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Orbital evolution of binary systems in dense stellar clusters is important ina variety of contexts: origin of blue stragglers, progenitors of compact objectmergers, millisecond pulsars, and so on. Here we consider the general problemof secular evolution of the orbital elements of a binary system driven by thesmooth tidal field of an axisymmetric stellar cluster (globular, nuclear, etc.)in which the binary orbits. We derive a secular Hamiltonian (averaged over boththe inner Keplerian orbit of the binary and its outer orbit within the cluster)valid to quadrupole order for an arbitrary cluster potential and explore itscharacteristics. This doubly-averaged 'tidal' Hamiltonian depends on just twoparameters, which fully absorb the information about the background clusterpotential and the binary's orbit within it: a dimensional parameter $A$ settingthe secular timescale, and a dimensionless parameter $\Gamma$ which determinesthe phase portrait of the binary's inner orbital evolution. We examine thedependence of $A$ and $\Gamma$ on cluster potential (both spherical andaxisymmetric) and on the binary orbit within the cluster. Our theory reproducesknown secular results - such as Lidov-Kozai evolution and the effect of theGalactic tide on Oort Cloud comets - in appropriate limits, but is moregeneral. It provides a universal framework for understanding dynamicalevolution of various types of binaries driven by the smooth tidal field of anyaxisymmetric potential. In a companion paper (Hamilton & Rafikov 2019b) weprovide a detailed exploration of the resulting orbital dynamics.

Secular dynamics of binaries in stellar clusters I: general formulation and dependence on cluster potential