  We study the classical ${SU(N)\otimes U(1)/SU(N-1)\otimes U(1)}$ Non Linear
Sigma model which is the continuous low energy effective field theory for $N$
component frustrated spin systems. The $\beta$ functions for the two coupling
constants of this model are calculated around two dimensions at two loop order
in a low temperature expansion. Our study is completed by a large $N$ analysis
of the model. The $\beta$ functions for the coupling constants and the mass gap
are calculated in all dimensions between 2 and 4 at order ${1/N}$. As a main
result we show that the standard procedure at the basis of the $1/N$ expansion
leads to results that partially contradict those of the weak coupling analysis.
We finally present the procedure that reconciles the weak coupling and large
$N$ analysis, giving a consistent picture of the expected scaling of frustrated
magnets.
