- #1

JackDP

- 7

- 0

## Homework Statement

Hi all, I'm trying to learn how to manipulate tensors and in particular to differentiate expressions. I was looking at a Lagrangian density and trying to apply the Euler-Lagrange equations to it.

## Homework Equations

Lagrangian density:

[tex]\mathcal{L} = -\frac{1}{2} \partial_{\alpha} \phi^{\beta} \partial^{\alpha} \phi_{\beta}

+ \frac{1}{2} \partial_{\alpha} \phi^{\alpha} \partial_{\beta} \phi^{\beta}

+ \frac{1}{2}\mu^2 \phi^{\alpha} \phi_{\alpha}[/tex]

Euler-Lagrange:

[tex]\frac{\partial \mathcal{L}}{\partial \phi^i} = \partial^k \frac{\partial \mathcal{L}}{\partial \phi^{i,k}}[/tex]

## The Attempt at a Solution

I have attempted to differentiation the expression several times; I can compute [tex]\frac{\partial \mathcal{L}}{\partial \phi^i}[/tex] with no problems and can compute [tex]\frac{\partial \mathcal{L}}{\partial \phi^{i,k}}[/tex] for the first and third terms.

However, I just cannot figure out how to differentiate the middle term. My attempt:

[tex]\mathcal{L}_2 = \frac{1}{2} \partial_{\alpha} \phi^{\alpha} \partial_{\beta} \phi^{\beta}

= \frac{1}{2} g_{\alpha \lambda} g_{\beta \sigma} \partial^{\lambda} \phi^{\alpha} \partial^{\sigma} \phi^{\beta}[/tex]

Hence

[tex] \frac{\partial \mathcal{L}_2}{\partial \phi^{i,k}} =

\frac{1}{2} g_{\alpha \lambda} g_{\beta \sigma} \left(

\delta_k^{\lambda} \delta_i^{\alpha} \partial^{\sigma} \phi^{\beta} +

\delta_k^{\sigma} \delta_i^{\beta} \partial^{\lambda} \phi^{\alpha}

\right)

= \frac{1}{2} \left(

g_{i k} \partial_{\beta} \phi^{\beta} +

g_{i k} \partial_{\alpha} \phi^{\alpha}

\right)

= g_{i k} \phi_i \phi^i

[/tex]

So as you can see, I have somehow picked up this additional factor of the metric. I'm not sure what to do with it, or where I have gone wrong!

Best wishes,

J