2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67281An action minimizing path between two given configurations, spatial or planar, of the $n$-body problem is always a true -- collision-free -- solution. Based on a remarkable idea of Christian Marchal, this theorem implies the existence of new "simple" symmetric periodic solutions, among which the Eight for 3 bodies, the Hip-Hop for 4 bodies and their generalizations.Dynamical Systems70F07, 70F10, 70F16, 34C14, 34C25text