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Calculus of Variations--Hanging String
A uniform string of length 2 meters hangs from two supports of the same height, 1 meter
apart. By minimizing the potential energy of the string, find the equation describing the
curve it forms and, in particular, find the vertical distance between the supports and the
lowest point of the string.
U = λ g [itex]\int y(x)dx[/itex]
0≤ x ≤ 1
S = [itex]\int\sqrt{dx^2+dy^2}[/itex]dt
Euler-Lagrange Equation
Been stuck on this one for awhile, tried solving naively by just solving the lagrange equation for potential energy and that obviously got me an answer of U = 0. I also tried to solve the arc length equation for y(x) and I got y'[x] = √3, so that is obviously wrong. I'm not sure how to add the constraints of the supports into the equation. I am probably going to have to parameterize y and x in terms of some other variable.
I have a feeling that the shape is going to be a cycloid...just having a hard time proving it.
Homework Statement
A uniform string of length 2 meters hangs from two supports of the same height, 1 meter
apart. By minimizing the potential energy of the string, find the equation describing the
curve it forms and, in particular, find the vertical distance between the supports and the
lowest point of the string.
Homework Equations
U = λ g [itex]\int y(x)dx[/itex]
0≤ x ≤ 1
S = [itex]\int\sqrt{dx^2+dy^2}[/itex]dt
Euler-Lagrange Equation
The Attempt at a Solution
Been stuck on this one for awhile, tried solving naively by just solving the lagrange equation for potential energy and that obviously got me an answer of U = 0. I also tried to solve the arc length equation for y(x) and I got y'[x] = √3, so that is obviously wrong. I'm not sure how to add the constraints of the supports into the equation. I am probably going to have to parameterize y and x in terms of some other variable.
I have a feeling that the shape is going to be a cycloid...just having a hard time proving it.