Global d-invariance in field theory

In a previous work the authors derived a remarkable local identity which allows the writing of any Lagrangian as a 'linear combination' of its field equations plus a divergence. Using this identity they were able to provide an alternative proof of the fact that a (higher-order) Lagrangian has identically vanishing field equations if and only if it is locally a divergence. The aim of this work is to investigate how far one can go in globalizing the previous results for (higher-order) Lagrangians. In the case of vector or affine bundles the previous results admit global generalizations in a natural way. The true obstacle is the topological structure of the fibre bundle (both of the basis manifold and of the fibres). As a general rule, it turns out that one can globalize in a non-unique way the previous results when the fibre bundle admits global sections and, moreover, it is contractible by fibred morphisms to one of its global sections. Uniqueness is lost at the level of affine bundles, and for 'non-trivial' topologies one loses the globality of the result.