Derivations of forms along a map: The framework for time-dependent second-order equations
The authors generalize the formalism of derivations of differential forms along the tangent bundle projection \(\tau: TM\to M\) proposed by \textit{E. Martínez}, \textit{J. F. Cariňena} and \textit{W. Sarlet} [Differ. Geom. Appl. 3, No. 1, 1-29 (1993; Zbl 0770.53018)]\ for the time- dependent case of the projection \(\pi: \mathbb{R}\times TM\to \mathbb{R}\times M\). They deal with tensor fields along \(\pi\), the selection of an appropriate vertical exterior derivative, the problem of classification of all derivatives (with a certain fixed choice of connection), covariant derivatives, torsion and curvature (by using a rather involved machinery of computators), and especially with the particular case that the connection is coming from a second-order equation. The theory leads to a large number of identities useful in calculations and can be applied to the theory of dynamical symmetries of second-order ordinary differential equations, and first-order inverse problems of the classical calculus of variations. The results remain valid in the more general setup if \(\mathbb{R} \times M\) is replaced by a fibre bundle \(E\to \mathbb{R}\) and \(\pi\) is replaced by the projection \(J^1 E\to E\).
scientific article; zbMATH DE number 778518
scientific article