Symmetry Integrability and Geometry-Methods and Applications | |
Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations | |
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
【 预 览 】
Files | Size | Format | View |
---|---|---|---|
RO202106300001527ZK.pdf | 453KB | download |