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Goldstein's Classical Mechanics has a puzzling few sentences in his discussion of orbits.

Referring to the case of orbit where the energy is low enough for the orbit to be bounded, he says :"This does not necessarily mean that the orbits are closed. All that can be said is that they are bounded, contained between two circles of radii $r_1$ and $r_2$ with turning points always lying on the circles."

Doesn't "bounded" automatically mean "closed"? The object cannot escape from the attractive force and hence returns over and over. At least, that is my understanding of the terms. Wikipedia says "The orbit can be open (so the object never returns) or closed (returning), depending on the total energy (kinetic + potential energy) of the system." But it also says "Orbiting bodies in closed orbits repeat their paths after a constant period of time." So the only way out I see is if a closed orbit is a special case of non-precessing bounded orbit.

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Goldstein's "closed" means the orbiting body will eventually return to some point with the same velocity it had there previously; i.e. that the path will repeat itself exactly. Note that this can occur even in the case of precession: if the ratio between radial period and angular period is rational, the orbit will be closed. There is no precession only if this ratio is unity.

What Goldstein is saying is that in this point in the text, all he has shown is that the objects' separation $r$ will satisfy $r_1 < r < r_2$ for all time, and that this does not necessarily imply that the orbit will ever repeat itself. For example, we have not ruled out the possibility that $r$ could be periodic with period $\pi$, while $\theta$ could repeat every 3 units of time.

Other authors use "closed" or "elliptical" to mean Goldstein's "bounded," and "open" or "hyperbolic" to mean "unbounded." Part of the degeneracy in terms probably comes from the fact that for two ideal point masses in Newtonian gravity, bounded orbits will be closed in Goldstein's sense. See the application of Bertrand's Theorem to the inverse square law.

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I'm not familiar with the book, but a light object orbiting two or more heavy objects will generally follow a chaotic path that is not closed. I'm not sure whether the orbit is necessarily bounded in Goldstein's sense.

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As I see it,- bounded means that object trajectory at any point in time is within fixed volume, which does not shrink or grow, while closed means that object will repeat path regularly with period $T$. So technically there may be open, but bounded systems and closed, but unbounded ones.

For the former one nice example is Double pendulum trajectory, which is constrained in area, but never repeats. It's hard to think example of the latter case, but if we could take two elementary masses spinning around their common COM with $R \to \infty$, then such system would approach unbounded closed one.

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This is one of those things interesting enough to trap yourself with words. But think about it: bounded doesn't automatically mean closed. Bounded means that there could be closed paths. Take a potential like this

Mexican hat potential

orbits doesn't have to be closed to be bounded, right?

On the other hand you can have closed, but not bounded trajectories if you take a sin/cos-like potential for a specific range of energies.

I could be wrong, but that's how I picture it.

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  • $\begingroup$ Very good illustration IMO. $\endgroup$
    – JHM
    Sep 14, 2022 at 16:12
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I simply think that by bounded he mean an upper or lower limit for r, closed means orbits in which particle retraces its motion.

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