4. Variational Structures

In this chapter, a complete treatment of the foundations of the calculus of variations on fibered manifolds is presented. Using the calculus of differential forms as the main tool, the aim is to study higher-order integral variational functionals of the orm \( \gamma \to {\int }J^{r} \gamma \ast \rho \), depending on sections γ of a fibered manifold Y, where ρ is a general differential form on the jet manifold \( J^{r} Y \) and \( J^{r} \gamma \) is the r-jet prolongation γ. The horizontal forms ρ are the Lagrangians. Variations (deformations) of sections of Y are considered as vector fields, permuting the set of sections, and their prolongations to the jet manifolds \( J^{r} Y \). They are applied to the variational functionals in a geometric way by means of the Lepage forms. The main idea can be introduced by means of the Cartan’s formula for the Lie derivative of a differential form η on a manifold Z, \( {\mathrm{\partial}}_{\xi } \eta = i_{\xi } {{d}}\eta + {{d}}i_{\xi } \eta \), where \( i_{\xi } \) is the contraction by a vector field ξ and d is the exterior derivative operator. Replacing Z with the r-jet prolongation \( J^{r} Y \) and η with ρ, it is proved that the form ρ in the variational functional \( \gamma \to {\int }J^{r} \gamma \ast \rho \) may be chosen in such a way that the Cartan’s formula forρ becomes a geometric version of the classical first variation formula. A structure theorem implies that for different underlying manifold structures and order of their jet prolongations, the concept of a Lepage formgeneralizes the well-known Cartan form in classical mechanics, the Poincaré-Cartan forms in the first-order field theory, the so-called fundamental forms, the second-order generalisation of the Poincaré-Cartan form, the Carathéodory form, and the Hilbert form in Finsler geometry.