Not homework, but something I'm interested in finding out. The setup is a flexible wire with left endpoint fixed at [itex]x=0[/itex] and right endpoint at [itex]x=L[/itex]. You push the right endpoint with some horizontal force directed towards the left endpoint which will move the right endpoint to a position [itex]x=L-a[/itex] and cause the wire to deform and bend into some sort of downwards-facing parabola thing. What I want to know is what function actually describes the shape of the wire.
Material from classical mechanics, hopefully.
The Attempt at a Solution
I believe that the solution might have something to do with the principle of stationary action. My original idea was to simply find the function of minimum arc length that joins two points, with the constraint that the arc length has to be greater than just the straight line distance between the two points using what little I know about calculus of variations. I couldn't figure out how to actually impose that constraint though, I just kept getting [itex]f(x)=mx+b[/itex]. Then I tried to use an approach similar to the derivation of the wave equation where you consider a small chunk of the wire, resolve the tensions, and then use some shrewd approximations to get an equation out of it. This didn't work though, because all I ended up doing was finding the wave equation again. I think there might be other forces besides the tension on a small chunk since the wire will want to return to a straight line so I could maybe throw in a spring-like force but I wouldn't really know where to put it.
Thanks for any suggestions.