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 binarys orbit within it: a dimensional parameter $A$ setting the secular timescale, and a dimensionless parameter $Gamma$ which determines the phase portrait of the binarys 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.
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