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.

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