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Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations

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