The representation of the variational sequence in field theory, submitted to

Abstract. The aim of this paper is to discuss some aspects of local and global properties of classical concepts of calculus of variations in the r−th order field theory on fibered manifolds within the framework of the variational sequence, which is the quotient of the De Rham sequence with respect to its subse-quence of contact differential forms. Such a discussion is, in general, based on the concept of sheaves of differential forms. In the paper a globally defined representation of the variational sequence by forms is constructed for its part closely related to the standard concepts of the calculus of variations. The ex-tended definition of the Euler-Lagrange form as a representative of the class of (n+2)-forms is considered and the definition of the so called Helmholtz-Sonin form as a representative of the class of (n+ 2)-forms is presented. The prop-erties of corresponding terms in the variational sequence, considered as the generalized Euler-Lagrange mapping and the Helmholtz-Sonin mapping, are studied. There is a close relationship between elements of the quotient sheaves (classes of forms) and the quotient mappings on one hand and the standard objects of the calculus of variations, such as lagrangian, Euler–Lagrange form and Helmholtz–Sonin expressions defining the so called Helmholtz–Sonin form, on the other hand. 1