  The simplest non-collision solutions of the N-body problem are the "relative
equilibria", in which each body follows a circular orbit around the centre of
mass and the shape formed by the N bodies is constant. It is easy to see that
the moment of inertia of such a solution is constant. In 1970, D. Saari
conjectured that the converse is also true for the planar Newtonian N-body
problem: relative equilibria are the only constant-inertia solutions. A
computer-assisted proof for the 3-body case was recently given by R. Moeckel.
We present a different kind of answer: proofs that several generalisations of
Saari's conjecture are generically true. Our main tool is jet transversality,
including a new version suitable for the study of generic potential functions.
