The pendulum using Lagrange in cartisian

[Ananthan9470]

I am newly learning Lagrange formalism and I learned how to get the equation of motion for a simple pendulum using Lagrange in the spherical coordinate system. But I am unable to derive the same using the Cartesian system. If someone can please tell me what is wrong with the following derivation, that would be great.
Sorry for the sloppy paint work.

 

[kontejnjer]

From what I can tell, you forgot to add a constraint to the problem in the beginning, which in your case would be of the form $x^2+y^2=r^2=const.$ since the point mass is assumed to be at a constant distance from the center. From there you can express one of the coordinates as a function of the other one, and solve the equation of motion (not sure it'll be pretty though).

 

[Ananthan9470]

From what I can tell, you forgot to add a constraint to the problem in the beginning, which in your case would be of the form $x^2+y^2=r^2=const.$ since the point mass is assumed to be at a constant distance from the center. From there you can express one of the coordinates as a function of the other one, and solve the equation of motion (not sure it'll be pretty though).

OK. Thanks a lot. And in polar, you don't have to worry about that because you are keeping your r coordinate a constant to be equal to the length of the pendulum so the constraint is automatically taken care of? Thanks.