1D Dynamics

By the way, this is a 2D problem (horizontal and vertical directions are involved in the problem).

I disagree. The problem is a single variable problem. There exists a co-ordinate system where the problem can be described only with one variable.

Whilst there may be a connection between the vertical coordinate of one mass with the horizontal coordinate of the other (effectively eliminating one of them), two variables are needed to completely describe the position of each of the two masses, i.e. the (x,y) coordinates, although the second variable may not be as interesting as the first since the horizontal surface will most probably be y = 0.

Concider the Euler Lagrande equations associated with the problem.

Theres no need to be considering lagrangians for such a simple dynamics question, constraints remove coordinate dependence but they don't remove the coordinate system. As I stated in my previous post, you need two coordinates to completely specify the position of any one mass in this problem; if the mass that starts on the horizontal table begins at (0,0) then its position sometime later will be (x,0) and if the string is of length l then the other mass starts at (l,0) and at some time later will be at (l,-y) (if you like your positive vertical axis to go upwards and positive horizontal axis to go to the right).