Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations (original) (raw)
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 1 π. 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 1 π), of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class Λ 1 S (J 1 π) 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 1 π), which is not a Poincaré-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.
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