Using a scaling transformation we exactly determine the dynamics of an harmonically confined Tonks-Girardeau gas under arbitrary time variations of the trap frequency. We show how during a one-dimensional expansion a ``dynamical fermionization'' occurs as the momentum distribution rapidly approaches an ideal Fermi gas distribution, and that under a sudden change of the trap frequency the gas undergoes undamped breathing oscillations displaying alternating bosonic and fermionic character in momentum space. The absence of damping in the oscillations is a peculiarity of the truly Tonks regime.
We study the momentum distributions and spatial correlations of few harmonically confined SU(
N) fermions using quantum Monte Carlo methods. In our study, we vary the spin degeneracy
N from 2 to 6 and the total number of particles from 6 to 18. Only balanced mixtures, with the same number of atoms per spin type, and repulsive unlike-spin contact interactions are considered. Going from
N=2 to
N=6, with the same occupancy of each spin state, we observe an increase of atom-atom correlations. This effect is particularly significant in the momentum distributions, which show fatter tails at large
k ($k\ensuremath{\sigma}\phantom{\rule{0.16em}{0ex}}>\phantom{\rule{0.16em}{0ex}}5, \ensuremath{\sigma}$ being the oscillator length) when
N grows, in agreement with experimental findings. Those tails also show the expected ${k}^{\ensuremath{-}4}$ decay related to the Tan contact for different values of the spin degeneracy. According to our results, the local spin ordering and the spin-spin correlations are mainly determined by
N via the Pauli exclusion principle, with minor influences from the particle-particle interactions, irrespective of the total number of confined atoms.