[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!

Best Answer

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.

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