The height of a string in a gravitational field in 2-dimensions is bounded by h(x0)=h(xl)=0 (nails in the wall) and also ∫l0ds=l. (h(0)=h(l)=0, if you take h as a function of arc length) .
What shape does it take?
My try so far: minimise potential energy of the whole string, J(x,h,˙h)=∫l0gh(x)ρdsl=gρl∫l0h(x)√1+˙h2dx
With the constraint ∫l0√1+˙h2dx−l=0
Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in ∫l0gρlh(x)√1+˙h2+λ(∫l0√1+˙h2dx−l)dx
Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.
But, in truth, by this point I am clueless. λ is worked through ∇J=λ∇(∫l0√1+˙h2dx−l). I have tried this , but get nonsensical answers.
Is this method the best? If so, in what ways am I going about it wrongly thusfar?
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