Action minimizing orbits in the $n$-body problem with simple choreography constraint

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 $q_1(t),\ldots,q_n(t)$ are the $n$ orbits then we impose the existence of $x \in H^1_{2 \pi}(\RR,\RR^d)$ such that \begin{equation*} q_i(t)=x(t+(i-1) \tau ),\,\,\,i=1,\ldots,n,\,\, t \in \RR, \end{equation*} 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 regular $n$-gon relative equilibrium. Next we deal with the problem in a rotating frame and we show a richer phenomenology: indeed while for some values of the angular velocity minimizers are still relative equilibrium, for others the minima of the action are not anymore rigid motions.