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

[Alex K]

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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.

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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?

Yes, although I'd like to take this opportunity to point out that it is not necessarily smallness which marks the transition into the quantum regime. If that were the case, we would expect a constant of nature which has units of length or time or energy which sets such a scale. But we don't (the Planck scale would satisy those criteria but it does not appear in quantum mechanics). The constant of nature which marks the transition to the quantum regime is Planck's constant which has units of Energy*time. In other words, quantum phenomena becone important whenever the energy transfer in some process times the time span it occupies are below this tiny constant. There is no fixed length below which "quantum" starts.

But I digress. Yes, Newton is good enough for low speeds if you don't measure too accurately. If you are doing GPS or are otherwise messing with atomic clocks, you may find yourself forced to use relativity at relatively low speeds. What is troubling is that the description of the world in classical mechanics, relativity and quantum mechanics differs so widely that it is not clear in which sense the *explanations* converge towards something. That is an issue for philosophers.