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The shape of a hanging chain

The shape of a hanging elastic string

The brachistochrone (the shape of a wire such that a bead running along it gets from A to B in the shortest time possible in a gravitational potential)

The elastica (shape of a toothpick being gently bent)

Profile curve of a bubble film suspended between two identical rings

Profile curve of a meniscus (water in a glass beaker sort of thing)

The last one I created a thread on a while ago, and think I got to the bottom of it. Let me know if you get a solution.

Wondered if anyone knew any other good ones.

If not, have a crack at these, some of them are pretty tricky (and watch out - there's no general analytic solution to the elastica problem, so don't pull your hair out over it).

Hope I've posted this in the right place. They're clearly not homework questions.