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## Homework Statement

Consider a point mass m attached to a string of slowly increasing length ##l(t)##. Them motion is confined to a plane. Find L and H. Is H conserved? Is H equal to the total energy? Is the total energy conserved? Assume ##|\dot{l}/l|<<\omega##

## Homework Equations

## The Attempt at a Solution

So I have $$L=\frac{1}{2}m(\dot{l}^2+l^2\dot{\theta}^2)+mglcos\theta$$. For H, for some reason the solution in the book doesn't calculate the ##p_l## but only ##p_\theta## and I am not sure why. The way I proceeded is: $$p_\theta=ml^2\dot{\theta}$$ and $$p_l=m\dot{l}$$ so $$H=\frac{1}{2}\frac{p_\theta^2}{ml^2}+\frac{1}{2}\frac{p_l^2}{m}-mglcos\theta$$. Is this correct by now? Now for the conservation I calculate $$\frac{dH}{dt}$$ which is not zero, as I will have terms such as ##\dot{l}## which are not 0 (is ##\dot{p_\theta}=0##?). Now by the form of ##H## it is clear that ##H=E_{tot}## so the total energy is not conserved either. Is my reasoning correct?