Metal Ball Rolling in a Parabolic Bowl in the presence of a magnet

[alikim]

TL;DR

How to modify the equation of motion of a ball in a parabolic bowl to include an extra external force?

I found this equation as a solution to the original problem of a point-like ball rolling without friction in a parabolic cavity from Lagrangian mechanics:

Is it possible to add an extra force to this equation? I want to add a "magnet" at a certain position that will affect the ball. The magnet here being a force that will have an inverse square distance dependency and pull the ball to the point where the magnet is located.

I would know how to do it better if I had a Newtonian solution with forces but I couldn't find any.

I'd appreciate it if anyone could point me to a Newtonian solution or explain how to modify the Lagrangian for this problem.

 

[anuttarasammyak]

Coming back to Lagrangian add -U to it where U is potential energy you set. This is a general prescription.