Hypothetical: Could modern day engineers move the Lower Mississippi River?

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If you have the time, space and money you can build an elevator to the moon.

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No, you can't.

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A space elevator is a proposed type of space transportation system.[1] Its main component is a ribbon-like cable (also called a tether) anchored to the surface and extending into space. It is designed to permit vehicle transport along the cable from a planetary surface, such as the Earth's, directly into space or orbit, without the use of large rockets. An Earth-based space elevator would consist of a cable with one end attached to the surface near the equator and the other end in space beyond geostationary orbit (35,800 km altitude). The competing forces of gravity, which is stronger at the lower end, and the outward/upward centrifugal force, which is stronger at the upper end, would result in the cable being held up, under tension, and stationary over a single position on Earth. Once the tether is deployed, climbers would repeatedly climb the tether to space by mechanical means, releasing their cargo to orbit. Climbers would also descend the tether to return cargo to the surface from orbit.[2]

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The concept of a space elevator was first published in 1895 by Konstantin Tsiolkovsky.[3] His proposal was for a free-standing tower reaching from the surface of Earth to the height of geostationary orbit. Like all buildings, Tsiolkovsky's structure would be under compression, supporting its weight from below. Since 1959, most ideas for space elevators have focused on purely tensile structures, with the weight of the system held up from above. In the tensile concepts, a space tether reaches from a large mass (the counterweight) beyond geostationary orbit to the ground. This structure is held in tension between Earth and the counterweight like an upside-down plumb bob. On Earth, with its relatively strong gravity, the required specific strength for the cable material is very high.

Current technology is not capable of manufacturing cable materials that are strong and light enough for a space elevator on Earth. However, in 2000, the recently discovered carbon nanotubes were first identified as possibly being able to meet the specific strength requirements for an Earth space elevator.[2] This sparked a surge of interest and development in space elevators focusing on carbon nanotubes and the similar boron nitride nanotubes. In 2014, diamond nanothreads were first synthesized.[4] Since they have strength properties similar to carbon nanotubes, diamond nanothreads were quickly seen as a candidate material as well.[5] Nanotubes and diamond nanothreads both hold promise as materials to make an Earth-based space elevator possible.

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Would it help slow down coastal erosion?

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Natural Alluviation of the Mississippi by letting it flow naturally would actually help rebuild much of what we've lost.

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