I'm focussing on the "repulsive gravity" case: the chain is pushed away from the center of the Earth by "gravity". Start with a small chain tied to the ground 6' apart. The chain will form a traditional (inverted) catenary. Then consider larger & larger chains, with the ends tied to the earth further & further apart. The chain is pushed into the air by the repulsive gravity until stopped by the ends tied to the ground. As we consider larger & larger chains, the end points get further & further apart until they are at opposite ends of the Earth. If we continue, the ends start getting closer & closer again, but on the opposite side of the Earth from the chain. We are now in an unstable equilibrium, because the entire chain can now be thrown off by simply moving away from the Earth in a perpendicular direction; let us ignore this an assume that something keeps this from happening. If we continue, eventually the entire Earth is encircled by the chain, and it touches the Earth at just one point. _Nothing is moving_, but the repulsive force of gravity keeps the chain taut in the air. At 08:26 AM 6/4/2011, Eugene Salamin wrote:
Is the chain, including its end points, supposed to be in orbit, or are you assuming the end points to be somehow held fixed? Inn the former case, I suspect the chain ends up as a tangle.
-- Gene
________________________________ From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Sent: Saturday, June 4, 2011 5:55 AM Subject: [math-fun] Really, really large catenaries
The usual catenary is formed by a hanging chain in a gravity field which is uniform -- i.e., the case for chains which are small relative to the size of the Earth.
Is there a closed form solution to "large" catenaries which have a size as large as, or larger than the Earth?
In particular, consider a chain whose links are _repulsed_ by a spherical (non-rotating) Earth (e.g., proportional to -1/r^2). What are the shapes of these curves?
A large enough chain would go completely around the Earth. I guess there would be solutions where such a closed chain would touch the Earth at just one point. I don't know if there would be stable solutions where the chain would not touch the Earth at all.
If rotation is introduced, things probably get a lot weirder.