In the present work, our main objective is to investigate the orbits of
spinning test particles around a Schwarzschild black hole under the influence
of a quintessence matter field (SQBH). We begin with the dynamics of the
spinning test particles around SQBH which is governed by the
Mathisson-Papapetrou-Dixon (MPD) equations under the pole-dipole approximation,
where the gravitational field and the higher multipoles of the particle are
neglected. Depending on the types of saddle points,the effective potential are
classified and the possibility of chaotic orbits is discussed. The inner most
stable circular orbits (ISCOs) of the spinning particle around SQBH are
addressed, as are the effects of the parameters $S$ (particles' spin) and
$\u03f5$ (equation of state parameter). Later, Periastron precession is
investigated up to the first-order spin correction for a spinning particle
moving in nearly circular orbits around SQBH. It is noted that the addition of
particle's spin revamps the results obtained for the non-spinning particles and
also articulates the some interesting observational properties of the SQBH.
Additionally, we discuss the ramifications of employing first-order spin
corrections for analysing ISCOs, as well as compare our results to the
Schwarzschild black hole to ensure that they are consistent in the limit when
equation of state parameter $\u03f5=-1/3$ and normalization factor $\alpha \to 0$ .