On the stability of the motion of a viscoelastic ring in a gravitational field

Abstract

The motion in a circular orbit of a thin, homogeneous, unstretched viscoelastic ring in a central Newtonian gravitational field is investigated. An approximate non-linear system of differential equations is written out which describe the quasistatic state of motion of the ring relative to the centre of mass. A stability condition for the rotation of the ring in the plane of the orbit at an angular velocity which decays in magnitude is obtained in the first approximation. The asymptotic stability of the relative equilibrium of the ring in the orbital coordinate system when the ring lies in the plane of the orbit is proved, together with the instability of this equilibrium when the plane of the ring is perpendicular to the velocity vector of the centre of mass.