%0 Journal Article
%T Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
%A Oana Constantinescu
%J Symmetry, Integrability and Geometry : Methods and Applications
%D 2012
%I National Academy of Science of Ukraine
%X We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-K hler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.
%K formal integrability
%K partial differential operators
%K Lagrangian semisprays
%K Helmholtz conditions
%U http://dx.doi.org/10.3842/SIGMA.2012.059