  We consider the limit $N\to +\infty$ of $N$-body type problems with weak
interaction, equal masses and $-\sigma$-homogeneous potential, $0<\sigma<1$. We
obtain the integro-differential equation that the motions must satisfy, with
limit choreographic solutions corresponding to travelling waves of this
equation. Such equation is the Euler-Lagrange equation of a corresponding
limiting action functional. Our main result is that the circle is the absolute
minimizer of the action functional among zero mean (travelling wave) loops of
class $H^1$.
