  Using molecular dynamics simulations, we study dynamics of a model polymer
melt composed of short chains with bead number N=10 in supercooled states. In
quiescent conditions, the stress relaxation function $G(t)$ is calculated,
which exhibits a stretched exponential relaxation on the time scale of the
$\alpha$ relaxation time $\tau_\alpha$ and ultimately follows the Rouse
dynamics characterized by the time $\tau_{\rm R} \sim N^2 \tau_\alpha$. After
application of shear $\gdot$, transient stress growth $\sigma_{xy}(t)/\gdot$
first obeys the linear growth $\int_0^t dt'G(t')$ for strain less than 0.1 but
saturates into a non-Newtonian viscosity for larger strain. In steady states,
shear-thinning and elongation of chains into ellipsoidal shapes take place for
shear $\gdot$ larger than $\tau_{\rm R}^{-1}$. In such strong shear, we find
that the chains undergo random tumbling motion taking stretched and compact
shapes alternatively. We examine the validity of the stress-optical relation
between the anisotropic parts of the stress tensor and the dielectric tensor,
which are violated in transient states due to the presence of a large glassy
component of the stress. We furthermore introduce time-correlation functions in
shear to calculate the shear-dependent relaxation times, $\tau_\alpha
(T,\gdot)$ and $\tau_{\rm R} (T,\gdot)$, which decrease nonlinearly as
functions of $\gdot$ in the shear-thinning regime.
