On mutual distances for a choreography with distinct masses

We state some simple properties of a configuration of N bodies whose masses are not all equal, and whose motion is a 'choreography'. In such a solution of Newton's equations, the bodies chase each other around the same curve, with the same phase shift between consecutive bodies. It follows from those properties that for any dimension of space, the masses of a choreography are the same for a logarithmic potential. A similar argument shows that the vorticities of a choreography are the same for N vortices which satisfy Helmholtz's equations (Philos. Mag. 33 (1858) 485-512). We prove a more general result for any potential. In particular, for a choreography with distinct masses, the ratio between the smallest and the largest mutual distances is bounded by a constant which does not depend on the masses. (C) 2003 Academie des sciences. Publie par Elsevier SAS. Tous droits reserves.