On the stability of the track of the space elevator

A string moving with geostationary angular velocity in its radial relative equilibrium configuration around the Earth, reaching from the surface of the Earth far beyond the geostationary height, could be used as track for an Earth to space elevator. This is an old dream of mankind, originating about 100 years ago in Russia. Besides the question of feasibility from a technological point of view also the question concerning the stability of such a configuration has not yet been completely solved. Under the assumption that a proper material (carbon nanotubes) is available, making the connection possible technologically, we address the question of existence and stability of the radial relative equilibrium of a tapered string on a circular geosynchronous trajectory around the Earth, reaching from the surface of the Earth far beyond the geostationary height.