- #1
Markus Kahn
- 112
- 14
Homework Statement
I'd like to derive the equations of motion for a system with Lagrange density
$$\mathcal{L}= \frac{1}{2}\partial_\mu\phi\partial^\mu\phi,$$
for ##\phi:\mathcal{M}\to \mathbb{R}## a real scalar field.
Homework Equations
$$\frac{\partial \mathcal{L}}{\partial\phi}-\partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}=0$$
The Attempt at a Solution
$$\begin{align*}\frac{\partial \mathcal{L}}{\partial\phi}-\partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}& = -\partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\\
&= - \frac{1}{2}\partial_\mu \frac{\partial}{\partial(\partial_\mu\phi)} \partial_\nu\phi\partial^\nu\phi\\
&=- \frac{1}{2}\partial_\mu \left( \partial^\nu \phi\frac{\partial}{\partial(\partial_\mu\phi)}\partial_\nu\phi + \partial_\nu\phi \frac{\partial}{\partial(\partial_\mu\phi)} \partial^\nu\phi\right)\\
&= -\frac{1}{2} \partial_\mu (\partial^\mu\phi+\partial_\mu\phi)\end{align*}$$
As far as I know I should get ##\Box \phi =0##, but for this to be true I need to show that
$$\frac{1}{2}(\partial^\mu\phi+\partial_\mu \phi)= \partial^\mu\phi$$
holds, and I honestly don't know why this should be the case.
Can somebody help?