  It is shown that the Lagrangian reduction, in which solutions of equations of
motion that do not involve time derivatives are used to eliminate variables,
leads to results quite different from the standard Dirac treatment of the first
order form of the Einstein-Hilbert action when the equations of motion
correspond to the first class constraints. A form of the first order
formulation of the Einstein-Hilbert action which is more suitable for the Dirac
approach to constrained systems is presented. The Dirac and reduced approaches
are compared and contrasted. This general discussion is illustrated by a simple
model in which all constraints and the gauge transformations which correspond
to first class constraints are completely worked out using both methods in
order to demonstrate explicitly their differences. These results show an
inconsistency in the previous treatment of the first order Einstein-Hilbert
action which is likely responsible for problems with its canonical
quantization.
