On Dumbbell Motions in the Generalized Circular Sitnikov Problem

Abstract

The translational–rotational motions of a symmetrical dumbbell are considered in a circular restricted three-body problem where two primaries of equal mass move in circular orbits about common barycenter. A new type of motion is described for the dumbbell whereby its barycenter moves along the normal to the plane of rotation of the two primaries, whilst the dumbbell itself rotates continuously around the normal, forming a constant angle of π/2 with it (the invariant manifold “gravitational propeller”). It is shown that this manifold includes several two-dimensional invariant submanifolds. The dynamics of a dumbbell on these submanifolds is described. Relative equilibria were found on a “gravitational propeller” when the dumbbell mass center rests at the barycenter of the system, while the dumbbell is oriented parallel to the axis connecting the primaries or perpendicular to it. The stability of these equilibria is investigated. Small spatial nonlinear oscillations on a “gravitational propeller” manifold in the vicinity of a stable relative equilibrium were studied for the dumbbell of infinitesimal length. It is shown that these oscillations have the nature of a nonlinear parametric resonance, which sets a “slow” amplitude modulation of “fast” harmonic oscillations along the angle of rotation of the dumbbell.