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

AI Thread Summary
The discussion focuses on modifying a Lagrangian mechanics equation for a point-like ball rolling in a parabolic bowl by introducing an additional force from a magnet. The magnet's force is described as having an inverse square distance dependency, which would attract the ball towards its position. The original poster seeks guidance on how to incorporate this magnetic force into the existing Lagrangian framework or to find a Newtonian solution for the problem. They mention that adding potential energy (U) to the Lagrangian is a general approach for such modifications. The conversation emphasizes the need for clarity on integrating external forces into the Lagrangian formulation.
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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:

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

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