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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..

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closed as not a real question by Manishearth Mar 2 '13 at 17:50

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

what closed orbit are you taking about. Give us the detailed information about what do you want to know – Dimensionless Mar 2 '13 at 15:18
@Akash There isn't a general rule to apply for any case? For example.... we can consider a mass point on a cone: what is the condition to have closed orbit? thanks! – sunrise Mar 2 '13 at 15:48
@sunrise Isn't it just a case of having the eccentricity less than 1? If $e=0$, the orbit is a circle, if $0 < e < 1$, the orbit is an ellipse, if $e=1$, the orbit is a parabola, is $e>1$, the orbit is a hyperbola. Only for the circle and ellipse are the orbits closed. – user12345 Mar 2 '13 at 19:11

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.

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Your answer is very very interesting! :) – sunrise Mar 2 '13 at 16:50

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.

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Thank you! mmmmh.. any idea to theorems about ratios? – sunrise Mar 2 '13 at 16:52
@sunrise No, not offhand. – joshphysics Mar 2 '13 at 16:55
I believe that there are also circular orbits in a $r^{-5}$ potential, but they go through the center of force where things are singular. It does not contradict Bertrand's theorem because it is not a general result for that power law. See Goldstein (Ed. 2) problem 3.6. – dmckee Mar 2 '13 at 17:10

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".

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I'm thinking to a quiet simple system, but I don't know the torus.. other analogous cases that can be answer to my question? – sunrise Mar 2 '13 at 16:49

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