[SubMareener]

Not a problem, the beauty of the space elevator is if it breaks at any point below its center of mass, which is at geosynchronous orbit, it just flies off into space. Also, you only have to do work to get to geosynchronous altitude, after that you need brakes to slow you down as you move up the elevator. With conducting carbon nanotube cables, and dynamic electrical braking, the cars going beyond geosynchronous altitude could provide power for the cars coming up from the earth. It really is a neat idea.

[RadioAstronomer]

Unfortunately, the Earth has a few things that can cause instability in such a system. The two main ones are, it is not a perfect sphere and has gravity wells which cause in track motion, and the Moon does not orbit on the equatorial plane which will cause crosstrack motion.