Equations of motion for Lagrangian of scalar QED

[BobaJ]

Homework Statement

I have the Lagrangian for scalar electrodynamics given by:
$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x)+(D_\mu\varphi(x))^*(D^\mu\varphi(x))-V(\varphi^*(x)\varphi(x)) $$
where $F_{\mu\nu}(x)=\partial_\mu A_\nu(x)-\partial_\nu A_\mu(x)$ is the electromagnetic field strength tensor, $D_\mu=\partial_\mu+ieA_\mu$ ist the covariant derivative, e is the electric charge and $V(\varphi^*\varphi)=m^2\varphi^*\varphi+\lambda(\varphi^*\varphi)^2$ is the potential of the scalar field.

I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Relevant Equations

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by:
$$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$ and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

For the electromagnetic field $A_\mu$ it should just be:
$$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\rho\frac{\partial \mathcal{L}}{\partial(\partial_\rho A_\mu)}=0.$$

Well, I started with the first equation of motion for the scalar field, but I'm really not sure if I'm doing it the right way.

\begin{equation}
\begin{split}
\frac{\partial \mathcal{L}}{\partial \varphi} &= \frac{\partial}{\partial \varphi} [(\partial_\mu \varphi^* - ieA_\mu\varphi^*) (\partial_\mu\varphi+ieA_\mu\varphi)-m^2\varphi^*\varphi-\lambda(\varphi^*\varphi)^2]\\
&= \frac{\partial}{\partial \varphi} [\partial_\mu\varphi^*\partial_\mu\varphi + ieA_\mu\varphi\partial_\mu\varphi^*-ieA_\mu\varphi^*\partial_\mu\varphi+e^2A_\mu\varphi^*\varphi-m^2\varphi^*\varphi-\lambda(\varphi^*\varphi)^2] \\
&= ieA_\mu\partial_\mu\varphi^*+e^2A_\mu\varphi^*-m^2\varphi^*-2\lambda(\varphi^*)^2\varphi
\end{split}
\end{equation}

and

\begin{equation}
\begin{split}
\partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \varphi} &= \partial_\mu\frac{\partial}{\partial (\partial_\mu\varphi)}[\partial_\mu\varphi^*\partial_\mu\varphi + ieA_\mu\varphi\partial_\mu\varphi^*-ieA_\mu\varphi^*\partial_\mu\varphi+e^2A_\mu\varphi^*\varphi-m^2\varphi^*\varphi-\lambda(\varphi^*\varphi)^2] \\
&= \partial_\mu [\partial_\mu\varphi^*-ieA_\mu\varphi^*]
\end{split}
\end{equation}

Does this at least go in the right direction? I'm really unsure. Thanks for your help. I appreciate it.

 

[Rosenthal]

This looks correct, except that the term in the last line of equation (1) proportional to A*phi should be A^2*phi.. The term involving the 4-divergence of the vector potential can be set to zero by adopting a particular gauge.