The oscillations of a body with an orbital tethered system

The motion about a centre of mass of a rigid body with a tethered system, designed to launch a re-entry capsule from a circular orbit is considered. In the deployment of the tethered system the direction and value of the tensile strength of the tether vary and, if the point of application of the tensile strength does not coincide with the centre of mass of the body, a moment occurs which leads to oscillations of the body with variable amplitude and frequency. A non-linear equation of the perturbed motion of the body about the centre of mass under the action of the tensile force of the tether and the gravitational moment is derived. Assuming that the change in the value and direction of the tensile force is slow and also that the gravitational moment is small, approximate and exact solutions of the non-linear differential equation of the unperturbed motion are obtained in terms of elementary functions and elliptic Jacobi functions. For perturbed motion, the action integral is expressed in terms of complete elliptic integrals of the first and second kind.