  The main tool to study a second order optimality problem is the Hessian
operator associated to the cost function that defines the optimization problem.
By regarding an orthogonal Stiefel manifold as a constraint manifold embedded
in an Euclidean space we obtain a concise matrix formula for the Hessian of a
cost function defined on such a manifold. We introduce an explicit local frame
on an orthogonal Stiefel manifold in order to compute the components of the
Hessian matrix of a cost function. We present some important properties of this
frame. As applications we rediscover second order conditions of optimality for
the Procrustes and the Penrose regression problems (previously found in the
literature). For the Brockett problem we find necessary and sufficient
conditions for a critical point to be a local minimum. Since many optimization
problems are approached using numerical algorithms, we give an explicit
description of the Newton algorithm on orthogonal Stiefel manifolds.
