  The secular approximation for the evolution of hierarchical triple
configurations has proven to be very useful in many astrophysical contexts,
from planetary to triple-star systems. In this approximation the orbits may
change shape and orientation, on time scales longer than the orbital time
scales, but the semi major axes are constant. For example, for highly inclined
triple systems, the Kozai-Lidov mechanism can produce large-amplitude
oscillations of the eccentricities and inclinations. Here we revisit the
secular dynamics of hierarchical triple systems. We derive the secular
evolution equations to octupole order in Hamiltonian perturbation theory. Our
derivation corrects an error in some previous treatments of the problem that
implicitly assumed a conservation of the z-component of the angular momentum of
the inner orbit (i.e., parallel to the total angular momentum of the system).
Already to quadrupole order, our results show new behaviors including the
possibility for a system to oscillate from prograde to retrograde orbits. At
the octupole order, for an eccentric outer orbit, the inner orbit can reach
extremely high eccentricities and undergo chaotic flips in its orientation. We
discuss applications to a variety of astrophysical systems, from stellar
triples to merging compact binaries and planetary systems. Our results agree
with those of previous studies done to quadrupole order only in the limit in
which one of the inner two bodies is a massless test particle and the outer
orbit is circular;our results agree with previous studies at octupole order for
the eccentricity evolution, but not for the inclination evolution.
