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, J1πJ1π. We prove that a system of time dependent SODE, identified with a semispray SS, is Lagrangian if and only if a special class, ΛS1(J1π), of semi-basic 11-forms is not empty. We provide global Helmholtz conditions to characterize the class ΛS1(J1π) of semi-basic 11-forms. Each such class contains the Poincaré–Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 11-form in ΛS1(J1π), which is not a Poincaré–Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.