  Recent observations constrained the tangential velocity of M31 with respect
to the Milky Way (MW) to be v_tan<34.4 km/s and the radial velocity to be in
the range v_rad=-109+/- 4.4 km/s (van der Marel et al. 2012). In this study we
use a large volume high resolution N-body cosmological simulation (Bolshoi)
together with three constrained simulations to statistically study this
kinematics in the context of the LCDM. The comparison of the ensembles of
simulated pairs with the observed LG at the 1-sigma level in the uncertainties
has been done with respect to the radial and tangential velocities, the reduced
orbital energy (e_tot), angular momentum (l_orb) and the dimensionless spin
parameter, lambda. Our main results are: (i) the preferred radial and
tangential velocities for pairs in LCDM are v_rad=-80+/-20 km/s, v_tan=50+/-10
km/s, (ii) pairs around that region are 3 to 13 times more common than pairs
within the observational values, (iii) 15%to 24% of LG-like pairs in LCDM have
energy and angular momentum consistent with observations while (iv) 9% to 13%
of pairs in the same sample show similar values in the inferred dimensionless
spin parameter. It follows that within current observational uncertainties the
quasi-conserved quantities that characterize the orbit of the LG, i.e. e_tot,
r_orb and lambda, do not challenge the standard LCDM model, but the model is in
tension with regard to the actual values of the radial and tangential
velocities. This might hint to a problem of the LCDM model to reproduce the
observed LG.
