Noether symmetries and the quantization of a Liénard-type nonlinear oscillator

Abstract

The classical quantization of a Liénard-type nonlinear oscillator is achieved by a quantization scheme (M. C. Nucci. Theor. Math. Phys., 168:994-1001, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. This method straightforwardly yields the Schrödinger equation in the momentum space as given in (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light on the apparently remarkable connection with the linear harmonic oscillator.

Quantization of the dynamics of a particle on a double cone by preserving Noether symmetries

“…In [10] the same method yielded the Schrödinger equation of an equation related to a Calogero's goldfish, and in [11] that of two nonlinear equations somewhat related to the Riemann problem [12]. In [13], and [14] it was shown that the preservation of the Noether symmetries straightforwardly yields the Schrödinger equation of a Liénard I nonlinear oscillator in the momentum space [15], and that of a family of Liénard II nonlinear oscillators [16], respectively.…”

“…Consequently if system (1) admits sl(N + 2, R) as Lie symmetry algebra then in [13] we reformulated the algorithm that yields the Schrödinger equation as follows:…”

Lax representation and quadratic first integrals for a family of non-autonomous second-order differential equations

“…Perhaps the most well-studied equation from (1) is the classical Liénard equation. For example, Lie and Noether symmetries of (2) were studied in [8,21,24,25]. Linearization and equivalence to some Painlevé-Gambier equations via the generalized Sundman transformations were considered in [6, 7, 10-12, 17, 20].…”

The nonlinear pendulum always oscillates

“…Consequently if system (4.1) admits sl(N + 2, R) as Lie point symmetry algebra then in [4] we reformulated the algorithm that yields the Schrödinger equation as follows:…”

“…Then, we determine three Lagrangians that admit the highest number of Noether point symmetries with the help of the Jacobi last multiplier [8]. Then, we recall the quantization method that preserves the Noether point symmetries as described for the first time in [19,20], reformulated in [4] for problems that are linearizable by Lie point symmetries (as in the present case), and successfully applied to various classical problems: second-order Riccati equation [21], dynamics of a charged particle in a uniform magnetic field and a non-isochronous Calogero's goldfish system [20], an equation related to a Calogero's goldfish equation [22], two nonlinear equations somewhat related to the Riemann problem [23], a Liénard I nonlinear oscillator [4], a family of Liénard II nonlinear oscillators [5], N planar rotors and an isochronous Calogero's goldfish system [24], a particle on a double cone [6]. Consequently, as a mathematical divertissement, we quantize the second-order differential equation determining the phase-space trajectories of the nonlinear pendulum.…”