  The Hamiltonian used in classical analyses of von Zeipel-Lidov-Kozai or ZLK
oscillations in hierarchical triple systems is based on the quadrupole
potential from a distant body on a fixed orbit, averaged over the orbits of
both the inner and the outer bodies ("double-averaging"). This approximation
can be misleading, because the corresponding Hamiltonian conserves the
component of angular momentum of the inner binary normal to the orbit of the
outer binary, thereby restricting the volume of phase space that the system can
access. This defect is usually remedied by including the effects of the
octopole potential, or by allowing the outer orbit to respond to variations in
the inner orbit. However, in a wide variety of astrophysical systems nonlinear
perturbations are comparable to or greater than these effects. The long-term
effects of nonlinear perturbations are described by an additional Hamiltonian,
which we call Brown's Hamiltonian. At least three different forms of Brown's
Hamiltonian are found in the literature; we show that all three are related by
a gauge freedom, although one is much simpler than the others. We argue that
investigations of ZLK oscillations in triple systems should include Brown's
Hamiltonian.
