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Mar 18, 2021 at 12:03 comment added Anthony Guillen The best thing I can do is trying to explain mike stone's comment under the question. For a purely classical mechanical system, you have $L = T - V$ and you can write your Lagrangian very easily. But since electromagnetism is related to special relativity, this approach does not work. Yet the Lagrangian is not arbitrary. It is built from symmetry requirements. For instance, you get the one of electromagnetism just by imposing Lorentz and local gauge invariance. And if you then take its non relativistic limit, you obtain what you want
Mar 17, 2021 at 14:51 comment added Noumeno I have made an edit to my question, explaining my perplexities. If you could give me a complete clarification it would help a lot.
Mar 17, 2021 at 11:58 history answered Anthony Guillen CC BY-SA 4.0