  Spin-orbit coupling (SOC), which is inherent to a Dirac particle that moves
under the influence of electromagnetic fields, manifests itself in a variety of
physical systems including non-relativistic ones. For instance, it plays an
essential role in spintronics developed in the past few decades, particularly
by controlling spin current generation and relaxation. In the present work, by
using an extended Caldeira-Leggett model, we elucidate how the interplay
between spin relaxation and momentum dissipation of an open system of a single
spin-$1/2$ particle with a Rashba type SOC is induced by the interactions with
a spinless, three-dimensional environment. Staring from the path integral
formulation for the reduced density matrix of the system, we have derived a set
of coupled nonlinear equations that consists of a quasi-classical Langevin
equation for the momentum with a frictional term and a spin precession
equation. The spin precesses around the effective magnetic field generated by
both the SOC and the frictional term. It is found from analytical and numerical
solutions to these equations that a spin torque effect included in the
effective magnetic field causes a spin relaxation and that the spin and
momentum orientations after a long time evolution are largely controlled by the
Rashba coupling strength. Such a spin relaxation mechanism is qualitatively
different from, e.g., the one encountered in semiconductors where essentially
no momentum dissipation occurs due to the Pauli blocking.
