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

[JuliaL]

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.

I remember, way back in the previous century, when I was taking fluid mechanics, the navier stokes equation was the thing only it couldn't be solved for real bodies and so was sort of a lovely theory with no real applications.
Then came computational fluid dynamics which was nothing but approximations but which could be implemented and now we've got things like spaceship one where really complex dynamics can be simulated and prototyping is minimized if not eliminated.

Sounds like your perturbation methodologies are something like that.

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.

I'm going to take that as a rhetorical (answer optional) question. Otherwise you're really asking the wrong person.
I get stuck on figuring out whether or not motion exists.
How about 'accurate enough?'