Decomposition of higher order tangent fields and calculus of variations

R. J. Alonso


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