Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories
Abstract
The integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle
J
1
E→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
J
1
E
). This result is applied to the particular case of multivector fields in the manifold
J
1
E
and connections in the bundle
J
1
E→M
(that is, jet fields in the repeated jet bundle
J
1
J
1
E
), 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.