"Landing" Launch Loop payloads without rockets
Imagine a long and relatively heavy orbiting tether, running vertically from 29900 km radius, through GEO (42164 km), to a counterweight above, perhaps at (50000 km). The tether has a thin, passive conductive rail on it, such that a magnet rail will be held off it by eddy current repulsion while travelling next to it, "held down" by coriolis acceleration.
The tether is made with not-excessively-tapered Kevlar. We will need many tons of it. More mass is better for this system. THIS is where we put the hotels, the radiationshielding, and the heavy buffet tables for obese tourists.
We launch a vehicle off the launch loop at 10148.7 m/s, and with very good radar and some trim thrusters, we "land" on the bottom of the rail at 29979.7 km altitude. At that point, we have the same angular frequency and circular orbital velocity, and a large vertical velocity component, 1746.8 m/s . As we slide upwards, we magnetically levitate/push against the east side of the rail with Coriolis acceleration (0.25 m/s2 at 1746.8 m/s, 0.03 m/s2 at 200 m/s, nicely proportional to eddy currents!) getting some angular acceleration from the rail while we slow down vertically from the small (0.28 m/s^2) and rapidly decreasing centripedal acceleration (gravity minus rotational centrifugal acceleration). We slide upwards, we slow down vertically, we speed up horizontally.
All those numbers result in us passing through GEO at about 200 m/s vertically, about 5.8 hours later. Hit a strip of passive braking magnets and stop, 2 kilometers at 1 gee.
The above assumes zero drag - given that there will be some eddy current drag, we launch from the loop and land on the tether rail a little faster. That speeds up transit time!
We can also launch a little faster (10154.1 m/s), hit the rail at 29988 km radius and 1804.8 m/s vertical velocity, and travel up the rail faster, cutting the vertical transit time significantly. 500 m/s gets us up the tether in 3.9 hours - we will need 13 kilometers to stop, though.
We can reverse the process, using a magnetic accelerator to launch a vehicle down a rail on the west side of the tether, falling off the end in a transfer orbit back to the upper atmosphere, restoring most of the momentum. We will probably need to make up some energy and momentum, certainly if we accumulate more upward vehicles than downward ones. But overall, the energy will be tiny compared to the the launch loop energies, and we can float a lot of solar cell around GEO.
If we deorbit our trash much faster than we receive payloads (aiming for the ocean, or the occasional email spammer) we can add more momentum by exploiting Coriolis force acceleration. We can also launch mass from the moon, or launch mass from the loop in slingshot orbits around it. However, the moon will not always be conveniently located for this, and vehicles may make many 20 day orbits before arriving at a suitable mean anomaly.
The rocket thrust needed will be pure velocity correction, centimeters per second if we've done our radar and orbital mechanics calculations correctly. If we miss, we just reenter normally - a delay, but not a disaster. Besides a little correction exhaust, and whatever we need to add to for momentum restoration, the system is mass conservative and mostly energy-recycling.
More rockets bite the dust ...