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

Post-Newtonian effects in N-body dynamics: Relativistic precession and conserved quantities in hierarchical triple systems