Finite order variational bicomplexes

Abstract

The theory of variational bicomplexes was established at the end of the seventies by several authors [2, 17, 23, 26, 29–32]. The idea is that the operations which take a Lagrangian into its Euler–Lagrange morphism [9, 10, 12, 24] and an Euler–Lagrange morphism into its Helmholtz' conditions of local variationality [1–3, 7, 11, 13, 18, 27] are morphisms of a (long) exact sheaf sequence. This viewpoint overcomes several problems of Lagrangian formulations in mechanics and field theories [21, 28]. To avoid technical difficulties variational bicomplexes were formulated over the space of infinite jets of a fibred manifold. But in this formalism the information relative to the order of the jet where objects are defined is lost.

We refer to the recent formulation of variational bicomplexes on finite order jet spaces [13]. Here, a finite order variational sequence is obtained by quotienting the de Rham sequence on a finite order jet space with an intrinsically defined sub-sequence, whose choice is inspired by the calculus of variations. It is important to find an isomorphism of the quotient sequence with a sequence of sheaves of ‘concrete’ sections of some vector bundle. This task has already been faced locally [22, 25] and intrinsically [33] in the case of one independent variable.

In this paper, we give an intrinsic isomorphism of the variational sequence (in the general case of n independent variables) with a sequence which is made by sheaves of forms on a jet space of minimal order. This yields new natural solutions to problems like the minimal order Lagrangian corresponding to a locally variational Euler–Lagrange morphism and the search of variationally trivial Lagrangians. Moreover, we give a new intrinsic formulation of Helmholtz' local variationality conditions, proving the existence of a new intrinsic geometric object which, for an Euler–Lagrange morphism, plays a role analogous to that of the momentum of a Lagrangian.