The family P-12 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, Ric hard Montgomery and Carles Simo through the minimisation of the action betw een suitable limit conditions. The three masses are equal and chase each ot her along the eight shape. This procedure can be generalized and leads to a family of three-dimensional periodic orbits with three equal masses and wi th 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 bei ng the classical Lagrangian solution with three equal masses and with a uni formly rotating equilateral triangle.