Método das prolongações e suas aplicações à física

Abstract

The method of prolongations, developed from the pioneering studies of Sophus Lie between the 1870s and 1890s, is a mathematical technique that allows the determination of generators of continuous transformations that act as symmetries, i.e., that map solutions onto solutions, of differential equations. Our aim in the present work will be to study the method and investigate how it can be applied to physics.

Firstly, we will study the method of prolongations in detail. After a necessary general review, we will construct the method in the context of differential equations, aiming at point transformations. Then, we will do the same in the context of variational problems, equipped with lagrangians, from which the so-called variational symmetries and very important associated results will arise.

Next, we will turn our attention to the applicability of the formalism to physics. We will do this in order to explore the results derived in the first chapters. The method will be applied to two equations of physical interest: the Burgers and Korteweg-De Vries equations. In addition, we will venture into the fields of general relativity and cosmology and see how the method of prolongations can help us solve problems in these areas. We will also have the opportunity to learn how we can apply our studies to the problem of quantization of gravitation.