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I know that the lagrangian is given by $$L = \frac{m}{2}\mathbf{\dot{r}}\cdot \mathbf{\dot{r}} -q\phi +q\mathbf{\dot{r}}\cdot \mathbf{A} $$

I can use this to derive the Lorentz force law using the Euler-Lagrange equations, but I was wondering how we formulate the lagrangian.

My understanding that the lagrangian takes the form ## L = T - U ## in nearly all situations

My issue is with the potential energy terms.

Typically, I understand that potential energy is the found by placing charge/mass in a scalar potential field, so the term ##-q\phi## is the contribution of the electric field.

The fact that the magnetic field is derived from a vector potential field is what's throwing me off right now. $$\mathbf{B}=\mathbf{\nabla\times \mathbf{A}}$$

Plus won't the electric field have some dependence on the vector potential as well due to presence of the magnetic field? $$ \mathbf{E}=-\mathbf{\nabla}\phi -\frac{\partial \mathbf{A}}{\partial t}$$

Any help will be much appreciated!