[T. P. Shestakova]{}[Department of Theoretical and Computational Physics, Southern Federal University,\
Sorge St. 5, Rostov-on-Don 344090, Russia\
E-mail: [shestakova@sfedu.ru]{}]{}

Introduction
============

In [@Shest1] we present the outline of Hamiltonian dynamics in extended phase space that can be constructed to be completely equivalent to a Lagrangian formulation for a system with constraints. As a rule, Hamiltonian formulation for gravity is constructed following to the Dirac scheme [@Dirac1; @Dirac2] and is a starting point for most attempts to quantize gravity. However, there are some reasons to doubt that Dirac Hamiltonian formulation for gravitational theory [@Dirac3] can be thought as an equivalent one to the original General Relativity. The reasons are closely connected with the role that given to gauge gravitational degrees of freedom $g_{0\mu}$, $\mu=0, 1, 2, 3$, fixing a reference frame, in these two formulations.

In Einstein (Lagrangian) formulation of General Relativity $g_{0\mu}$ components of metric tensor are treated on an equal basis with the rest of components, $g_{ij}$, $i, j=1, 2, 3$, defining 3-space geometry. The theory is invariant under gauge transformations, an infinitesimal form of which is $$\label{g_transf}
\delta g_{\mu\nu}
 =\eta^{\lambda}\partial_{\lambda}g_{\mu\nu}
 +g_{\mu\lambda}\partial_{\nu}\eta^{\lambda}
 +g_{\nu\lambda}\partial_{\mu}\eta^{\lambda}$$ and which are true for all components of metric tensor ($\eta^{\mu}$ are infinitesimal parameters). In the 