In this paper, we provide controllability tests and motion control algorith
ms for underactuated mechanical control systems on Lie groups with Lagrangi
an equal to kinetic energy. Examples include satellite and underwater vehic
le control systems with the number of control inputs less than the dimensio
n of the configuration space. Local controllability properties of these sys
tems are characterized, and two algebraic tests are derived in terms of the
symmetric product and the Lie bracket of the input vector fields. Perturba
tion theory is applied to compute approximate solutions for the system unde
r small-amplitude forcing; in-phase signals play a crucial role in achievin
g motion along symmetric product directions. Motion control algorithms are
then designed to solve problems of point-to-point reconfiguration, static i
nterpolation and exponential stabilization. We illustrate the theoretical r
esults and the algorithms with applications to models of planar rigid bodie
s, satellites and underwater vehicles.