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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Goldstein H., Poole C.P. Jr, Safko J.L.: Classical Mechanics. Pearson Education, Gurgaon (2011)
Greiner W.: Classical Mechanics: Systems of Particles and Hamiltonian Dynamics. Springer, New York (2004)
Arnold V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)
Carinena J.F., Ranada M.F., Santander F.: Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability. J. Math. Phys. 46, 062703 (2005)
Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: Unusual Liénard-type nonlinear oscillator. Phys. Rev. E 72, 066203 (2005)
Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator. J. Math. Phys. 48, 032701 (2007)
Gladwin Pradeep R., Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: Nonstandard conserved Hamiltonian structures in dissipative/damped systems: nonlinear generalizations of damped harmonic oscillator. J. Math. Phys. 50, 052901 (2009)
Mathews P.M., Lakshmanan M.: On a unique nonlinear oscillator. Q. Appl. Math. 32, 215 (1974)
Musielak Z.E.: Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. J. Phys. A: Math. Theor. 41, 055205 (2008)
Musielak Z.E., Roy D., Swift L.D.: Method to derive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients. Chaos Solitons Fractals 38, 894 (2008)
Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: On the general solution for the modified Emden-type equation \({\ddot{x}+\alpha x \dot{x}+\beta x^3=0}\). J. Phys. A: Math. Theor. 40, 4717 (2007)
Chithiika Ruby V., Senthilvelan M., Lakshmanan M.: Exact quantization of a PT-symmetric (reversible) Liénard type nonlinear oscillator. J. Phys. A: Math. Theor. 45, 382002 (2012)
Chithiika Ruby V., Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Lienard type nonlinear oscillators. J. Math. Phys. 56, 012103 (2015)
Choudhury A.G., Guha P.: Quantization of the Liénard II equation and Jacobi’s last multiplier. J. Phys. A: Math. Theor. 46, 165202 (2013)
Gubbiottia G., Nucci M.C.: Noether symmetries and the quantization of a Liénard-type nonlinear oscillator. J. Nonlinear Math. Phys. 21, 248–264 (2014)
Corichi A., Ryan M.P. Jr: Quantization of non-standard Hamiltonian systems. J. Phys. A: Math. Gen. 30, 3553 (1997)
Choudhury A.G., Guha P., Khanrad B.: On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification. J. Math. Anal. Appl. 360, 651–664 (2009)
Jacobi C.G.J.: Sul principio dellultimo moltiplicatore e suo uso come nuovo principio generale di meccanica. Giornale Arcadico di Scienze Lettere ed Arti 99, 129 (1844)
Jacobi, C.G.J.: Vorlesungen über Dynamik. Nebst fünf hinterlassenen Abhandlungen desselben herausgegeben von A. Clebsch, Druck und Verlag von Georg Reimer, Berlin (1886)
Nucci M.C., Leach P.G.L.: Lagrangians galore. J. Math. Phys. 48, 123510 (2007)
Nucci M.C., Leach P.G.L.: The Jacobi last multiplier and its applications in mechanics. Phys. Scr. 78, 065011 (2008)
Nucci M.C.: Jacobi last multiplier and Lie symmetries: a novel application of an old relationship. J. Nonlinear Math. Phys. 12, 284 (2005)
Chandrasekar V.K., Pandey S.N., Senthilvelan M., Lakshmanan M.: A simple and unified approach to identify integrable nonlinear oscillators and systems. J. Math. Phys. 47, 023508 (2006)
Mohanasubha R., Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations. Proc. R. Soc. A 470, 20130656 (2014)
Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations. Proc. R. Soc. A 461, 2451 (2005)
Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: On the complete integrability and linearization of nonlinear ordinary differential equations. II. Third-order equations. Proc. R. Soc. A 462, 1831–1852 (2006)
Tiwari, A.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: On the complete Lie point symmetries classification of the mixed quadratic-linear Liénard type equation \({\ddot{x}+f(x)\dot{x}^2+g(x)\dot{x}+h(x)=0}\). Nonlinear Dyn. 82, 1953–1968 (2015)
Gladwin Pradeep R., Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: On certain new integrable second order nonlinear differential equations and their connection with two dimensional Lotka-Volterra system. J. Math. Phys. 51, 033519 (2010)
Chandrasekar V.K., Senthilvelan M., Lakshmanan M.: A unification in the theory of linearization of second order nonlinear ordinary differential equations. J. Phys. A: Math. Gen. 39, L69–L76 (2006)
Olver P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)
Mohanasubha R., Sabiya Shakila M.I., Senthilvelan M.: On the linearization of isochronous centre of a modified Emden equation with linear external forcing. Commun. Nonlinear Sci. Numer. Simul. 19, 799–806 (2014)
Musielak Z.E.: General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems. Chaos Solitons Fractals 42, 2645–2652 (2009)
Cieślinski J.L., Nikiciuk T.: A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients. J. Phys. A 43, 175205 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-016-1602-9