  A novel approach for Lagrange formulation for field theories is proposed in
terms of Kawaguchi geometry (areal metric space). On the extended configuration
space M for classical field theory composed of spacetime and field
configuration space, one can define a geometrical structure called Kawaguchi
areal metric K from the field Lagrangian and (M,K) can be regarded as Kawaguchi
manifold. The geometrical action functional is given by K and the dynamics of
field is determined by covariant Euler-Lagrange equation derived from the
variational principle of the action. The solution to the equation becomes a
minimal hypersurface on (M,K) which has the same dimension as spacetime. We
propose that this hypersurface is what we should regard as our real spacetime
manifold, while the usual way to understand spacetime is to consider it as the
parameter spacetime (base manifold) of a fibre bundle. In this way, the
dynamics of field and spacetime structure is unified by Kawaguchi geometry. The
theory has the property of strong covariance, which means that it allows one to
choose arbitrary parameter spacetime from the total extended configuration
space. In other words, we can take more wider class of coordinate
transformations such that mix spacetime and field variables which are forbidden
to be exchanged in conventional formalism. In this aspect, we call our
formalism fibration-free. This aspect also simplifies the discussion on
symmetries such as Noether's theorem.
