New methods for the integration of distributions of vector fields

Abstract

The thesis explores a new mathematical framework in order to obtain closed-form expressions for the integral manifolds of involutive distributions of vector fields. In particular, it is focused on the important particular case of distributions generated by the vector fields associated to ordinary differential equations (ODEs). We introduce the concept of Cinf-structure, which generalizes the notion of solvable structure for involutive distributions. The generalization is rooted in the extension of Lie point symmetries to Cinf-symmetries for ODEs. We adapt this notion to the case of distributions of vector fields and use it to define Cinf-structures. The main result of the thesis is that the knowledge of a Cinf-structure for a given involutive distribution allows us to integrate the distribution by solving a sequence of completely integrable Pfaffian equations. This result is applied to the particular case of distributions generated by the vector fields associated to ODEs. In this case, a Cinf-structure allows us to split the integration of an mth-order ODE into m completely integrable Pfaffian equations. The thesis also explores the applicability of Cinf-structures to integrate distributions in several cases where standard methods fail. In particular, we show that the method yields closed-form solutions for ODEs that cannot be addressed using classical Lie symmetry methods. We also provide a method for searching for a C-structure for a given ODE. These results are then applied to practical scenarios, mainly in mathematical physics. The algorithms for the search of Cinf-structures and their use to integrate involutive distributions are implemented in the CinfStructures package for the computer algebra system Maple. A detailed description of the package has been included.