  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 Fr\"olicher-Nijenhuis theory on the first jet bundle,
$J^1\pi$. We prove that a system of time dependent SODE, identified with a
semispray $S$, is Lagrangian if and only if a special class,
$\Lambda^1_S(J^1\pi)$, of semi-basic 1-forms is not empty. We provide global
Helmholtz conditions to characterize the class $\Lambda^1_S(J^1\pi)$ of
semi-basic 1-forms. Each such class contains the Poincar\'e-Cartan 1-form of
some Lagrangian function. We prove that if there exists a semi-basic 1-form in
$\Lambda^1_S(J^1\pi)$, which is not a Poincar\'e-Cartan 1-form, then it
determines a dual symmetry and a first integral of the given system of SODE.
