  We establish the general phonon dynamics of magnetic solids by incorporating
the Mead-Truhlar correction in the Born-Oppenheimer approximation. The
effective magnetic-field acting on the phonons naturally emerges, giving rise
to the phonon Hall effect. A general formula of the intrinsic phonon Hall
conductivity is obtained by using the corrected Kubo formula with the energy
magnetization contribution incorporated properly. The resulting phonon Hall
conductivity is fully determined by the phonon Berry curvature and the
dispersions. Based on the formula, the topological phonon system could be
rigorously defined. In the low temperature regime, we predict that the phonon
Hall conductivity is proportional to $T^{3}$ for the ordinary phonon systems,
while that for the topological phonon systems has the linear $T$ dependence
with the quantized temperature coefficient.
