  The exact formulation of the path integral centroid dynamics is extended to
include composites of the position and momentum operators. We present the
generalized centroid dynamics (GCD), which provides a basis to calculate
Kubo-transformed correlation functions by means of classical averages. We
define various types of approximate GCD, one of which is equivalent to the ring
polymer molecular dynamics (RPMD). The RPMD and another approximate GCD are
tested in one-dimensional harmonic system, and it is shown that the RPMD works
better in the short time region.
