Post-Newtonian effects in N-body dynamics: Relativistic precession and
  conserved quantities in hierarchical triple systems
  Conventional approaches to incorporating general relativistic effects into
the dynamics of N-body systems containing central black holes, or of
hierarchical triple systems with a relativistic inner binary, may not be
adequate when the goal is to study the evolution of the system over a timescale
related to relativistic secular effects, such as the precession of the
pericenter. For such problems, it may necessary to include post-Newtonian
"cross terms" in the equations of motion in order to capture relativistic
effects consistently over the long timescales. Cross terms are post-Newtonian
(PN) terms that explicitly couple the two-body relativistic perturbations with
the Newtonian perturbations due to other bodies in the system. In this paper,
we show that the total energy and the normal component of total angular
momentum of a hierarchical triple system is manifestly conserved to Newtonian
order over the relativistic pericenter precession timescale of the inner binary
if and only if PN cross-term effects in the equations of motion are taken
carefully into account.
