# Condition for closed orbit [closed]

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..

• what closed orbit are you taking about. Give us the detailed information about what do you want to know – Deiknymi 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

## 3 Answers

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.

• 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.

• 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 --- ex-moderator kitten 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".

• 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