Two different Hamiltonian formulations of the metric gravity are discussed
and applied to describe a free gravitational field in the $d$ dimensional
Riemann space-time. Theory of canonical transformations, which relate
equivalent Hamiltonian formulations of the metric gravity, is investigated in
details. In particular, we have formulated the conditions of canonicity for
transformation between the two sets of dynamical variables used in our
Hamiltonian formulations of the metric gravity. Such conditions include the
ordinary condition of canonicity known in classical Hamilton mechanics, i.e.,
the exact coincidence of the Poisson (or Laplace) brackets which are determined
for the both new and old dynamical Hamiltonian variables. However, in addition
to this any true canonical transformations defined in the metric gravity, which
is a constrained dynamical system, must also guarantee the exact conservation
of the total Hamiltonians $H_t$ (in the both formulations) and preservation of
the algebra of first-class constraints. We show that Dirac’s modifications of
the classical Hamilton method contain a number of crucial advantages, which
provide an obvious superiority of this method in order to develop various
non-contradictory Hamiltonian theories of many physical fields, when a number
of gauge conditions are also important. Theory of integral invariants and its
applications to the Hamiltonian metric gravity are also discussed. For
Hamiltonian dynamical systems with first-class constraints this theory leads to
a number of peculiarities some of which have been investigated.
