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 $J^{1}E\to E\to M$, it is shown that integrable multivector fields in $E$ are equivalent to integrable connections in the bundle $E\to M$ (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 $J^{1}E\to M$ (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.

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. Phys

References in corpus (1)

• Canonical Structure of Classical Field Theory in the Polymomentum Phase Space

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