classical mechanics - 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 $p=2$.