  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 M\odot - 2
\times 10^11 M\odot with a mean value of ~ 2 \times 10^10 M\odot. 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.
