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](http://faculty.ifmo.ru/butikov/Projects/Collection.html) of Eugene Butikov.
Only a few periodic solutions are known;
see the nice [article](http://www.ams.org/samplings/feature-column/fcarc-orbits1) 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.



2- and 3-body problems when gravity is not inverse-square
The answers to question (1) for the 2 body problem are fine, and complete enough.


Regarding (2). The 3 body problem (and N-body) with p =3 is significantly simpler
than with $p \ne 3$. The added simplicity is due to the
occurenc of an additional integral which comes out of the Lagrange Jacobi identity
for the evolution of the total moment of inertia $I$. This identity asserts
that $d^2 I/ dt^2 = 4 H + (4 - 2(p-1)) U$ where $H = K - U$ is the total energy,
with $K$ the kinetic energy and $U$ the NEGATIVE of the total potential energy,
a function which is homogeneous of degree $p-1$. When $p =3$ we get
$d^2 I/ dt^2 = 4H = const.$!.
(The total moment of inertia
is a the squared norm relative to the ``mass inner product' and as such
measures the total size of the system. )


For details on this Lagrange-Jacobi identity and its 
use see the first sections of my paper
`Hyperbolic Pants fit a three-body problem', Ergodic Theory and Dynamical Systems, Volume 25, - June 2005, 921-947, which you can also find on my web site 
<http://count.ucsc.edu/~rmont/papers/list.html>
or on the arXivs. Also see the references there.


For a study of choreographies with various $p$- potentials see the paper by Fujiwara et al.
`Choreographic Three Bodies on the Lemniscate':J. Phys. A: Math. Gen. 36 (21 March 2003) 2791-2800, available on his web site ( or the ArXivs).
<http://www.clas.kitasato-u.ac.jp/~fujiwara/nBody/nbody.html>


The discoverer of the figure eight, Cris Moore, in his beautiful
2 page paper `Braids and Classical Gravity'
(which Casselman should have a ref. to) found numerically, and argues
convincingly that as one increases $p$ more and more ``braid types'' (and hence choreographies) appear. It is known that
all possible braid types (and so choreography types) occur as soon as $p =2$. 


The cases $p \ge 2$ are often called ``strong-force potentials'' and from the
variational perspective are much simpler than $p < 2$ for the reason that
with the strong force potentials all collision paths have infinite action.
This fact regarding action is surprising, since with the strong force condition in
force it seems that almost all bounded solutions end in collision. This "seems" is a
theorem for the 2-body problem, and for the negative energy three body problem
when $p=2$. 


