Question: Suppose humans of the late 21st century or beyond began constructing a solar shield as a solution to climate change. Suppose for some reason (open to suggestions) they choose to build the shield in geosynchronous orbit, rather than at the L1 point. Suppose the shield is opaque, and large enough that it casts a daily eclipse on the Earth's surface. Is this plausible?

If so, how much thrust would be required for the sunshield to maintain its position? Would it be possible to supply that thrust using only solar energy and electric propulsion, via an electrodynamic tether system for example, such that the shield could remain in orbit without ongoing deliveries of chemical propellant?

By my calculations, if the shield was proportional to the moon in terms of eclipse potential, but at the distance of geosynchronous orbit, it would be approximately 340 km in diameter. Without knowing much about the potential progress of materials science in the next century, I assume, even if shield was very thin, the mass of it would make it's construction practical only to a civilization with functional space elevators. But, perhaps astroid mining could also provide the necessary materials without the launch costs.

Answer: 
Without going into the plausibility of the scenario, objects in orbit will tend to remain in orbit, unless acted upon by an outside force. Once you are clear of the Earth's atmosphere, you can effectively stay in orbit for geological ages (note, the atmosphere actually does go beyond the 100km mark, the ISS needs to be boosted every so often because air drag is an issue even at it's orbit).

A large, thin object such as you describe would essentially be a solar sail. It can be made out of many different materials, including metal (in layers measured in molecules), carbon fibre, nanotubes etc. IT can be extremely light for its size, and the pressure of sunlight would actually become a noticeable force on it.

During the part of the orbit where it blocks the sun, it will experience a net pressure on the side facing the sun towards the Earth, gradually pushing the sail into a lower orbit. On the opposite side of the orbit, it will be receiving solar pressure on the opposite side of the structure, effectively pushing it away from the Earth.

This will gradually move it out of position as the orbit is asymmetrically changed (the push away from the Earth will stop once the sail is in the Earth's shadow). I don't have the kills to calculate ow long it would take for the object to be pushed out of any useful orbit, but over a period of time, it's effects will become erratic and less predictable on the Earth's climate.

To at least partially counter that, the sail might be built with a series of smaller sails along the edge. These sails could be independently steered, and used to manipulate the orbit of the main sail, perhaps turning edge on to the sun on the lit side of the earth, and deploying to increase the sail area when the sail needs a push away from the Earth. This would serve to keep the sail in position much longer, although it would still require someone to come up with a rocket to make larger course correction when needed.
![enter image description here](https://i.stack.imgur.com/zy8s0.jpg)

Just put these highly reflective obsolete CD's in orbit spaced 6 inches apart, tied to fishing nets and spanned across some sort of frame. Each net could be spanned on its own frame and the distance between nets could be increased and decreased to control temperature