Saari’s homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian n-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for n ≥ 3.
Diacu, Florin; Fujiwara, Toshiaki; Pérez-Chavela, Ernesto; and Santoprete, Manuele, "Saari’s Homographic Conjecture of the Three-Body Problem" (2008). Mathematics Faculty Publications. 14.