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

Domenii publicaţii > Matematica + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: Bucataru I, Constantinescu O

Editorial: JOURNAL OF GEOMETRY AND PHYSICS , Volume: 60 Issue: 11 , p.1710-1725 , 2010.


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, $J^1pi$. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian if and only if a special class, $Lambda^1_S(J^1pi)$ of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class $Lambda^1_S(J^1pi)$ 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 $Lambda^1_S(J^1pi)$ , which is not a Poincare-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.

Cuvinte cheie: Semi-basic forms, Poincare lemma, Helmholtz conditions, Inverse problem, Dual symmetry, First integral