> One last thing to mention is that even though these orbits can in principle be solved numerically, they can't in practice.
This is true in the general case, but there are numerous examples of N-body systems that are both of real-world interest and remarkably stable. The Solar System, for instance. The reason is ultimately anthropic, of course - if it weren't stable, we probably wouldn't be here observing it.
Many systems (including the Solar System) are stable over short time periods, but the stability of the Solar System is actually unknown over longer time periods. As an example, one of the numerical experiments Scott Tremaine has done has been to simulate the solar system, but to perturb the initial conditions of the planets by 1mm (that's one millimeter!). Over ~100 million years, in ~1% of systems Mercury actually crashes into Venus! [1]
This is true in the general case, but there are numerous examples of N-body systems that are both of real-world interest and remarkably stable. The Solar System, for instance. The reason is ultimately anthropic, of course - if it weren't stable, we probably wouldn't be here observing it.