Stephen Wolfram writes, at a mathematical level, the three-body problem is about solving a system of three ordinary differential equations that give the positions of the three bodies as a function of time. post 
But having discovered that particular solutions to the three-body problem could have a kind of instability, Poincaré decided to look not at individual solutions, but at the space of all possible solutions. For the three-body problem, this was very complicated—though in his efforts to analyze it he invented the field of topology.
It’s remarkable how similar the issues are across so many different fields. For example, the whole idea of using “perturbation theory” and series expansions that has existed since the 1700s in celestial mechanics is now also core to quantum field theory.
But if one looks, say, at the total energy of the three-body system—which one can prove from the equations should stay constant—then one will typically see an error slowly build up in it. One can avoid this if one effectively does a change of variables in the equations to “factor out” energy. And one can imagine doing a whole hierarchy of algebraic transformations that in a sense give the numerical scheme as much help as possible.
There are a few other effects that are also important in the precise observed timing of the eclipse. The most obvious is geo elevation. In doing astronomical computations, the Earth is assumed to be an ellipsoid.
Right at the beginning or end of totality one typically sees points of light on the rim of the Moon. Figuring out exactly when all these rays are extinguished requires taking geo elevation data for the Moon, and effectively doing full 3D ray tracing.
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I enjoyed Wolfram's history even though everything he writes has a hint of advertising. It was when I started poking nearby town names into his website calculator that I decided to seek totality myself. site
See Wolfram Coding where he plays with these calculations and shares his work online. workbook
See Eclipse Trip Report for my own quest.