  The usual treatment of a (first order) classical field theory such as
electromagnetism has a little drawback: It has a primary constraint submanifold
that arise from the fact that the dynamics is governed by the antisymmetric
part of the jet variables. So it is natural to ask if there exists a
formulation of this kind of field theories which avoids this problem, retaining
the versatility of the known approach. The following paper deals with a family
of variational problems, namely, the so called non standard variational
problems, which intends to capture the data necessary to set up such a
formulation for field theories; moreover, we will formulate a multisymplectic
structure for the family of non standard variational problems, and we will
relate this with the (pre)symplectic structure arising on the space of sections
of the bundle of fields. In this setting the Dirac theory of constraints will
be studied, obtaining among other things a novel characterization of the
constraint manifold which arises in this theory, as generators of an exterior
differential system associated to the equations of motion and the chosen
slicing. Several examples of application of this formalism are discussed: Two
of them motivated from the physical point of view, that is, electromagnetism
and Poisson sigma models, and two examples of mathematical application. In the
case of electromagnetism, it is shown that this formulation avoids the problems
arising in the usual approach.
