Slight Detour -- Part #1
So, I know I still haven't finished the second half of that earlier physics post, where we'll derive the Schrodinger equation from classical mechanics, but I thought I would take a little detour first. There was a problem that's been bugging me since the middle of last semester. I asked Asma about it the other night, and she gave me some very nice
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As you probably saw in the page you linked,
means
and both sides of that can be integrated.
(I imposed the restriction after I found the form of the solution. We are looking for a value of the constant C so that we have cosh(Cx)/C = 1 at the ends of the bubble, so solve arccosh(C)/C = x for C; as far as I know there's no nice solution, but it can be done ok numerically.)
I wonder if soap bubbles really behave like this? We have a quantitative prediction: they should break when the half-distance is at the maximum of arccosh(C)/C, which is about 0.6627 times hoop radius. The radius of the middle part, just before breaking, should be about 0.5524 times hoop radius. It seems as though there is no possible (minimum energy) bubble that connects the hoops if they are farther than that!
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