A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)

TL;DR

This paper introduces a unified method for constructing Poisson-Hopf deformations of Lie-Hamilton systems with a focus on those with a Vessiot-Guldberg Lie algebra isomorphic to sl(2), covering various classical systems and providing constants of motion.

Contribution

It develops a novel unified framework for Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2), enabling systematic generation and analysis of deformed systems.

Findings

Applied to three classes of sl(2) Lie-Hamilton systems.

Covered deformations of classical systems like Ermakov, Milne-Pinney, and Riccati equations.

Provided explicit constants of motion for deformed systems.

Abstract

Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems, a novel unified approach to nonequivalent deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to $\mathfrak{sl}(2)$ is proposed. This, in particular, allows us to define a notion of Poisson-Hopf systems in dependence of a parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of $\mathfrak{sl}(2)$ Lie-Hamilton systems. Our results cover deformations of the Ermakov system, Milne-Pinney, Kummer-Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with $t$-dependent coefficients). Furthermore $t$-independent constants of motion are given as well. Our methods can be employed to generate other Lie-Hamilton systems and their deformations for other Vessiot-Guldberg Lie algebras and their deformations.