Differential Geometry

We discuss intrinsic aspects of Krupka's approach to nite{order variational sequences. We recover in an intrinsic way the second{order varia-tional calculus for aane Lagrangians by means of a natural generalisation of rst{ordertheories. Moreover, we nd an intrinsic expressionfor the Helmholtz morphism using a technique introduced by Koll a r that we have adapted to our context. Introduction The theory of variational bicomplexes was established at the end of the sev-enties by several authors AnDu80], OlSh78], Tak79], Tul77], Vin77]. In these works there is the idea that one can give a geometric formulation of the calculus of variations without using integrals. But, except of AnDu80], in these works the variational bicomplex is built over the space of innnite jets of a bred manifold. This procedure is suggested by the relatively simple structure of such spaces. In this paper, we start from Krupka's setting of variational sequences on nite{ order jet spaces Kru90]. A nite{order bicomplex is produced when one quotients the de Rham sequence on a nite{order jet space by means of an intrinsically deened subsequence. This nite{order approach has been fruitfully applied to a concrete relativistic theory in MoVi95]; the idea of a study of the variational bicomplex by means of intrinsic techniques has its origin in this work. Here, we restrict ourselves to the case of the rst{order variational bicomplex on a bred manifold whose base is one{dimensional. We give isomorphisms of the quotient sheaves of the bicomplex with subsheaves of the sheaves of forms on a jet space of suitable order. This order is always found as the minimal among all possible candidates; this aspect is not present in the innnite jet formalism. As a by{product, we nd that the rst{order variational bicomplex allows to deal with second{order aane Lagrangians and the related Euler{Lagrange morphism,