Lie Systems of Differential Equations and Connections in Fibre Bundles

Abstract

The study of systems of differential equations admitting a super-position function allowing us to write the general solution in terms of a set of arbitrary, but independent, particular solutions, and some constants determining each solution, can be reduced to that of an equation on a Lie group. It will be shown that all these systems of differential equations can be seen as the systems determining the horizontal curves on an appropriate connection and we will show how the theory of reduction can be used to simplify the problem of finding the general solution of such Lie systems. The theory will be illustrated with several physical applications