  Principles of successful Hamiltonian approaches, which were developed to
describe free gravitational field(s) in the metric gravity, are formulated and
discussed. By using the standard $\Gamma-\Gamma$ Lagrangian ${\cal
L}_{\Gamma-\Gamma}$ of the metric GR we properly introduce all momenta of the
metric gravitational field and derive the both canonical $H_C$ and total $H_t$
Hamiltonians of the metric GR. We also developed an effective method which is
used to determine various Poisson brackets between analytical functions of the
basic dynamical variables, i.e., generalized coordinates $g_{\alpha\beta}$ and
momenta $\pi^{\mu\nu}$. In general, such variables can be chosen either from
the straight $\{ g_{\alpha\beta}, \pi^{\mu\nu} \}$, or dual $\{
g^{\alpha\beta}, \pi_{\mu\nu} \}$ sets of symplectic dynamical variables which
always arise (and complete each other) in any Hamiltonian formulation developed
for the coupled system of tensor fields. By applying canonical
transformation(s) of dynamical variables we reduce the canonical Hamiltonian
$H_C$ to its natural form. The natural form of canonical Hamiltonian provides
numerous advantages in actual applications to the metric GR, since the general
theory of dynamical systems with such Hamiltonians is well developed.
Furthermore, many analytical and numerically exact solutions have been found
and described in detail for dynamical systems with the Hamiltonians already
reduced to their natural forms. In particular, reduction of the canonical
Hamiltonian $H_C$ to its natural form allows one to derive the Jacobi equation
for the free gravitational field(s), which takes a particularly simple form.
