I'd like to know when an orbit is closed. I know that, to have a closed orbit, there is a ratio that must be a rational number, but I don't know other things..
closed as not a real question by Manishearth Mar 2 '13 at 17:50
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There's a branch of study called dynamical systems theory that deals with questions like this. The answer is not straightforward, except in a handful of cases. Trajectories can appear to be highly chaotic and yet, when observed for a sufficiently long time, will turn out to be perfectly periodic. The notion of Poincairé recurrence time arises, which suggests that over a long enough time, any orbit must be considered periodic. This, in turn, leads to a debate over whether there is such a thing as an irrational number in a physical system with lower limits to measurable lengths and times.
The straightforward answer to this very broad question is that there isn't a straightforward answer.
To add to KDN's nice answer, there is a theorem called Bertrand's theorem which states that in the case of a particle moving in a central potential, the only potentials that produce stable, closed orbits are the inverse square and radial harmonic oscillator potentials.
Your reference to rational ratios suggests that you might be thinking of the simple dynamical system on the torus---a favourite toy example of mathematicians. See, for example, the wikipedia article "Linear flow on the torus".