The authors present a comparison between two approaches to the geometric formulation of calculus of variations: from one point of view the approach based on the variational complex and its representation in terms of differential forms, from the other the description in terms of variational morphisms. In both cases there are decomposition formulas which allows for a geometric description of integration by parts and the authors show how to relate the two decompositions. As a derived result, the authors introduce a recursive formula for the derivation of the Krupka-Betounes equivalents of a Lagrangian form for first-order field theories and generalize it to second-order ones. Let us outline with more details the contents of the work.
The paper is divided into 5 sections, the first one being a short introduction. The second one is a brief summary containing the definitions of the main objects which will be used in the central part of the manuscript. In Section 3 and 4 the authors present their main results, which follow from the comparison between the two approaches previously mentioned. The first and fundamental observation in Section 3 consists in the fact that any 1-contact \((n+1)\)-form can be seen as a variational morphism of codegree 0 and vice versa. Then, the authors show that the decomposition of variational morphisms in terms of a volume and a boundary terms due to Fatibene and Francaviglia, is the counterpart of the decomposition of the form using the interior Euler operator and the residual operator.
In Section 4 the previous comparison is extended to the more complicated case of 1-contact \((n-s+1)\)-form. Using the methods employed in Section 3, the authors provide a decomposition formula for these forms and introduce a local interior Euler operator and residual operator for lower degree forms. These forms can be identified with variational morphisms of codegree s and due to this identification, it is possible to compare the decomposition of variational morphisms in terms of volume and boundary terms with the one derived from the local interior Euler and residual operators: in general, they are different and further analysis is required in order to understand uniqueness and global properties of these alternative integration by parts formulas (it is important to remark that the boundary term in the decomposition of a variational morphism is not uniquely determined so that all the results presented in the paper are coherent with previous ones). Eventually, Section 5 is devoted to the presentation of a recursive formula which allows to derive the Krupka-Betounes equivalent of a Lagrangian form for first-order field theories and to extend it to second-order field theories.
The paper is well organized since the subdivision in sections and subsections is properly adapted to the contents of the work. There are introductory remarks at the beginning of each section which facilitate the reading and allow to rapidly spot the principal results. The authors have tried to make the paper self-contained adding a section containing basic notions and definitions. Nevertheless, a full understanding of the motivation and the results presented can be achieved only after reading of the main references of the paper, which are correctly mentioned whenever required. In particular, [L. Fatibene and M. Francaviglia, Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1138.81303); D. Krupka, Introduction to global variational geometry. Amsterdam: Atlantis Press (2015; Zbl 1310.49001); M. Palese et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 045, 45 p. (2016; Zbl 1347.70043)] are highly recommended.
The aim of the authors is to provide a local comparison between two different approaches to geometric integration by parts, leaving global considerations to forthcoming analysis. Therefore, the proofs are mainly based on local descriptions of forms and operators. The theoretical discussion is supported by examples which are clearly presented and help the reader to connect the novelties with known results.