Derivations of forms along a map: The framework for time-dependent second-order equations
type: Item
instance of: scholarly article
title: Derivations of forms along a map: The framework for time-dependent second-order equations
zbMATH Open document ID: 0831.58003
DOI: 10.1016/0926-2245(95)00013-T
author: A. Vandecasteele
author: F. Cantrijn
author: W. Sarlet
author: Eduardo Martínez
published in: Differential Geometry and its Applications
publication date: 1996-02-13T00:00:00Z
review text: 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 \(J^1 E\to E\).
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 58A30
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58A99
zbMATH DE Number: 778518
zbMATH Keywords: tangent bundle
zbMATH Keywords: derivations
zbMATH Keywords: connections
zbMATH Keywords: second-order equations
Wikidata QID: Q115362563
reviewed by: Jan Chrastina
MaRDI profile type: Publication
cites work: Generalized Bianchi identities for horizontal distributions
cites work: A geometrical version of the Helmholtz conditions in time-dependent Lagrangian dynamics
cites work: Towards a geometrical understanding of Douglas's solution of the inverse problem of the calculus of variations
cites work: Q4009045
cites work: Q3244523
cites work: Structure presque tangente et connexions. I
cites work: Derivations of differential forms along the tangent bundle projection
cites work: Derivations of differential forms along the tangent bundle projection. II
cites work: Geometric characterization of separable second-order differential equations
cites work: Q3719916
cites work: The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics
cites work: Q3774596
cites work: Adjoint symmetries for time-dependent second-order equations
cites work: Q3994886
cites work: Q4012517
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description: scientific article; zbMATH DE number 778518