Lie systems: theory, generalisations, and applications

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.Comment: 161 pages, 2 figure

Foliated Lie systems: Theory and applications

A $\mathcal{F}$- foliated Lie system is a first-order system of ordinary differential equations whose particular solutions are contained in the leaves of the foliation $\mathcal{F}$ and all particular solutions within any leaf can be written as a certain function, a so-called foliated superposition rule, of a family of particular solutions of the system within the same leaf and several parameters. We analyse the properties of such systems and we illustrate our results by studying Lax pairs and a class of $t$-dependent Hamiltonian systems. We finally study foliated Lie systems through Poisson structures and $r$-matrices.Comment: 24 page