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We present a formalism for constructing schematic diagrams to depict chaoticthree-body interactions in Newtonian gravity. This is done by decomposing eachinteraction in to a series of discrete transformations in energy- and angularmomentum-space. Each time a transformation is applied, the system changes stateas the particles re-distribute their energy and angular momenta. These diagramshave the virtue of containing all of the quantitative information needed tofully characterize most bound or unbound interactions through time and space,including the total duration of the interaction, the initial and final stablestates in addition to every intervening temporary meta-stable state. As shownvia an illustrative example for the bound case, prolonged excursions of one ofthe particles, which by far dominates the computational cost of thesimulations, are reduced to a single discrete transformation in energy- andangular momentum-space, thereby potentially mitigating any computationalexpense. We further generalize our formalism to sequences of (unbound)three-body interactions, as occur in dense stellar environments during binaryhardening. Finally, we provide a method for dynamically evolving entirepopulations of binaries via three-body scattering interactions, using a purelyanalytic formalism. In principle, the techniques presented here are adaptableto other three-body problems that conserve energy and angular momentum.

Illustrating chaos: A schematic discretization of the general three-body problem in Newtonian gravity