  We study a singular Hamiltonian system with an $\al$-homogeneous potential
that contains, as a particular case, the classical $N$--body problem. We
introduce a variational Morse--like index for a class of collision solutions
and, using the asymptotic estimates near collisions, we prove the
non-minimality of some special classes of colliding trajectories under suitable
spectral conditions provided $\al$ is sufficiently away from zero. We then
prove some minimality results for small values of the parameter $\al$.
