  The increasing role of general relativity in the dynamics of stellar systems
with central massive black holes and in the evolution of hierarchical triple
systems inspires a close examination of how post-Newtonian effects are
incorporated into N-body dynamics. The majority of approaches incorporate
relativity by adding to the Newtonian N-body equations the standard two-body
post-Newtonian terms for a given star around the black hole or for the close
binary in a triple system. We argue that, for calculating the evolution of such
systems over timescales comparable to the relativistic pericenter advance
timescale, it is essential to include "cross terms" in the equations of motion.
These are post-Newtonian terms that represent a coupling between the potential
of the central black hole and the potential due to other stars in the system.
For hierarchical triple systems, these are couplings between the potential of
the inner binary and that of the distant third body. Over pericenter precession
timescales, the effects of such terms can actually be "boosted" to amplitudes
of Newtonian order. We write down the post-Newtonian N-body equations of motion
including a central black hole in a truncated form that includes all the
relevant cross terms, in a format ready to use for numerical implementation. We
do the same for hierarchical triple systems, and illustrate explicitly the
effects of cross terms on the orbit-averaged equations of evolution for the
orbit elements of the inner binary for the special case where the third body is
on a circular orbit. We also describe the inspiration for this investigation:
the motion of a test body about a central body with a Newtonian quadrupole
moment, including the relativistic pericenter advance, whose correct solution
for the conserved total Newtonian energy requires including PN cross terms
between the mass monopole and quadrupole potentials.
