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2 Update based on following two of the suggested references.

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:

1. 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 $p$?

2. 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.

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# 2- and 3-body problems when gravity is not inverse-square

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:

1. 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 $p$?

2. 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!