  Nonequilibrium mode-coupling theory (MCT) for uniformly sheared underdamped
systems is developed, starting from the microscopic thermostatted SLLOD
equation, and the corresponding Liouville equation. Special attention is paid
to the translational invariance in the sheared frame, which requires an
appropriate definition of the transient time-correlators. The derived MCT
equation satisfies the alignment of the wavevectors, and is manifestly
translationally invariant. Isothermal condition is implemented by the
introduction of the current fluctuation in the dissipative coupling to the
thermostat. This current fluctation grows in the $\alpha$-relaxation regime,
which generates a deviation of the yield stress in the glassy phase from the
overdamped case. Response to a perturbation of the shear rate demonstrates an
inertia effect which is not observed in the overdamped case. Our theory turns
out to be a non-trivial extension of the theory by Fuchs and Cates (J. Rheol.
53(4), 2009) to underdamped systems. Since our starting point is identical to
that by Chong and Kim (Phys. Rev. E 79, 2009), the contradictions between
Fuchs-Cates and Chong-Kim are resolved.
