# Can stable non-circular orbits exist in 2 or 3 dimensional hyperbolic space

I know that stable non-circular orbits in euclidean space exist only in 3 spatial dimensions but what about if the spatial dimensions are hyperbolic instead? Are there any stable non-circular orbits in 2 or 3 dimensional hyperbolic space?

The gravitational field for any space of dimension > 1 scales inversely with the area of a sphere in that space. In hyperbolic 3-space, the area of a sphere is given by $A = 4\pi\sinh(r)^2$; the same as the formula in euclidean space, but substituting $\sinh(r)$ for the euclidean $r$.
Any curved space is locally equivalent to flat euclidean space on a small enough scale. The corresponds to the fact that $\sinh(r) \approx r$ when $r < 1$ (if $r$ is measured in units where the curvature of the space is equal to -1, setting the absolute scale for the space). Thus, for orbits where the apoapsis is much less than 1 in the absolute scale of the space, everything will work pretty much identically to orbits in normal euclidean space, with only minor perturbations, and thus, yes, there are stable orbits.
Above that scale, the $\sinh$ function fairly rapidly blows up, and the potential acts like that of the gravitational potential in a series of every higher-dimensional euclidean spaces, for which there are no non-circular bounded orbits. A minor perturbation from circularity will result in a satellite flying off to infinity, or being drawn to the center. That seems to suggest that there would be no stable, bounded non-circular orbits with sizes above the absolute scale of the space, except that perturbations resulting in infalling paths will eventually pass through the pseudo-Euclidean region before actually reach the center, where the centrifugal pseudo-force would start to dominate over a force which does not increase as quickly towards the center as it would in a higher-dimensional euclidean space, forcing the satellite outwards again. It is not, however, immediately obvious whether that results in bounded baths, or merely converting inwardly-unbounded paths into outwardly-unbounded ones, and I'm not certain how to prove that.
More rigorously, in order to show that there are bounded orbits, we need to show that the effective potential parameterized by angular momentum $L$ has a minimum. If my thinking on how centrifugal force works in hyperbolic space is correct, then the effective potential should look something like $U(r)_\text{eff} = \frac{L^2}{2m\sinh(r)^2} - k\coth(r)$. Playing around with this for a while in Wolfram Alpha, there are global minima to the function in $r$ for values of $L$ less than about $0.7$. Above that, surprisingly, you start to get series of multiple shallow local minima, which somewhat surprisingly seems to indicate that not only do bounded orbits exist at all scales, but that at high enough momenta, there are multiple bounded orbits at different energies!