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.
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