Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 059, 17 pages      arXiv:1203.1716      http://dx.doi.org/10.3842/SIGMA.2012.059

Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

Oana Constantinescu
Faculty of Mathematics, Alexandru Ioan Cuza University, Bd. Carol no. 11, 700506, Iasi, Romania

Received March 16, 2012, in final form September 03, 2012; Published online September 06, 2012

Abstract
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-Kähler 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.

Key words: formal integrability; partial differential operators; Lagrangian semisprays; Helmholtz conditions.

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