  We present a method that is optimized to explicitly obtain all the
constraints and thereby count the propagating degrees of freedom in (almost
all) manifestly first order classical field theories. Our proposal uses as its
only inputs a Lagrangian density and the identification of the a priori
independent field variables it depends on. This coordinate-dependent, purely
Lagrangian approach is complementary to and in perfect agreement with the
related vast literature. Besides, generally overlooked technical challenges and
problems derived from an incomplete analysis are addressed in detail. The
theoretical framework is minutely illustrated in the Maxwell, Proca and
Palatini theories for all finite $d\geq 2$ spacetime dimensions. Our novel
analysis of Palatini gravity constitutes a noteworthy set of results on its
own. In particular, its computational simplicity is visible, as compared to
previous Hamiltonian studies. We argue for the potential value of both the
method and the given examples in the context of generalized Proca and their
coupling to gravity. The possibilities of the method are not exhausted by this
concrete proposal.
