We present a study of the polarization of the $\Upsilon(1S)$ and $\Upsilon(2S)$ states using a $1.3 $ fb$^{-1}$ data sample collected by the D0 experiment in 2002--2006 during Run II of the Fermilab Tevatron Collider. We measure the polarization parameter $\alpha=(\sigma_{T}-2\sigma_{L})/(\sigma_{T}+2\sigma_{L})$, where $\sigma_{T}$ and $\sigma_{L}$ are the transversely and longitudinally polarized components of the production cross section, as a function of the transverse momentum ($p_{T}^{\Upsilon}$) for the $\Upsilon(1S)$ and $\Upsilon(2S)$. Significant $p_{T}^{\Upsilon}$-dependent longitudinal polarization is observed for the $\Upsilon(1S)$. A comparison with theoretical models is presented.