What I understand is that we have two kinds of coordinates when working with the Lagrangian formalism with different zeros (which may happen to coincide) to measure from, those are the Cartesian coordinates and the generalized coordinates. To solve a problem we specify the zero of the will be used Cartesian coordinates and use a set of generalized coordinates to specify exactly the positions of each body in the system. In the last part we write down the Cartesian coordinates of each body in the system interns of the specified generalized coordinates in a preparation for the calculation of the kinetic energy of the system. Now, here is my question, do we calculate the potential energy of each body in the system with respect to the chosen zero of the Cartesian coordinates or do we calculate it with respect to the zero of respective generalized coordinates? It seems to me in the solution of the problem in the attached photo that he calculated the potential of the mass m, the mass on the incline, with respect to the zero of the generalized coordinate x prime since he calculated that to be negative mgx_prime sinθ whereas the potential energy of the mass M is found to be zero which also seems to be calculated with respect to the zero of the generalized coordinate x which happens to coincide, in this case, 
with the zero of the Cartesian coordinates!
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2$\begingroup$ Any zero you like, as long as it's the same place! $\endgroup$– levitopherCommented Nov 3, 2022 at 15:57
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$\begingroup$ Thank you! But what do you mean by as long as it's the same place? $\endgroup$– JackCommented Nov 3, 2022 at 16:08
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1$\begingroup$ The position vector $~\vec R~$ to the mass, must be given in (x,y) system. The potential energy is $~U=-M\,g\,(\vec R)_y~$ $\endgroup$– EliCommented Nov 3, 2022 at 16:57
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I think to get the answer we want to make the most physical statement we can. The only thing that matters is the change in the potential energy, so we can say "the change in the potential energy is the mass times the gravitational acceleration times the change in the height". The height is unequivocally "the y component" of that mass. So if you pick y=0 at the location you have indicated, the potential energy is either
$$U=mgy$$ or $$U=mgx'sin(\theta)$$
depending on the coordinate system you are using.
answered Nov 3, 2022 at 18:39