  This paper studies the reduction by symmetry of variational problems on Lie
groups and Riemannian homogeneous spaces. We derive the reduced equations of
motion in the case of Lie groups endowed with a left-invariant metric, and on
Lie groups that admits a bi-invariant metric. We repeated this analysis for
Riemannian homogeneous spaces, where we derive the reduced equations by
considering an alternative variational problem written in terms of a connection
on the horizontal bundle of the underlying Lie group. We study also the case
that the underlying Lie group admits a bi-invariant metric, and consider the
special case that the homogeneous space is in fact a Riemannian symmetric
space. These ideas are applied to geodesics for a rigid body on $SO(3)$ to
derive geodesic equations on the dual of its Lie algebra (a vector space), the
heavy-top in $SE(3)$ to derive reduced equations of motion on the unit sphere
$S^2$, geodesics on $S^2$ as a Riemannian symmetric space endowed with a
bi-invariant metric and optimal control problems for applications to robotic
manipulators.
