We present a SINFONI integral field kinematical study of 33 galaxies at z~3 from the AMAZE and LSD projects which are aimed at studying metallicity and dynamics of high-redshift galaxies. The number of galaxies analyzed in this paper constitutes a significant improvement compared to existing data in the literature and this is the first time that a dynamical analysis is obtained for a relatively large sample of galaxies at z~3. 11 galaxies show ordered rotational motions (~30% of the sample), in these cases we estimate dynamical masses by modeling the gas kinematics with rotating disks and exponential mass distributions. We find dynamical masses in the range 2 times 10^9 Modot - 2 times 10^11 Modot with a mean value of ~ 2 times 10^10 Modot. By comparing observed gas velocity dispersion with that expected from models, we find that most rotating objects are dynamically hot, with intrinsic velocity dispersions of the order of ~90 km s-1. The median value of the ratio between the maximum disk rotational velocity and the intrinsic velocity dispersion for the rotating objects is 1.6, much lower than observed in local galaxies value (~10) and slightly lower than the z~2 value (2 - 4). Finally we use the maximum rotational velocity from our modeling to build a baryonic Tully-Fisher relation at z~3. Our measurements indicate that z~3 galaxies have lower stellar masses (by a factor of ten on average) compared to local galaxies with the same dynamical mass. However, the large observed scatter suggests that the Tully-Fisher relation is not yet in place at these early cosmic ages, possibly due to the young age of galaxies. A smaller dispersion of the Tuly-Fisher relation is obtained by taking into account the velocity dispersion with the use of the S_0.5 indicator, suggesting that turbulent motions might have an important dynamical role.
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