Sections along a map applied to higher-order Lagrangian mechanics. Noether's theorem (Q1182676)

Let \(\pi: E\to M\) be a fibre bundle, \(f: N\to M\) a differentiable map. A differentiable map \(\sigma: N\to E\) such that \(\pi\circ\sigma=f\) is called a section of \(E\) along \(f\). The main part of the paper is devoted to thorough study of various formal aspects of the latter concept within the framework of the higher-order jet theory, especially relations to various canonical (natural) tensor fields, liftings and prolongations, vertical endomorphisms, and Frölicher-Nijenhuis theory are involved. The (not quite simple) results are applied to one-dimensional Lagrangian and Hamiltonian mechanics. Owing to an appropriately adapted definition of symmetries of variational problems, a version of Noether's theorem admitting the converse is derived as the main achievement.