  We discuss the total collision singularities of the gravitational N-body
problem on shape space. Shape space is the relational configuration space of
the system obtained by quotienting ordinary configuration space with respect to
the similarity group of total translations, rotations, and scalings. For the
zero-energy gravitating N-body system, the dynamics on shape space can be
constructed explicitly and the points of total collision, which are the points
of central configuration and zero shape momenta, can be analyzed in detail. It
turns out that, even on shape space where scale is not part of the description,
the equations of motion diverge at (and only at) the points of total collision.
We construct and study the stratified total-collision manifold and show that,
at the points of total collision on shape space, the singularity is essential.
There is, thus, no way to evolve the solutions through these points. This
mirrors closely the big bang singularity of general relativity, where the
homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an
essential singularity at the big bang. A simple modification of the
general-relativistic model (the addition of a stiff matter field) changes the
system into one whose shape-dynamical description allows for a deterministic
evolution through the singularity. We suspect that, similarly, some
modification of the dynamics would be required in order to regularize the total
collision singularity of the N-body model.
