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Lagrangian of charged particle in magnetic field

I am aware that this question has been asked before, but the answer uses a formula I haven't seen before, and I was wondering if there is another more intuitive way to solve this problem. I wish to ...
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Lagrangian of Non-Relativistic Charged Particle in a Magnetic Field

I'm trying to derive the Lagrangian for a non-relativistic charged particle under the influence of a magnetic potential. I'm assuming that $F=-\textrm{grad}(V)$ and so by the Lorentz force we have $-\...