Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing as $1/d^p$ for distance separation $d$ and some power $p$. Two questions:

Presumably the 2-body problem still factors into two independent 1-body problems, results in planar motion, and can be solved. Have the orbits (the equivalents of elliptical and parabolic orbits for $p=2$) been worked out for other (perhaps specific) values of $p$?

In some sense, the 3-body problem for $p=2$ cannot be solved. Most systems are choatic; see this interesting collection of Eugene Butikov. Only a few periodic solutions are known; see the nice article by Bill Casselman on the discovery of "choreographies." Is the situation simpler for other values of $p$? Perhaps $p=1$?

References and pointers would be appreciated. Thanks!

**Edit**. Thanks to Agol, Ken, and José. I've now looked at Arnolʹd's *Huygens* and Needham
(but not yet Arnolʹd's *Classical Mechanics*). Indeed, as the commenters say, there is a remarkable
2-body result for $p=1$: the orbits for a linearly attractive force are ellipses, for a linearly repulsive force, hyperbolae. This depends on the Kasner-Arnolʹd theorem stating that for each power law, there is a *dual* power law that maps orbits of one to orbits of the other. Newton proved in *Principia* that elliptical orbits result if and only if the force is inverse-linear or inverse-square. The Kasner-Arnolʹd theorem explains why.

Huygens and Barrow, Newton and Hooke, but I see that my college library has it, so I can retrieve it later. $\endgroup$ – Joseph O'Rourke Aug 18 '10 at 17:09