Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum

Abstract

Geometric phase can explain the rotation of a dynamical system independent of angular momentum. The canonical example of such is a cat (a non-rigid body with an inbuilt control system), falling from an inverted position, being able to re-orient itself with negligible total angular momentum so as to land on its feet. The system of three bodies moving under mutual gravitation is similarly non-rigid, capable of changing size and shape under the dynamics of that force. Using coordinates that reduce by translations and rotations and simultaneously regularise all binary collisions, which separate shape dynamics from rotational dynamics, we show how certain discrete symmetries (including both reversing and non-reversing symmetries of the equations of motion) can force the geometric phase of motion periodic to vanish. This result is illustrated with periodic orbits discovered in a numerical survey, many of which are heretofore unknown, and the findings of this survey are discussed in detail, including stability, geometric phase, and classification of orbits.