The ADM Hamiltonian at the postlinear approximation

The ADM Hamiltonian for a many-particle system is calculated up to the postlinear approximation, i.e., to the approximation that both the equations of motion for the particles and the equations of motion for the gravitational field in case of no-incoming radiation correctly result up to the postlinear approximation. The relation of this Hamiltonian to the ADM Hamiltonian obtained by a post-Newtonian approximation scheme which was applied up to the first radiation-reaction and radiation levels is discussed. From here the standard formulas for the mechanical angular momentum and energy losses as well as the radiated energy and angular momentum are deduced. Background logarithmic and logarithmic radiative terms are shown to be not present at our approximation if the condition of no-incoming radiation is fulfilled.