Geometric models for Lie--Hamilton systems on $\mathbb{R}^2$

TL;DR

This paper introduces geometric models for Lie--Hamilton systems on r^2, using symplectic leaves and quotient spaces, enabling a unified analysis and extension to higher dimensions.

Contribution

It presents two novel geometric models for Lie--Hamilton systems on r^2, connecting symplectic leaves and quotient spaces, and offers a framework for higher-dimensional generalizations.

Findings

Models recover known results naturally.

Framework facilitates analysis of Lie--Hamilton systems.

Potential extension to higher-dimensional manifolds.

Abstract

This paper provides a geometric description for Lie--Hamilton systems on $\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie--Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie--Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give rise to a natural framework for the analysis of Lie--Hamilton systems on $\mathbb{R}^2$ while retrieving known results in a natural manner. Our methods may be extended to study Lie--Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.