What are the generalized coordinates for the suspended rod Lagrangian?

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SUMMARY

The discussion focuses on deriving the Lagrangian for a suspended rod system using generalized coordinates. The rod, of length 2b, is described with coordinates x, y1, and y2, where x represents the longitudinal displacement and y1 and y2 denote the horizontal displacements of the rod's ends. The small angle approximation is applied, leading to the relationship θ = (x^2 + y^2)/l, with height above equilibrium approximated as l(θ^2). The participants clarify the need to use the coordinates directly rather than angles for accurate modeling.

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Homework Statement


A thin rod of length 2b is suspended by 2 light strings both attached to the ceiling. Using x, y1, y2 as your generalized coordinates right down the lagrangian of the system. Where x is the longitudinal displacement of the rod and y1 and y2 are the horizontal displacements of the ends.


Homework Equations



The strings remain taught and displacements from equilibrium are small

The Attempt at a Solution


If θ Is the angle that one of the strings makes with the vertical and we make a small angle approximation then θ= (x^2 +y^2)/l but and the height above the equilibrium position is equal to l(1-cos θ) which is approximately equal to l(θ^2), but this gives me a bunch of very small terms which I don't think is correct
 
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. I know I am missing something, do I take the coordinates as the generalized coordinates instead of the angle?
 

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