##
Decomposition of higher order tangent fields and
calculus of variations

##
*R. J. Alonso*

**E-mail:** ricardo@gugu.usal.es
**Abstract.** Given a linear connection on a differentiable manifold
$M$, we obtain a method for decomposing higher order tangent fields (in
$\T^\infty$) by means of ordinary tangent fields and differential operators.
As a consequence, we can find primitives for the result of the action of
$\T^\infty$ on the module of differential forms on $M$ of maximum degree.
With certain restrictions, the same model applies to forms with values
in a $\C^\infty(M)$-module. When the aforementioned module is the contact
module of a fibered manifold, we can apply this result to the calculus
of variations and then we get the Poincar\'e-Cartan operators associated
to the decomposition obtained. This method clarifies the role played by
the linear connection in the higher order variational theory. Finally we
give an outline of the application to the variational bicomplex.

**AMSclassification.** 58E30, 32C38, 58G05

**Keywords.** Differential operators, calculus of variations, Lagrangian,
Euler-Lagrange operator, Poincar\'e-Cartan operators, jets, contact system,
variational bicomplex