  About 50 years ago, in 1958, Dirac published his formulation of generalized
Hamiltonian dynamics for gravitation. Several years later Arnowitt, Deser and
Misner (ADM) proposed their description of the dynamics of General Relativity
which became a basis of the Wheeler - DeWitt Quantum Geometrodynamics. There
exist also other works where the Hamiltonian formulation of gravitational
theory was discussed. In spite of decades passed from the famous papers by
Dirac and ADM, there are unsolved problems. Namely, are the Dirac and ADM
formulations equivalent to each other? Are these formulations equivalent to the
original (Lagrangian) Einstein theory? Is the group of transformation in phase
space the same as the group of gauge transformation of the Einstein theory?
What are rules according to which a generator of transformations in phase space
should be constructed? Let us mention also another approach based on extended
phase space where gauge degrees of freedom are treated on the equal ground with
physical degrees of freedom. Our purpose is to review the above questions and
to demonstrate advantages of the extended phase space approach by the example
of a simple model with finite number degrees of freedom.
