  The goal of the present account is to review our efforts to obtain and apply
a ``collective'' Hamiltonian for a few, approximately decoupled, adiabatic
degrees of freedom, starting from a Hamiltonian system with more or many more
degrees of freedom. The approach is based on an analysis of the classical limit
of quantum-mechanical problems. Initially, we study the classical problem
within the framework of Hamiltonian dynamics and derive a fully self-consistent
theory of large amplitude collective motion with small velocities. We derive a
measure for the quality of decoupling of the collective degree of freedom. We
show for several simple examples, where the classical limit is obvious, that
when decoupling is good, a quantization of the collective Hamiltonian leads to
accurate descriptions of the low energy properties of the systems studied. In
nuclear physics problems we construct the classical Hamiltonian by means of
time-dependent mean-field theory, and we transcribe our formalism to this case.
We report studies of a model for monopole vibrations, of $^{28}$Si with a
realistic interaction, several qualitative models of heavier nuclei, and
preliminary results for a more realistic approach to heavy nuclei. Other topics
included are a nuclear Born-Oppenheimer approximation for an {\em ab initio}
quantum theory and a theory of the transfer of energy between collective and
non-collective degrees of freedom when the decoupling is not exact. The
explicit account is based on the work of the authors, but a thorough survey of
other work is included.
