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 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 nonautonomous 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 nonuniqueness of solutions, the integrability problem and Noether’s theorem, giving insights into the differences between mechanics and field theories.