  To circumvent some technical difficulties faced by the geometric Lagrangian
approach to the physical degree of freedom count presented in the work of Diaz,
Higuita, and Montesinos, J. Math. Phys. 55, 122901 (2014) that prevent its
direct implementation to field theory, in this paper, we slightly modify the
geometric Lagrangian approach in such a way that its resulting version works
perfectly for field theory (and for particle systems, of course). As in
previous work, the current approach also allows us to directly get the
Lagrangian constraints, a new Lagrangian formula for the counting of the number
of physical degrees of freedom, the gauge transformations, and the number of
first- and second-class constraints for any action principle based on a
Lagrangian depending on the fields and their first derivatives without
performing any Dirac's canonical analysis. An advantage of this approach over
the previous work is that it also allows us to handle the reducibility of the
constraints and to get the off-shell gauge transformations. The theoretical
framework is illustrated in 3-dimensional generalized general relativity
(Palatini and Witten's exotic actions), Chern-Simons theory, $4$-dimensional BF
theory, and $4$-dimensional general relativity given by Palatini's action with
cosmological constant.
