Free gravitational field in the metric gravity as a Hamilton system

The closed system of Hamilton equations is derived for all tensor components of the free gravitational field $g_{\alpha \beta }$ and corresponding momenta ${\pi }^{\gamma \delta }$ in the metric General Relativity. The Hamilton-Jacobi equation for the free gravitational field $g_{\alpha \beta }$ is also derived and discussed. In general, all methods and procedures based on the Hamilton and Hamilton-Jacobi approaches are very effective in actual applications to many problems known in the metric GR.