If someone asks if you can move a big boulder, and the answer is that it is too heavy, it really matters if it is in a location where it can roll down, like on

If someone asks if you can move a big boulder, and the answer is that it is too heavy, it really matters if it is in a location where it can roll down, like on the side of a mountain. In that case, heavy may not be such a problem. It so happens that for every elliptical orbit, there is a circular orbit with the same angular momentum but a lower energy. The amount of energy excess available from this orbital change is 0.5mv^2*(eccentricity)^2. The moon's orbit has an eccentricity of 0.055, so 0.055^2 or 0.3% of the moons orbital energy could be extracted when changing its orbit from elliptical to circular. That is about 1.1e26 J, more than 1000 times the energy from burning all fossil fuel reserves. The orbital energy could be gathered by sending a swarm of satellites connected by long cables into a circular orbit around the Earth, matching the period and phase of the Moon. The satellites would be spun around a common axis, like a Klemperer rosette, but using cables instead of gravity to keep them in place. The moon would then pass through the middle of this ring twice each month. Each time it did, the moon's gravity would pull the satellites closer to the center, and the cables would be reeled in. When the moon passed by, the cables would be reeled out again to their original position, allowing for generators to gather the energy. Gathered energy could be used to put more or heavier satellites into place using mass from the moon, making the process rate increase exponentially. It may not alter the moon's orbit 'significantly', but it is doable with our current state of technology.