  In 1999 Chenciner and Montgomery found a remarkably simple choreographic
motion for the planar 3-body problem (see \cite{CM}). In this solution 3 equal
masses travel on a eight shaped planar curve; this orbit is obtained minimizing
the action integral on the set of simple planar choreographies with some
special symmetry constraints. In this work our aim is to study the problem of
$n$ masses moving in $\RR^d$ under an attractive force generated by a potential
of the kind $1/r^\alpha$, $\alpha >0$, with the only constraint to be a simple
choreography: if $q_1(t),...,q_n(t)$ are the $n$ orbits then we impose the
existence of $x \in H^1_{2 \pi}(\RR,\RR^d)$ such that $q_i(t)=x(t+(i-1) \tau),
i=1,...,n, t \in \RR, $ where $\tau = 2\pi / n$. In this setting, we first
prove that for every $d,n \in \NN$ and $\alpha>0$, the lagrangian action
attains its absolute minimum on the planar circle. Next we deal with the
problem in a rotating frame and we show a reacher phenomenology: indeed while
for some values of the angular velocity minimizers are still circles, for
others the minima of the action are not anymore rigid motions.
