Lie systems: theory, generalisations, and applications

TL;DR

This paper provides a comprehensive overview of Lie systems, detailing their mathematical properties, generalizations, and diverse applications in mathematics and physics, highlighting recent theoretical advancements.

Contribution

It offers a self-contained guide to the theory of Lie systems, including new findings, their generalizations, and broad applications across disciplines.

Findings

Enhanced understanding of the geometrical features of Lie systems

Development of new generalizations of Lie systems

Illustration of applications in mathematics and physics

Abstract

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.