Integrability of Dynamical Systems: A Geometrical Viewpoint

TL;DR

This paper explores the geometric approach to understanding the integrability of dynamical systems, emphasizing the role of geometric symmetries and developing a new method to compute first integrals using differential geometry.

Contribution

It introduces a novel geometric method to determine first integrals of dynamical systems by leveraging symmetries of the kinetic metric.

Findings

Established a link between geometric symmetries and first integrals.

Developed a new geometric technique for computing first integrals.

Enhanced understanding of integrability through differential geometry.

Abstract

The physical phenomena are described by physical quantities related by specific physical laws. In the context of a Physical Theory, the physical quantities and the physical laws are described, respectively, by suitable geometrical objects and relations between these objects. These relations are expressed with systems of (mainly second order) differential equations. The solution of these equations is frequently a formidable task, either because the dynamical equations cannot be integrated by standard methods or because the defined dynamical system is non-integrable. Therefore, it is important that we have a systematic and reliable method to determine their integrability. This has led to the development of several (algebraic or geometric) methods, which determine if a dynamical system is integrable/superintegrable or not. Most of these methods concern the first integrals (FIs), that is, quantities that are constant along the evolution of the system. FIs are important, because they can be used to reduce the order of the system of the dynamical equations and, if there are `enough' of them, even to determine its solution by means of quadratures. In the latter case, the dynamical system is said to be Liouville integrable and it is associated with a canonical Lagrangian, whose kinetic energy defines a metric tensor known as kinetic metric. It is proved that there is a close relation between the geometric symmetries (collineations and Killing tensors) of this metric and the quantities defining the FIs. This correspondence makes it possible to use the results of Differential Geometry in the study of the integrability of dynamical systems. In this thesis, we study this correspondence and geometrize the determination of FIs by developing a new geometric method to compute them.