First collisions at the LHC with unprecedented Energy! (Ask a particle physisicist)

[Pyrrho]

Higher number body systems can sometimes be treated in perturbation theory, i.e. a systematic approximation. Newton did not understand this, but Laplace and colleagues did, and this is how they finally were able to fully describe the solar system including the effects planets have on each other.

It is very easy to find scenarios where all analytic description fails and all we can do is try and run a computer simulation, which is also always an approximation.

The equations of motion of the standard model have not been solved exactly analytically either, and need to be treated e.g. with such a perturbation theory.
In fact, Feynman diagrams are exactly that - a graphic representation of a systematic approximation taking more and morr virtual particle effects into account.

But Julia, here's a follow up question to your idea, how do you measure the accuracy of a theory? How accurate is newtonian physics? I can contemplate situations where it fails with arbitrary severity - just go close enough to the speed of light.

Isn't Newtonian physics "good enough," as long as one stays away from very fast things and very tiny things?  Or in other words, isn't Newtonian physics adequate for anything a regular person is going to need?