Variational first-order partial differential equations

Geometrical and variational properties of systems of first-order partial differential equations (PDE) on fibered manifolds are studied. Existence of Lagrangians is shown to be equivalent with the existence of a closed form which is global and unique; an explicit construction of this form is given. A bijective map between a set of dynamical forms on J1Y, representing first-order PDE, and forms on the total space Y is found, providing a geometric description of the equations by means of a (not necessarily closed) ideal generated by a system of n-forms on Y (n = dimension of the base manifold). Conditions for this ideal to be closed are studied. Relations with Hamiltonian structures and with multisymplectic forms are discussed.