We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frolicher-Nijenhuis theory on the first jet bundle, $J^1pi$. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian if and only if a special class, $Lambda^1_S(J^1pi)$ of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize