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 Frolicher-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 Poincare-Cartan 1-form of some Lagrangian function. We prove that if there exists a semibasic 1-form in Lambda(1)(S)(J(1)pi) , which is not a Poincare-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE. (c) 2010 Elsevier B.V. All rights reserved.