We present a geometric algorithm for obtaining consistent solutions to
systems of partial differential equations, mainly arising from singular
covariant first-order classical field theories. This algorithm gives an
intrinsic description of all the constraint submanifolds.
The field equations are stated geometrically, either representing their
solutions by integrable connections or, what is equivalent, by certain kinds of
integrable m-vector fields. First, we consider the problem of finding
connections or multivector fields solutions to the field equations in a general
framework: a pre-multisymplectic fibre bundle (which will be identified with
the first-order jet bundle and the multimomentum bundle when Lagrangian and
Hamiltonian field theories are considered). Then, the problem is stated and
solved in a linear context, and a pointwise application of the results leads to
the algorithm for the general case. In a second step, the integrability of the
solutions is also studied.
Finally, the method is applied to Lagrangian and Hamiltonian field theories
and, for the former, the problem of finding holonomic solutions is also
analized.Comment: 30 pp. Presented in the International Workshop on Geometric Methods
in Modern Physics (Firenze, April 2005