On the Hamilton approach for the metric GR

Basic principles of the Hamilton approach developed for the metric General Relativity (Einstein`s GR) are discussed. In particular, we derive the Hamiltonian of the metric GR in the explicit form. This Hamiltonian is a quadratic function of the momenta $\pi^{mn}$ conjugate to the spatial components $g_{mn}$ of the metric tensor $g_{\alpha\beta}$. The Hamilton approach is used to analyze some problems of metric GR, including the internal structure of propagating gravitational waves and quantization of the metric GR. We also derive the Schr\"{o}dinger equation for the free Gravitational field and show that actual gravitational field cannot propagate as pure harmonic oscillations, or harmonic gravitational waves. A number of inequalities useful in applications to the metric GR are derived.