We construct a SCUBA-2 450-$mu$m map in the COSMOS field that covers an area of 300 arcmin$^{2}$ and reaches a 1$sigma$ noise level of 0.65 mJy in the deepest region. We extract 256 sources detected at 450 $mu$m with signal-to-noise ratio $>$ 4.0 and analyze the physical properties of their multi-wavelength counterparts. We find that most of the sources are at $zlesssim3$, with a median of $z = 1.79^{+0.03}_{-0.15}$. About $35^{+32}_{-25}$% of our sources are classified as starburst galaxies based on their total star-formation rates (SFRs) and stellar masses ($M_{ast}$). By fitting the far-infrared spectral energy distributions, we find that our 450-$mu$m-selected sample has a wide range of dust temperatures (20 K $ lesssim T_{rm d} lesssim$ 60 K), with a median of ${T}_{rm d} = 38.3^{+0.4}_{-0.9}$ K. We do not find a redshift evolution in dust temperature for sources with $L_{rm IR}$ > $10^{12}$ $rm L_odot$ at $z<3$. However, we find a moderate correlation where dust temperature increases with the deviation from the SFR-$M_{ast}$ relation. The increase in dust temperature also correlates with optical morphology, which is consistent with merger-triggered starbursts in sub-millimeter galaxies. Our galaxies do not show the tight IRX-$beta_{rm UV}$ correlation that has been observed in the local Universe. We construct the infrared luminosity functions of our 450-$mu$m sources and measure their comoving SFR densities. The contribution of the $L_{rm IR}$ > $10^{12}$ $rm L_odot$ population to the SFR density rises dramatically from $z$ = 0 to 2 ($propto$ ($1+z$)$^{3.9pm1.1}$) and dominates the total SFR density at $z gtrsim 2$.
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