Reduced Hamiltonian descriptions

Suppose that the evolution of some large, isolated system is Hamiltonian, but that one is interested only in the evolution of a smaller piece of the system, i.e., a subsystem. Alternatively, suppose that one is interested in the evolution of two halves of a single Hamiltonian system, but that one is not interested in the details of their mutual interaction. It is well known that, in either case, if the degrees of freedom of the two pieces are not completely decoupled, an exact reduced description of the individual components cannot be Hamiltonian. It is, however, shown here that, if one only allows for the average effects of each piece on the other, neglecting detailed correlations between the two components in a generalized self-consistent field approximation, one is always led to an approximate reduced description which is Hamiltonian.