Deriving the equations of motion from an effective Lagrangian density

I feel a bit embarrassed that I can't figure this out on my own, but I am trouble deriving the equations of motion for the Lagrangian $$\mathcal{L}_{eff}=(\omega+eA_0)^2f(r)^2-(\partial_r f(r))^2-V(f)+\frac{1}{2}(\partial_r A_0(r))^2.$$ This effective Lagrangian comes from taking a Maxwell-scalar field theory and assuming a spherical ansatz for the scalar field and the $$A_0$$ component of the gauge field. If I treat $$f(r)$$ and $$A_0(r)$$ as the dynamical variables, then the Euler-Lagrange equations that yield the equations of motion should be \begin{align} 0&=\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu f)}-\frac{\partial \mathcal{L}}{\partial f}\\ 0&=\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_0)}-\frac{\partial \mathcal{L}}{\partial A_0}\\ \end{align} and since the only dependence in the Lagrangian is on the radial coordinate $$r$$, this should just reduce to \begin{align}0&=\frac{d}{dr}\left(\frac{\partial \mathcal{L}}{\partial f'}\right)-\frac{\partial \mathcal{L}}{\partial f}\\ 0&=\frac{d}{dr}\left(\frac{\partial \mathcal{L}}{\partial A_0'}\right)-\frac{\partial \mathcal{L}}{\partial A_0}. \end{align} The terms are $$\frac{\partial \mathcal{L}}{\partial f'}=-2f'$$ $$\frac{\partial \mathcal{L}}{\partial f}=(\omega+eA_0)^2\cdot 2f-\frac{dV(f)}{df}$$ and $$\frac{\partial \mathcal{L}}{\partial A_0'}=A_0'$$ $$\frac{\partial \mathcal{L}}{\partial A_0}=2ef(r)^2(\omega+eA_0).$$

Plugging these terms into the Euler-Lagrange equations just gives

\begin{align} 0&=f''(r)+f(r)(\omega+eA_0)^2-\frac{1}{2}\frac{dV(f)}{df}\\ 0&=A_0''(r)-2ef(r)^2(\omega+eA_0). \end{align}

The problem is that the correct equations of motion are apparently

\begin{align} 0&=f''(r)+\color{red}{2\frac{f'(r)}{r}}+f(r)(\omega+eA_0)^2-\frac{1}{2}\frac{dV(f)}{df}\\ 0&=A_0''(r)+\color{red}{2\frac{A_0'(r)}{r}}-2ef(r)^2(\omega+eA_0). \end{align} For example, this result is given as equations (7), (8) of of this paper (Gulamov 2015) and further corroborated by equations (4.1), (4.2) of this famous paper (Lee 1989).

I am obviously missing the $$1/r$$ terms, but I cannot see where this would come from. Can anyone provide see what I am missing in this simple derivation?

• I am not accustomed to this notation, but isn't the $\partial_\mu \frac{\cdot}{\cdot}$ a divergence? Results of vector calculus operations in spherical coordinates are not trivial, maybe check the usual 3D versions? Nov 3 '19 at 18:55
• The terms in red are additional terms due to the curvature in spherical coordinates, i.e $\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{df}{dr}\right)=\frac{d^2f}{dr^2}+\frac{2}{r}\frac{df}{dr}$, perhaps that gives you a lead Nov 3 '19 at 23:30

1. Recall that the action $$S=\int \! \mathbb{L}$$ is an integral over the Lagrangian 4-form $$\mathbb{L}~=~ d^4x~ {\cal L}$$, where $${\cal L}$$ is the Lagrangian density. It's a density in the sense that under a coordinate transformation $$x\to x^{\prime}$$, it transforms as $${\cal L}^{\prime}={\cal L}/J$$ with the inverse Jacobian $$J:=\det\frac{\partial x^{\prime}}{\partial x}$$. It this way the action $$S$$ is a scalar/invariant under coordinate transformations.
2. The inverse Jacobian from 3D rectangular to 3D spherical coordinates is $$J^{-1}=\color{red}{r^2\sin\theta}$$, so in spherical coordinates the Lagrangian density reads $${\cal L}^{\prime}=\color{red}{r^2\sin\theta} ~{\cal L}$$.
3. Now because of OP's spherical ansatz, the factor $$\color{red}{\sin\theta}$$ is not important, but the factor $$\color{red}{r^2}$$ is, and it leads to the sought-for correct EL equations.