Lagrangian of Two Masses on a Table

[Parmenides]

This is a problem from the Goldstein text. It gives two point masses $m_1$ and $m_2$ connected by a string (negligible mass), where $m_2$ is suspended by the string through a hole in a smooth table; $m_1$ rests on the table. It is important to note that $m_2$ only travels in a vertical line.

I only need assistance in determining the setup for the Lagrangian. In my first attempt, I had thought that the only generalized coordinate necessary was $y$, the vertical direction. Since the length of the string $l$ is constant, I figured that the distance traveled from $m1$ to the hole could just be determined by something like $(l - y)$. But this is apparently not the case. It seems that the generalized coordinates used in the problem are the length of the string from the hole to $m_1$ ($r$) and the angle that $m_1$ makes as it moves around on the table ($\theta$).

The length $r$ I could understand, but why should the angle have any physical significance? If $m_2$ just moves in a vertical line, it shouldn't move the other mass around such that it rotates or anything. Why not just view the problem as if we were just looking at the table from the side, rather than focus on anything that happens on the top of the table? I'm clearly misinterpreting the representation of the problem, but I don't understand how. Why do we care about the angle $\theta$?

 

[Matterwave]

I think you are thinking about the problem too simplistically. m1 can rotate around the table without forcing m2 to move off of the vertical line. If m1 rotates then it exerts a centrifugal force which will make m2 fall even slower than one would expect if this were simply a pulley problem.