Pendulum motions of extended lunar space elevator

In the usual everyday life, it is well known that the inverted pendulum is unstable and is ready to fall to “all four sides,” to the left and to the right, forward and backward. The theoretical studies and the lunar experience of moon robots and astronauts also confirms this property. The question arises: Is this property preserved if the pendulum is “very, very long”? It turns out that the answer is negative; namely, if the pendulum length significantly exceeds the Moon radius, then the radial equilibria at which the pendulum is located along the straight line connecting the Earth and Moon centers are Lyapunov stable and the pendulum does not fall in any direction at all. Moreover, if the pendulum goes beyond the collinear libration points, then it can be extended and manufactured from cables. This property was noted by F. A. Tsander and underlies the so-called lunar space elevator (e.g., see 1]). In the plane of the Earth and Moon orbits, there are some other equilibria which turn out to be unstable. The question is, Are there equilibria at which the pendulum is located outside the orbital plane? In this paper, we show that the answer is positive, but such equilibria are unstable in the secular sense. We also study necessary conditions for the stability of lunar pendulum oscillations in the plane of the lunar orbit. It was numerically discovered that stable and unstable equilibria alternate depending on the oscillation amplitude and the angular velocity of rotation. The study of the lunar elevator dynamics originates in 2]. The concept of lunar elevator was developed in detail in 3, 4]. Several classes of equilibria with the finiteness of the Moon size taken into account were studied in 5]. The possibility of location of an orbital station fixed to the Moon surface by a pair of tethers was investigated in 6]. The problem of orientation of the terminal station of the lunar space elevator was studied in 7]. The influence of the tether length variations on the motion of the lunar tether system was considered in 8]. The alternation of stable and unstable flat oscillations is well known in the problem of satellite oscillations in a circular orbit 9, 10].