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The inverse problem of a mixed Liénard-type nonlinear oscillator equation from symmetry perspective

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Abstract

In this paper, we discuss the inverse problem for a mixed Liénard-type nonlinear oscillator equation \({\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0}\), where \({f(x), g(x)}\) and h(x) are arbitrary functions of x. Very recently, we have reported the Lie point symmetries of this equation. By exploiting the interconnection between Jacobi last multiplier, Lie point symmetries and Prelle–Singer procedure, we construct a time-independent integral for the case exhibiting maximal symmetry from which we identify the associated conservative nonstandard Lagrangian and Hamiltonian functions. The classical dynamics of the nonlinear oscillator is also discussed, and certain special properties including isochronous oscillations are brought out.

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Author notes

  1. Ajey K. Tiwari

    Present address: Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban, 4000, South Africa

Authors and Affiliations

  1. Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli, 620 024, India

    Ajey K. Tiwari, M. Senthilvelan & M. Lakshmanan

  2. Department of Physics, Motilal Nehru National Institute of Technology, Allahabad, 211 004, India

    S. N. Pandey

  3. Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA University, Thanjavur, 613 401, India

    V. K. Chandrasekar

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  1. Ajey K. Tiwari
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  2. S. N. Pandey
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  3. V. K. Chandrasekar
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  4. M. Senthilvelan
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  5. M. Lakshmanan
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Correspondence to M. Senthilvelan.

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Tiwari, A.K., Pandey, S.N., Chandrasekar, V.K. et al. The inverse problem of a mixed Liénard-type nonlinear oscillator equation from symmetry perspective. Acta Mech 227, 2039–2051 (2016). https://doi.org/10.1007/s00707-016-1602-9

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  • DOI: https://doi.org/10.1007/s00707-016-1602-9

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