The masses in a symmetric centered solution of the 𝑛-body problem

Abstract

We prove that if a planar solution of the n n -body problem has a symmetry axis, fixed with respect to the considered frame, and the center of mass of the particle system lies on this axis during the motion, then the symmetric masses must be equal. We also show that the set of initial conditions leading to symmetric solutions has measure zero and is nowhere dense relative to the set of all initial conditions that define solutions in the space.