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 $h(x_0)=h(x_l)=0$ (nails in the wall) and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, 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, $$J(x,h, \dot{h})=\int_0^l gh(x) \rho \frac{ds}{l}=\frac{g \rho }{l}\int_0^l h(x) \sqrt{1+\dot{h}^2} dx$$


With the constraint $$\int_0^l \sqrt{1+\dot{h}^2} dx- l=0$$ If it helps, it's evident that $\dot{h}(\frac{l}{2})=0$.


Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in $$\int_0^l \frac{g \rho }{l}h(x) \sqrt{1+\dot{h}^2} +\lambda(\int_0^l \sqrt{1+\dot{h}^2} dx- l)dx$$


Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.


But, in truth, by this point I am clueless. $\lambda$ is worked through $\nabla J = \lambda \nabla(\int_0^l \sqrt{1+\dot{h}^2} dx- l)$. 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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