## Relative equilibria in the 3-dimensional curved n-body problem    [PDF]

Florin Diacu
We consider the 3-dimensional gravitational $n$-body problem, $n\ge 2$, in
spaces of constant Gaussian curvature $\kappa\ne 0$, i.e.\ on spheres ${\mathbb S}_\kappa^3$, for $\kappa>0$, and on hyperbolic manifolds ${\mathbb H}_\kappa^3$, for $\kappa<0$. Our goal is to define and study relative
equilibria, which are orbits whose mutual distances remain constant in time. We
also briefly discuss the issue of singularities in order to avoid impossible
configurations. We derive the equations of motion and define six classes of
relative equilibria, which follow naturally from the geometric properties of
${\mathbb S}_\kappa^3$ and ${\mathbb H}_\kappa^3$. Then we prove several
criteria, each expressing the conditions for the existence of a certain class
of relative equilibria, some of which have a simple rotation, whereas others
perform a double rotation, and we describe their qualitative behaviour. In
particular, we show that in ${\mathbb S}_\kappa^3$ the bodies move either on
circles or on Clifford tori, whereas in ${\mathbb H}_\kappa^3$ they move either
on circles or on hyperbolic cylinders. Then we construct concrete examples for
each class of relative equilibria previously described, thus proving that these
classes are not empty. We put into the evidence some surprising orbits, such as
those for which a group of bodies stays fixed on a great circle of a great
sphere of ${\mathbb S}_\kappa^3$, while the other bodies rotate uniformly on a
complementary great circle of another great sphere, as well as a large class of
quasiperiodic relative equilibria, the first such non-periodic orbits ever
found in a 3-dimensional $n$-body problem. Finally, we briefly discuss other
research directions and the future perspectives in the light of the results we
present here.
View original: http://arxiv.org/abs/1108.1229