So a bead slides down a frictionless parabolic wire of shape y=ax^2. I have to express the Lagrangian in terms of x and y. Then I have to use the constraint equation to express this solely in terms of x. Then I have to find the equations of motion, and simplify them for small oscillations.(adsbygoogle = window.adsbygoogle || []).push({});

How do I get from

L=1/2m(x')^2 + 1/2m(y')^2 - mgy

to

L=1/2m(x')^2(1 + 4a^2x^2) -mgax^2 ???

And in this specific example I am having trouble seeing what the constraint force or constraint potential is. It seems to that we know that the force of gravity is present and acting in the negative y direction, and we are told there is no force of friction, so what is the constraint?

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# Lagrangian Constraint Problem

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