  The dynamics of small perturbations around sphaleron and black hole solutions
in the Einstein-Yang-Mills theory for the gauge group $SU(2)$ is investigated.
The perturbations can be split into the two independent sectors in accordance
with their parity; each sector contains negative modes. The even-parity
negative modes are shown to correspond to the negative second variations of the
height of the potential energy barrier near the sphaleron. For the odd-parity
sector, the existence of precisely $n$ (the number of nodes of the sphaleron
solution gauge field function) negative modes is rigorously proven. The same
results hold for the Einstein-Yang-Mills black holes as well.
