Lie--Hamilton systems: theory and applications

TL;DR

This paper introduces Lie--Hamilton systems, a new class of Lie systems on Poisson manifolds, and develops methods to analyze their symmetries, constants of motion, and linearisability, with applications in physics and mathematics.

Contribution

It defines and studies Lie--Hamilton systems, providing new geometric tools and criteria for their analysis and applications.

Findings

Developed methods for superposition rules and constants of motion.

Identified conditions for linearisability of Lie--Hamilton systems.

Illustrated results with examples from physics and mathematics.

Abstract

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.