$\partial^{\nu} \partial_{\nu}$ vs. $\partial_{\nu} \partial^{\nu}$ I was doing a problem regarding field theory. I am given the following lagrangian density: $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi_i\partial^\mu\phi_i-\frac{m^2}{2}\phi_i\phi_i$$ for three scalar fields. I want to determine the equations of motion for the field $\phi_i$. I used the Euler Lagrange equation:$$\frac{\partial\mathcal{L}}{\partial\phi_j}-\partial_\mu[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi_j)}]=0.$$ My question is, is there a difference between using the Euler Lagrange equation the way I wrote above and the Euler Lagrange equation with an upper $\mu$ index: $$\frac{\partial\mathcal{L}}{\partial\phi_j}-\partial^\mu[\frac{\partial\mathcal{L}}{\partial(\partial^\mu\phi_j)}]=0~?$$ 
I ask this because by the end of my calculations, if I use the former equations , I get $$\partial_\nu\partial^\nu\phi_j+m^2\phi_j=0$$ while the latter equations give:
$$\partial^\nu\partial_\nu\phi_j+m^2\phi_j=0.$$
Are both the Klein-Gordon Equation or just the last one?
 A: As mentioned in the comments by WarreG and user2723984, since the full expression of the two in terms of the metric tensor is
$$
\partial_\nu \partial^\nu=\partial_\nu g^{\mu\nu}\partial_\mu=\partial^\mu\partial_\mu,
$$
the two equations are identical.
A: I would say the answer to your question is: it depends on the metric. Yes, because if your metric is Minkowski and Cartesian then $$\partial_{\mu}\partial^{\mu} = \partial_{\mu}\left(g^{\mu\nu}\partial_{\nu}\right)=\underbrace{\left(\partial_{\mu}g^{\mu\nu}\right)}_{0}\partial_{\nu}+g^{\mu\nu}\left(\partial_{\mu}\partial_{\nu}\right)=g^{\mu\nu}\partial_{\mu}\partial_{\nu}=\partial^{\nu}\partial_{\nu}.$$ But for example, if your metric is Minkowski and spherical? In this case $\partial_{\mu}g^{\mu\nu}\neq0$ and $\partial^{\mu}\partial_{\mu}\neq\partial_{\mu}\partial^{\mu}$.
As you can see, $\partial_{\mu}\partial^{\mu}$ and $\partial^{\mu}\partial_{\mu}$ are equivalent as long as you understand that you are dealing with Cartesian coordinates.
