Exterior Jet Bundles

[Rock Brentwood]

For field theory, when the field is based on natural objects such as
differential forms, a more suitable way to encapsulate its Lagrangian
appears to be a kind of bundle I've not seen discussed in the
literature anywhere -- the exterior jet bundle.

Thus, for a 1-form field the configuration variables would comprise a
1-form, q, its first order gradients would be the configuration
"velocity" 2-forms v = dq (where the exterior differential operator is
taken with respect to the coordinate dependence q = q(x)), and the
Lagrangian L = L(x, q(x), dq(x)) would be a function of the "exterior
jet extension" x |-> (x, q(x), dq(x)) of a section x |-> (x, q(x)).

This makes for a field-theoretic generalization of mechanics that is
more closely parallel to the univariate Lagrangian case. The analogue
of the Legendre bundle would be played by the (n-1) forms f, the (n-2)
forms p, with the variational expressed as delta(L) = delta(q) ^ f +
delta(v) ^ p.