  We study laser cooling of $^{24}$Mg atoms in dipole optical trap with pumping
field resonant to narrow $(3s3s)\,^1S_0 \rightarrow \, (3s3p)\,^{3}P_1$
($\lambda = 457$ nm) optical transition. For description of laser cooling of
atoms in the optical trap with taking into account quantum recoil effects we
consider two quantum models. The first one is based on direct numerical
solution of quantum kinetic equation for atom density matrix and the second one
is simplified model based on decomposition of atom density matrix over
vibration states in the dipole trap. We search pumping field intensity and
detuning for minimum cooling energy and fast laser cooling.
