When writing the equations of motion for the simple pendulum, why do textbooks always choose $\theta$ to be the generalized coordinate? The force of gravity is in the y-direction so wouldn't it be natural to write everything in terms of y instead of $\theta$? Since the string is of fixed length $l$, we can write $x=\sqrt{l^2-y^2}$ and so shouldn't we be able to write the Equations of Motion completely in terms of $y$?
Answer
Peter Green's answer already showed you the error ($x=\sqrt{l^2-y^2}$ isn't generally true), but you can also directly see that $y$ isn't a sufficient coordinate:
No matter how fast the pendulum moves, at the bottom we always have $y=-l$ and $\dot{y}=0$. Therefore, you can't describe the state of the system just by $y$ and $\dot{y}$.
Edit: It's also worth pointing out that the other answers are indeed correct that $\theta$ is used instead of cartesian coordinates also because it actually is the choice that gives the simplest (and, subjectively, most natural) equations.
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