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.
It is just a physical fact that lagrangian does not explicitly depend on the other partial derivatives.