The integrability of multivector fields in a differentiable manifold is
studied. Then, given a jet bundle J1E→E→M, it is shown that integrable
multivector fields in E are equivalent to integrable connections in the
bundle E→M (that is, integrable jet fields in J1E). This result is
applied to the particular case of multivector fields in the manifold J1E and
connections in the bundle J1E→M (that is, jet fields in the repeated jet
bundle J1J1E), in order to characterize integrable multivector fields and
connections whose integral manifolds are canonical lifting of sections. These
results allow us to set the Lagrangian evolution equations for first-order
classical field theories in three equivalent geometrical ways (in a form
similar to that in which the Lagrangian dynamical equations of non-autonomous
mechanical systems are usually given). Then, using multivector fields; we
discuss several aspects of these evolution equations (both for the regular and
singular cases); namely: the existence and non-uniqueness of solutions, the
integrability problem and Noether's theorem; giving insights into the
differences between mechanics and field theories.Comment: New sections on integrability of Multivector Fields and applications
to Field Theory (including some examples) are added. The title has been
slightly modified. To be published in J. Math. Phy