Relative Equilibria of Dumbbells Orbiting in a Planar Newtonian Gravitational System

Abstract

In the cosmos, any two bodies share a gravitational attraction. When in proximity to one another in empty space, their motions can be modeled by Newtonian gravity. Newton found their orbits when the two bodies are infinitely small, the so-called two-body problem. The general situation in which the bodies have varying shapes and sizes, called the full two-body problem, remains open. We find relative equilibria (RE) and their stability for an approximation of the full two-body problem, where each body is restricted to a plane and consists of two point masses connected by a massless rod, a dumbbell. In particular, we find symmetric RE in which the bodies are arranged colinearly, perpendicularly, or trapezoidally. When the masses of the dumbbells are pairwise equal, we find asymmetric RE bifurcating from the symmetric RE. And while we find that only the colinear RE have nonlinear/energetic stability (for sufficiently large radii), we also find that the perpendicular and trapezoid configurations have radial intervals of linear stability. We also provide a geometric restriction on the location of RE for a dumbbell body and any number of planar rigid bodies in planar orbit (an extension of the Conley Perpendicular Bisector Theorem).