# [Math] Euler-Lagrange equations of the Lagrangian related to Maxwell’s equations

classical-mechanicspartial differential equationsphysics

Clarification on Lagrangian mechanics would be much appreciated:

Suppose
$$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$

Are the corresponding Euler-Lagrange equations then:
$$c=0$$ by considering $\phi$, and
$$2(\dot\phi_{,i}+\ddot A_i)+d_i=0$$
by considering $A_i$?

I am confused by the dependent variables in this Lagrangian — they are differentiated wrt to different variables, namely $\phi$ wrt spatial elements, whereas $A_i$ wrt time. Moreover, shouldn't $L$ also be a function of $A_{i,j}$?

Help would be much appreciated!

It might help you to understand the physical significance of the lagrangian here. This is the lagrangian for Maxwell's equations in terms of the potentials. $\phi$ and $A$ are the scalar and vector potentials, and $c$ and $d$ are the charge and current distributions. The first term $|\dot A+\nabla\phi|^2$ is the electric field, second term magnetic, and the remaining terms the coupling between charges and fields.