Order and chaos in the rotation and revolution of a line segment and a point mass

We study the classical dynamics of two bodies, a massive line segment or slash (/) and a massive point or dot (.), interacting gravitationally. For this slash-dot (/.) body problem, we derive algebraic expressions for the force and torque on the slash, which greatly facilitate analysis. The diverse dynamics include a stable synchronous orbit, generic chaotic orbits, sequences of unstable periodic orbits, spin-stabilized orbits, and spin-orbit coupling that can unbind the slash and dot. The extension of the slash provides an extra degree of freedom that enables the interplay between rotation and revolution. In this way, the slash-dot body problem exhibits some of the richness of the three body problem with only two bodies and serves as a valuable prototype for more realistic systems. Applications include the dynamics of asteroid-moonlet pairs and asteroid rotation and escape rates.