Secular dynamics of binaries in stellar clusters I: general formulation
  and dependence on cluster potential
  Orbital evolution of binary systems in dense stellar clusters is important in
a variety of contexts: origin of blue stragglers, progenitors of compact object
mergers, millisecond pulsars, and so on. Here we consider the general problem
of secular evolution of the orbital elements of a binary system driven by the
smooth tidal field of an axisymmetric stellar cluster (globular, nuclear, etc.)
in which the binary orbits. We derive a secular Hamiltonian (averaged over both
the inner Keplerian orbit of the binary and its outer orbit within the cluster)
valid to quadrupole order for an arbitrary cluster potential and explore its
characteristics. This doubly-averaged 'tidal' Hamiltonian depends on just two
parameters, which fully absorb the information about the background cluster
potential and the binary's orbit within it: a dimensional parameter $A$ setting
the secular timescale, and a dimensionless parameter $\Gamma$ which determines
the phase portrait of the binary's inner orbital evolution. We examine the
dependence of $A$ and $\Gamma$ on cluster potential (both spherical and
axisymmetric) and on the binary orbit within the cluster. Our theory reproduces
known secular results - such as Lidov-Kozai evolution and the effect of the
Galactic tide on Oort Cloud comets - in appropriate limits, but is more
general. It provides a universal framework for understanding dynamical
evolution of various types of binaries driven by the smooth tidal field of any
axisymmetric potential. In a companion paper (Hamilton & Rafikov 2019b) we
provide a detailed exploration of the resulting orbital dynamics.
