ON STABILITY OF RELATIVE EQUILIBRIA FOR AN INVERTED PENDULUM ATTACHED TO THE EQUATOR

Relative equilibria of a pendulum attached to the equator of the rotating Earth are studied. The length of the inverted pendulum corresponding to change in the degree of instability is determined. Family of relative equilibria appearing with the change of the degree of instability is found, and the stability of these equilibria is investigated. Everyday experience tells us that the inverted pendulum of a small length is unstable. The degree of instability in this case is equal to two, which means the pendulum topples for any small deviation from local vertical in any direction. However, as is known from investigations in dynamics of the so called orbital elevator, 1 the radial relative equilibria are stable in the Lyapunov sense for the inverted pendulum if its free end is located behind the geosynchronous orbit. In this case the degree of instability for this equilibrium equals to zero. By virtue of continuity there exist values of the inverted pendulum length such that its degree of instability is equal to one. Determination of a range of such lengths is one of the goals of this paper. Another goal relates to determination of oblique relative equilibria and investigation of their stability. The relative equilibria and their stability are studied with the use of redundant Cartesian coordinates. They are described using with the Routh equation for the critical points of the Routh function. This description allows to easily determine the constraint reaction.