  The usefulness in control theory of the geometric theory of motion on Lie
groups and homogeneous spaces will be shown. We quickly review some recent
results concerning two methods to deal with these systems, namely, a
generalization of the method proposed by Wei and Norman for linear systems, and
a reduction procedure. This last method allows us to reduce the equation on a
Lie group $G$ to that on a subgroup $H$, provided a particular solution of an
associated problem in $G/H$ is known. These methods are shown to be very
appropriate to deal with control systems on Lie groups and homogeneous spaces,
through the specific examples of the planar rigid body with two oscillators and
the front-wheel driven kinematic car.
