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 three-body problem (see[11]). In this solution, three equal masses travel on an figure-of-eight shaped planar curve; this orbit is obtained by 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 R-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 is an element of H-2pi(1) (R, R-d) such that q(i)(t) = x(t + (i - 1)tau), i = 1, ..., n, t is an element of R, where tau = 2pi/n. In this setting, we first prove that for every d, n is an element of N 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 show a richer phenomenology: indeed, while for some values of the angular velocity, the minimizers are still relative equilibria, for others, the minima of the action are no longer rigid motions.