  This paper develops a novel approach to necessary optimality conditions for
constrained variational problems defined in generally incomplete subspaces of
absolutely continuous functions. Our approach involves reducing a variational
problem to a (nondynamic) problem of constrained optimization in a normed space
and then applying the results recently obtained for the latter class using
generalized differentiation. In this way, we derive necessary optimality
conditions for nonconvex problems of the calculus of variations with velocity
constraints under the weakest metric subregularity-type constraint
qualification. The developed approach leads us to a short and simple proof of
the First-order necessary optimality conditions for such and related problems
in broad spaces of functions, including those of class C^k.
