  We present a comprehensive study of the linear response of interacting
underdamped Brownian particles to simple shear flow. We collect six different
routes for computing the response, two of which are based on the symmetry of
the considered system and observable with respect to the shear axes. We include
the extension of the Green-Kubo relation to underdamped cases, which shows two
unexpected additional terms. These six computational methods are applied to
investigate the relaxation of the response towards the steady state for
different observables, where interesting effects due to interactions and a
finite particle mass are observed. Moreover, we compare the different response
relations in terms of their statistical efficiency, identifying their relative
demand on experimental measurement time or computational resources in computer
simulations. Finally, several measures of breakdown of linear response theory
for larger shear rates are discussed.
