Jacobi-Lie systems: Fundamentals and low-dimensional classification

TL;DR

This paper introduces and classifies Jacobi-Lie systems, a class of differential equations with Lie algebraic structure related to Jacobi manifolds, providing foundational insights and low-dimensional classifications.

Contribution

It defines Jacobi-Lie systems, analyzes their properties, and classifies them in low dimensions, expanding the understanding of Lie systems with Hamiltonian structures on Jacobi manifolds.

Findings

Classification of Jacobi-Lie systems on $\\mathbb{R}$ and $\mathbb{R}^2$

Examples illustrating physical and mathematical applications

Foundational analysis of Hamiltonian vector fields in Jacobi-Lie systems

Abstract

A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on $\mathbb{R}$ and $\mathbb{R}^2$. Our results shall be illustrated through examples of physical and mathematical interest.