ON PLANAR PERIODIC MOTIONS OF AN ORBITAL ELEVATOR

We consider the motion of a dumbbell in a central Newtonian field of gravity. The dumbbell consists of two massive points connected by a massless rod. The third massive point, a cabin, slides along the rod according to a prescribed rule. One can consider such a mechanical system as a simplified model for an orbital tethered elevator. We study the most interesting case when the elevator cabin moves periodically between the dumbbell end masses. For the sake of simplicity, only planar motions of the system are analyzed. Assuming the cabin mass small compared to the masses of the endpoints, we use the Poincare theory to determine the conditions of exis- tence for families of periodic motions that depend analytically on the appropriate small parameter. These mo- tions tend to corresponding stable radial relative equilibria as the small parameter tends to zero. Each one of the two existing relative equilibria generates exactly one family of such periodic motions if the parameter is small enough. Stability of these periodic motions is investigated in the linear approximation. The solutions are computed up to the terms of the first order with respect to the small parameter.