  We identified the z~2 Lyman break galaxies using deep HST ultraviolet
(F275W/F336W) imaging of Abell 1689. Because of the imaging depth and the large
magnification provided by the cluster, we detect galaxies 100x fainter (-19.5<
M_1500 <-13) than previous surveys at this redshift. We are able to calculate
the intrinsic sensitivity of the observations as a function of source plane
position, allowing determinations of effective volume as a function of
luminosity. We fit the faint-end slope of the luminosity function to be alpha =
-1.74 +/-0.08, consistent with the values obtained for 2.5 < z < 6. There is no
turnover in the luminosity function down to MUV = -13. The trend of
increasingly redder UV spectral slopes with luminosity at higher redshifts is
observed in our sample, but with redder slopes at all luminosities and average
reddening of < E(B - V) >= 0.15. We assume the stars in these galaxies are
metal poor (0.2Z_sun) compared to their brighter counterparts (Z_sun),
resulting in bluer assumed intrinsic UV slopes and larger derived dust
extinction. The total UV luminosity density at z ~ 2 is 4.31x10^26
erg/s/Hz/Mpc^3, more than 70% of which is emitted by galaxies in the luminosity
range of our sample. We determine the star formation rate density at z ~ 2
(assuming constant dust extinction correction of 4.2 over all luminosities and
a Kroupa IMF) of 0.148 M/yr/Mpc^3, significantly higher than previous
determinations because of the additional population of fainter galaxies and the
larger dust correction factors.[abridged]
