  We performed path integral simulations of spin evolution controlled by the
Rashba spin-orbit interaction in the semiclassical regime for chaotic and
regular quantum dots. The spin polarization dynamics have been found to be
strikingly different from the D'yakonov-Perel' (DP) spin relaxation in bulk
systems. Also an important distinction have been found between long time spin
evolutions in classically chaotic and regular systems. In the former case the
spin polarization relaxes to zero within relaxation time much larger than the
DP relaxation, while in the latter case it evolves to a time independent
residual value. The quantum mechanical analysis of the spin evolution based on
the exact solution of the Schroedinger equation with Rashba SOI has confirmed
the results of the classical simulations for the circular dot, which is
expected to be valid in general regular systems. In contrast, the spin
relaxation down to zero in chaotic dots contradicts to what have to be expected
from quantum mechanics. This signals on importance at long time of the
mesoscopic echo effect missed in the semiclassical simulations.
