  We present a formalism for constructing schematic diagrams to depict chaotic
three-body interactions in Newtonian gravity. This is done by decomposing each
interaction in to a series of discrete transformations in energy- and angular
momentum-space. Each time a transformation is applied, the system changes state
as the particles re-distribute their energy and angular momenta. These diagrams
have the virtue of containing all of the quantitative information needed to
fully characterize most bound or unbound interactions through time and space,
including the total duration of the interaction, the initial and final stable
states in addition to every intervening temporary meta-stable state. As shown
via an illustrative example for the bound case, prolonged excursions of one of
the particles, which by far dominates the computational cost of the
simulations, are reduced to a single discrete transformation in energy- and
angular momentum-space, thereby potentially mitigating any computational
expense. We further generalize our formalism to sequences of (unbound)
three-body interactions, as occur in dense stellar environments during binary
hardening. Finally, we provide a method for dynamically evolving entire
populations of binaries via three-body scattering interactions, using a purely
analytic formalism. In principle, the techniques presented here are adaptable
to other three-body problems that conserve energy and angular momentum.
