Tuesday, 11 February 2020

homework and exercises - Shape of a string/chain/cable/rope/wire?




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=gρll0h(x)1+˙h2dx


With the constraint l01+˙h2dxl=0

If it helps, it's evident that ˙h(l2)=0.


Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in l0gρlh(x)1+˙h2+λ(l01+˙h2dxl)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(l01+˙h2dxl). 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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