A new geometric approach to Lie systems and physical applications

The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei--Norman method and the reduction method, which is very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.