Lie systems: theory, generalisations, and applications

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.Comment: 161 pages, 2 figure

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

“…Among the low dimensional Lie algebras, the case r 2 sl (2, R) plays a special role, as it is related to various of the most relevant and best studied cases of SODE Lie systems (see, e.g., [2,4,5] and the references therein).…”

“…For a long time considered as a particular technique for differential equations, the importance of Lie systems in physical applications, particularly in the context of integrable systems, as well as in control theory, has motivated extensive studies on the subject in the last few decades (see, e.g., [2,3]) that have led to natural generalizations of the notion of Lie systems [4][5][6]. These generalizations of the classical analytical formulation allow an elegant and effective description of Lie systems in terms of distributions in the sense of Fröbenius and the theory of fiber bundles, hence offering a much wider spectrum of applications, as well as their adaptation to quantum systems (the excellent treatise on the geometrical foundations of Lie systems [2] contains an extensive and updated list of references enumerating these applications).…”

“…It is immediate to verify that there is a one-to-one correspondence between systems and vector fields of this type. The basic criterion concerning the existence of fundamental systems of solutions is due to Lie himself [1] and can be formulated as follows [2,11]:…”

“…The existence of such an algebra hence ensures the existence of a (nonlinear) superposition principle [2,12,13], which can (at least formally) be derived in terms of the differential invariants associated with the generators of a Vessiot-Guldberg-Lie algebra.…”

“…Actually, the same system Equation (1) may admit different superposition rules, hence non-isomorphic Vessiot-Guldberg-Lie algebras that can have different dimensions. The Riccati equation and other ODEs related to it constitute the canonical examples for this pathology [2]. In this sense, the problem differs considerably from the construction of differential equations admitting certain types of symmetry algebras [14].…”