Quantization of quadratic Liénard-type equations by preserving Noether symmetries

Abstract

The classical quantization of a family of a quadratic Liénard-type equation (

Noether Symmetries Quantization and Superintegrability of Biological Models

“…Also, we have shown that if we replace the independent variable t with τ = it, then Equation (8) is transformed into Equation (20), which is one of the isochronous Liénard II equations [33]. Its corresponding Schrödinger equation was derived in [23,34]. The eigenfunctions and energy eigenvalues are given in (23) and (24), respectively.…”

“…This method was reformulated in [19] for problems that are linearizable by Lie point symmetries, and also successfully applied to various classical problems: second-order Riccati equation [20], dynamics of a charged particle in a uniform magnetic field and a non-isochronous Calogero's goldfish system [18], an equation related to a Calogero's goldfish equation [21], two nonlinear equations somewhat related to the Riemann problem [22], a Liénard I nonlinear oscillator [19], a family of Liénard II nonlinear oscillators [23], N planar rotors and an isochronous Calogero's goldfish system [24], the motion of a free particle and that of a harmonic oscillator on a double cone [25].…”

“…The quantization of Equation (21) was given in [34]. In [23], the quantization that preserves the Noether symmetries was applied to all equations (21). Equation (20) gives rise to the following Schrödinger equation…”

The nonlinear pendulum always oscillates

“…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.…”

Position-dependent mass approach and quantization for a torus Lagrangian

“…At the same time, some of these models lead to applications in nano-ribbons [5]. When it comes to the mathematical methods for the symmetries of the nonlinear systems, it is known that any second order non-linear differential equation admits eight parameter Lie point symmetries, thus they are solvable through point transformations [6], [7], [8], [9], [10]. On the other hand, the integrable and super-integrable classical oscillators and their relationships with a nonlinear Liénard type nonlinear equations are given in [11], [12], [13].…”