  We analyze the dynamical evolution of binary stars that interact with a
static background of single stars in the environment of a massive black hole
(MBH). All stars are considered to be single mass, Newtonian point particles.
We follow the evolution of the energy E and angular momentum J of the center of
mass of the binaries with respect to the MBH, as well as their internal
semi-major axis a, using a Monte Carlo method. For a system like the Galactic
center, the main conclusions are the following: (1) The binary fraction can be
of the order of a few percent outside 0.1 pc, but decreases quickly closer to
the MBH. (2) Within ~0.1 pc, binaries can only exist on eccentric orbits with
apocenters much further away from the MBH. (3) Far away from the MBH, loss-cone
effects are the dominant mechanism that disrupts binaries with internal
velocities close to the velocity dispersion. Closer to the MBH, three-body
encounters are more effective in disrupting binaries. (4) The rate at which
hard binaries become tighter is usually less than the rate at which a binary
diffuses to orbits that are more bound to the MBH. (5) Binaries are typically
disrupted before they experience an exchange interaction; as a result, the
number of exchanges is less than one would estimate from a simple "nv\sigma
estimate''. We give applications of our results to the formation of X-ray
binaries near MBHs and to the production rates of hyper-velocity stars by
intermediate mass MBHs.
