In this paper we analyze and generalize, from the point of view of the
maximum principle, a class of nonlinear optimal control problems orig
inally introduced in Brockett (1994). The optimal control problems dis
cussed here lie on the adjoint orbits of compact Lie groups or on Gras
smann manifolds and the structure of the equations that arise depends
on the metric used - the so-called normal metric. We consider some spe
cial cases of the general problem and their meaning from the point of
view of Riemannian geometry and Hamiltonian mechanics.