A non-linear oscillator with quasi-harmonic behaviour: two- andn-dimensional oscillators

Abstract

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a twodimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere S 2 and hyperbolic plane H 2 , is discussed.MSC Classification: 37J35, 34A34, 34C15, 70H06 a)

An update on the classical and quantum harmonic oscillators on the sphere and the hyperbolic plane in polar coordinates

“…Note that in the λ > 0 case, such a minimum only exists for J values such that |J| < α/λ, thereby showing that bounded trajectories are restricted to low angular momentum values. It is worth pointing out that such a limitation on bounded motions for λ > 0 was not reported in [1] and that for J = 0, one may set V eff,min = 0. In Figs.…”

“…Having completed the solution of the Euler-Lagrange equations in polar coordinates, we may connect it with that in cartesian coordinates presented in [1]. This is the purpose of the next section.…”

“…During many years, there has been a continuing interest for some generalizations [1,2,3,4] of a classical nonlinear oscillator [5,6], which was introduced as a one-dimensional analogue of some quantum field theoretical models, and for the corresponding extensions [2,3,7,8,9,10,11] of its quantum version [12,13]. Such a model is indeed an interesting example of a system with nonlinear oscillations with a frequency showing amplitude dependence.…”

“…In 2004, Cariñena, Rañada, Santander, and Senthilvelan introduced a two-dimensional (and more generally n-dimensional) classical generalization [1] of the one-dimensional model of [5,6]. They established that the nonlinearity parameter λ, entering the definition of the potential and the position-dependent mass, can be interpreted as −κ, where κ is the curvature of the two-dimensional space, so that their model actually describes a harmonic oscillator on the sphere (for λ = −κ < 0) and on the hyperbolic plane (for λ = −κ > 0).…”

“…In [1], the Euler-Lagrange equations in cartesian coordinates were so complicated that they could not be directly solved in a simple way. Some particular expressions were then assumed for the solutions and the undetermined parameters they contained were shown to satisfy some constraints by inserting such expressions in the equations.…”