Um novo uso de simetrias de Lie não-locais na obtenção de quantidades conservadas

Abstract

In this thesis we present an efficient method to use a non-local symmetry admitted by a second-order rational ordinary differential equation (rational 2EDO) to determine a Liouvillian first integral. We build an algorithm to compute a non-local symmetry of a rational 2EDO and then show that it is possible to use this symmetry in an alternative way to the usual Lie method. Basically, from a non-local symmetry, we construct three polynomial vector fields (in R2), which ‘share’ the Liouvillian first integral with the rational 2EDO. These 2D polynomial vector fields can be used to determine an integrating factor for the rational 2EDO. The main advantages of the proposed method are: obtaining the non-local symmetry is semi-algorithmic and very efficient and, in addition, its use to find an integrating factor is a sequence of very efficient linear or quasi linear processes. In a last step, we implemented the algorithm in a computational package written in Maple.