Higher-order differential equations represented by connections on prolongations of a fibered manifold. (English) Zbl 0992.34006

Vondra, Alexandr

Extr. Math. 15, No. 3, 421-512 (2000).

Here, the author continues his studies on the geometry of ordinary differential equations extending his previous ideas, constructions and results from the first- and second-order situation to that of an arbitrary finite-order one. The main subjects are: jet prolongations of sections and morphisms, total derivatives, prolongations of vector fields, the contact structure and the Cartan distribution, repeated jets, \((k+1)\)-connections on fibered manifolds, higher-order equations represented by connections, prolongations and fields of paths, symmetries and vertical prolongations, connections on affine bundle \(J^{k+1}\pi \rightarrow J^k \pi\) for \(\pi :Y \rightarrow X\) an arbitrary fibered bundle, the method of characteristics, the method of fields of paths, strong horizontal distributions, semispray connections, time-dependent higher-order differential equations. The notions are clearly explained and the proofs are given in all details. Several examples are included.

Reviewer: M.Anastasiei (Iaşi)

MSC:

34A26	Geometric methods in ordinary differential equations
53C05	Connections (general theory)
35A30	Geometric theory, characteristics, transformations in context of PDEs
34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.

Keywords:

fibered manifolds; jet bundles; higher-order differential equations; semispray connections