  In this letter we study the dynamics of a first order phase transition from
nucleonic to quark matter in neutron stars. Using standard equations of state
for these two phases we find the density range where such a transition is
possible. Then we study the transformation of the star assuming that the quark
core is formed via a spherical shock wave. The thermodynamical conditions in
the quark core are found from the conservation laws across the transition
region. Their dependence on the density and velocity of the incoming nuclear
matter are studied. It is found that the shock is especially violent in the
beginning of the conversion process when the velocity of the infalling matter
is especially high. As the shock propagates further from the center the front
velocity first increases and reaches a maximum value when the incoming velocity
is around $0.2$. Finally, the front velocity quickly goes to zero when incoming
matter velocity approaches zero. We have shown that the density and pressure
jumps are especially large in the begining of the transition process.
