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Classification of Lie point symmetries for quadratic Liénard type equation

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10.1063/1.4803455

### Abstract

In this paper we carry out a complete classification of the Lie point symmetry groups associated with the quadratic Li nard type equation, , 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 nard 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.

© 2013 AIP Publishing LLC

Received 20 September 2012
Accepted 11 April 2013
Published online 07 May 2013

Acknowledgments: A.K.T. and S.N.P. are 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 UGC. The work forms part of a Department of Science and Technology, Government of India IRHPA project and a Ramanna Fellowship project of M.L. He also acknowledges the financial support provided through a DAE Raja Ramanna Fellowship.

Article outline:

I. INTRODUCTION

II. DETERMINING EQUATIONS FOR THE INFINITESIMAL SYMMETRIES

III. GENERAL FORM OF THE EQUATION FOR *b* ≠ 0 CASE – EIGHT PARAMETER SYMMETRIES

IV. ISOCHRONOUS CONDITION, LINEARIZABILITY, AND THE NATURE OF SOLUTION OF EQ. (25)

A. Isochronicity condition

B. Linearizability condition

V. SPECIAL CASES OF MAXIMAL SYMMETRY GROUP

A. *f*(*x*) = λ = constant

B.

VI. NON-MAXIMAL SYMMETRY: CASE *b* = 0

A. Three parameter symmetry (case (i) and (*c* *G* _{2} + *d* *G* _{3}) ≠ 0)

B. Two parameter symmetry (case (ii) and (*c* *G* _{2} + *d* *G* _{3}) ≠ 0)

C. One parameter symmetry (*c* *G* _{2} + *d* *G* _{3} = 0)

VII. LIE SYMMETRIES OF EQ. (2) WITH *f*(*x*) = 0 or *g*(*x*) = 0

VIII. EQUIVALENCE TRANSFORMATIONS

IX. CONCLUSION

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2013-05-07

2014-09-30

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