The family P12 of the three-body problem -: The simplest family of periodic orbits, with twelve symmetries per period

A beautiful plane eight-shaped orbit has been found by Alain Chenciner, Richard Montgomery and Carles Simo through the minimisation of the action between suitable limit conditions. The three masses are equal and chase each other along the eight shape. This procedure can be generalized and leads to a family of three-dimensional periodic orbits with three equal masses and with 12 space-time symmetries per period. The property of a unique orbit for the three masses is conserved in a suitable uniformly rotating set of axes. The eight-shaped orbit represents the end of the family, its beginning being the classical Lagrangian solution with three equal masses and with a uniformly rotating equilateral triangle.