The infinite spin problem concerns the rotational behavior of total collision
orbits in the $n$-body problem. It has long been known that when a solution
tends to total collision then its normalized configuration curve must converge
to the set of normalized central configurations. In the planar n-body problem
every normalized configuration determines a circle of rotationally equivalent
normalized configurations and, in particular, there are circles of normalized
central configurations. It's conceivable that by means of an infinite spin, a
total collision solution could converge to such a circle instead of to a
particular point on it. Here we prove that this is not possible, at least if
the limiting circle of central configurations is isolated from other circles of
central configurations. (It is believed that all central configurations are
isolated, but this is not known in general.) Our proof relies on combining the
center manifold theorem with the Lojasiewicz gradient inequality.
