  A starting point for the present work was the statement recently discussed in
the literature that two Hamiltonian formulations for the theory of gravity, the
one proposed by Dirac and the other by Arnowitt - Deser - Misner, may not be
related by a canonical transformation. In its turn, it raises a question about
the equivalence of these two Hamiltonian formulations and their equivalence to
the original formulation of General Relativity. We argue that, since the
transformation from components of metric tensor to the ADM variables touches
gauge degrees of freedom, which are non-canonical from the point of view of
Dirac, the problem cannot be resolved in the limits of the Dirac approach. The
proposed solution requires the extension of phase space by treating gauge
degrees of freedom on an equal footing with other variables and introducing
missing velocities into the Lagrangian by means of gauge conditions in
differential form. We illustrate with a simple cosmological model the features
of Hamiltonian dynamics in extended phase space. Then, we give a clear proof
for the full gravitational theory that the ADM-like transformation is canonical
in extended phase space in a wide enough class of possible parametrizations.
