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

Symmetry Integrability and Geometry-Methods and Applications

article

Oana Constantinescu¹ 

[1] Faculty of Mathematics, Alexandru Ioan Cuza University

关键词: formal integrability;    partial dif ferential operators;    Lagrangian semisprays;    Helmholtz conditions;   

DOI  :  10.3842/SIGMA.2012.059

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown