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

Abstract

In this paper, we discuss the inverse problem for a mixed Liénard type nonlinear oscillator equationẍ + f (x)ẋ 2 + g(x)ẋ + 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 non-standard Lagrangian and Hamiltonian functions. The classical dynamics of the nonlinear oscillator is also discussed and certain special properties including isochronous oscillations are brought out.Keywords Inverse problem · Liénard type nonlinear oscillator equation · Jacobi last multiplier · Prelle-Singer procedure · Lie point symmetries IntroductionIn general, the Lagrangian of an autonomous dynamical system is defined as the difference between kinetic and potential energies [1,2,3]. Systems which have Lagrangians of this form are termed as natural/standard Lagrangians. However, recent studies reveal that certain autonomous dynamical systems do admit other forms of Lagrangians in which no such clear identification of kinetic and potential energy terms can be made. In other words these new forms of Lagrangians do not have separate kinetic or potential energy terms. However, they provide the same equation of motion as

New non-standard Lagrangians for the Liénard-type equations

“…The same Lagrangian as (9) was obtained in [7][8][9]14]. Therefore, we have seen that the family of Liénard-type equations (1) defined by correlation (10) shares Lagrangian ( 5) with (4).…”

“…One can see that condition (16) does not coincide with conditions for the existence of a Lagrangian for (1) that were found in [7][8][9]14]. Therefore, we obtain a new family of the Liénard-type equations that admits a nonstandard Lagrangian.…”

“…We also obtain autonomous first integrals for each member of this equations family. Moreover, we show that some of the conditions for the existence of nonstandard Lagrangian for equation (1) that were found in [7][8][9]14] follows from linearizability via nonlocal transformations of the corresponding family of Liénard-type equations. To the best of our knowledge, our results have not been reported previously.…”

“…Criterion of linearizability ( 6) is completely equivalent to the criterion for the existence of a non-standard Lagrangian obtained in [7,8]. Indeed, assuming that σ = 1 and κ = 0 and changing notation to the one of works [7,8] (see also [9,14]) and differentiate the result with respect to y we get the criterion of the existence of a non-standard Lagrangian obtained in [7,8]. We get the corresponding Lagrangian as follows…”

“…Authors of [10] constructed Lagrangians for many of the Painlevé-Gambiér equations by means of the Jacobi last multiplier. Results obtained in [7][8][9] were rederived via the Jacobi approach in [14].…”