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This question has been bugging me for some time. First a little background. We know that a space elevator is essentially a (very) long rope with a heavy asteroid attached at the other end well beyond geostationary orbit. This orbit has some angular momentum associated with it, notably $L=M_{ast}\omega h$ where $h$ is the height above the center of the earth and $\omega$ is the angular velocity of the earth.

But wait! If we want to send something to geostationary orbit via the space elevator, we increase the angular momentum of the system by $L_{new}=M_{ast}\omega h + M_{obj}\omega h_{GSO}$. So clearly the angular momentum must be supplied by something.

But if it was supplied by the asteroid, its angular momentum must decrease and this would lead to instability in the system as $\omega$ would have to decrease. It can't be supplied only by earth (unless we use a very long rigid pole).

So my question is thus: What methods have been proposed to preserve the angular momentum of a space elevator?

Final addendum: I found on Wikipedia that the GSO orbit velocity is 3.03km/s. Even with LH2/LOX fuel, this works out to a mass fraction of $\approx 2$. So if we wanted to send a 20 ton payload to GSO, we would need a minimum of 20 tons of fuel, and that neglects any equipment to store and use this fuel or to lift 40 tons of mass off the surface of the earth. I also sense that packing 20 tons of explosives next to our (very expensive!) space elevator is not the most appealing of solutions.

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    $\begingroup$ If the elevator is anchored on earth's surface, the planet earth is the momentum bank. An ascending or descending payload would exert a westward or eastward Coriolis force. This shove induces a wave. When this wave reaches the earth, it will move the huge planetary mass only a minute distance. $\endgroup$ – HopDavid Jul 28 '17 at 22:11
  • $\begingroup$ @HopDavid: except ascent/descent would be much slower than the wave - it would be more of a continuous force. But yeah, that's the gist and answer you should post: momentum of Earth pulling the elevator and forcing it to keep up. One thing to remember; the elevator is not kept in 0g equilibrium at GEO - it's actually pulling outwards, the trailing end exerting more outwards force than the inner side creates through weight, the difference being passed through anchor into Earth and pulling the elevator taut. $\endgroup$ – SF. Jul 29 '17 at 14:05
  • $\begingroup$ @SF. "except ascent/descent would be much slower than the wave" I in no way said otherwise. $\endgroup$ – HopDavid Jul 29 '17 at 15:46
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An excellent question, rarely dealt with in descriptions of space elevators.

In theory, the extra angular momentum can be supplied in part by a reduction in earth´s momentum. There is, however, no reason why the station at the end of the cable should not also experience a loss in momentum. This would be especially true at the end of the ascent.

The energy needed to provide the angular momentum could be taken from down-wards moving mass (which, conversely, needs to slow down), but since this mass would not be a continuous movement (like liquid in a pipe) but discrete carriages, we would probably get some interesting wave phenomena. Handling this oscillation seems like a huge challenge.

A more practical approach would be using horizontal rockets on the lift carriage, firing continuously, but this would sabotage much of the economics of this technology, as described in the original post. You would need the same rockets firing for the descent, by the way... (In contrast with returning spacecraft, we cannot use friction to reduce angular velocity.)

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