
- Lagrangian mechanics
- 8.0
- Integral equations
- 7.0
- Equations of motion
- 6.0

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