Using averaging techniques and developing proper algebraic formalisms, we study the limiting process of ordinary differential equations with highly oscillatory right-hand sides. We give sufficient conditions, generalizing earlier work by Kurzweil and Jarnik, for a sequence {u(j) = (u(1)(j),..., u(m)(j))} subset of or equal to L-1 ([0,T], R-m) to be such that, for every choice of smooth vector fields f(k), k = 1,..., m, on a smooth manifold, the trajectories of (x) over dot = Sigma(k=1)(m) u(k)(j)(t)f(k)(x) converge to the trajectories of an "extended system" (x) over dot = Sigma(k=1)(r) upsilon(k)(t)f(k)(x), where the new directions f(m+1),..., f(r) are Lie brackets of f(1),..., f(m).