  We present an action principle formulation for the study of motion of an
extended body in General Relativity in the limit of weak gravitational field.
This gives the classical equations of motion for multipole moments of arbitrary
order coupling to the gravitational field. In particular, a new force due to
the octupole moment is obtained. The action also yields the gravitationally
induced phase shifts in quantum interference experiments due to the coupling of
all multipole moments.
