Generalized Li\'enard systems and isochronous connections

TL;DR

This paper investigates the classical and quantum properties of a generalized Lie9nard system, revealing its isochronicity, bi-Hamiltonian structure, and potential for bound states through canonical quantization.

Contribution

It introduces a generalized Lie9nard equation derived from Levinson-Smith type equations, maps it to a harmonic oscillator, and explores its bi-Hamiltonian and quantum characteristics.

Findings

The system can be mapped to a harmonic oscillator, demonstrating isochronicity.

It exhibits a bi-Hamiltonian structure with two distinct Hamiltonians.

One Hamiltonian has an equispaced spectrum, indicating quantized bound states.

Abstract

In this paper, we explore some classical and quantum aspects of the nonlinear Li\'enard equation $\ddot{x} + k x \dot{x} + \omega^2 x + (k^2/9) x^3 = 0$, where $x=x(t)$ is a real variable and $k, \omega \in \mathbb{R}$. We demonstrate that such an equation could be derived from an equation of the Levinson-Smith kind which is of the form $\ddot{z} + J(z) \dot{z}^2 + F(z) \dot{z} + G(z) = 0$, where $z=z(t)$ is a real variable and $\{J(z), F(z), G(z)\}$ are suitable functions to be specified. It can further be mapped to the harmonic oscillator by making use of a nonlocal transformation, establishing its isochronicity. Computations employing the Jacobi last multiplier reveal that the system exhibits a bi-Hamiltonian character, i.e., there are two distinct types of Hamiltonians describing the system. For each of these, we perform a canonical quantization in the momentum representation and explore the possibility of bound states. While one of the Hamiltonians is seen to exhibit an equispaced spectrum with an infinite tower of states, the other one exhibits branching but can be solved exactly in closed form for certain choices of the parameters.