# Homework Help: Comparing Lagrangians to find sets of values

1. Oct 15, 2014

### Dixanadu

1. The problem statement, all variables and given/known data
Hey guys, so this is a follow up from my previous post.

So I have this Lagrangian for two coupled fields:

$\begin{split} \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2})(\partial^{\mu}\phi_{2})-\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1}) -\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})-g(\phi_{1}\phi_{2})^{2} \end{split}$

I was given the transformation $\phi_{1}\rightarrow\phi_{1}+\epsilon\phi_{2}, \phi_{2}\rightarrow\phi_{2}-\epsilon\phi_{1}$, which (if I've done it right) yields the following:

$\begin{split} \mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2})(\partial^{\mu}\phi_{2})-\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2}) \\ -\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})-g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2} \end{split}$

Now I have to "Find the most general values for m_1, m_2 and g such that this transformation is a symmetry".

2. Relevant equations
Apparently there is no coordinate transformation, so I've been told that I need to impose the condition

$\mathcal{L}(\phi',\partial_{\mu}\phi')=\mathcal{L}(\phi,\partial_{\mu}\phi)$

3. The attempt at a solution
So err...am I just meant to equate both Lagrangians? If I do that I end up with the equations:
$\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2})=\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1})$
$\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})=\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})$
$g(\phi_{1}\phi_{2})^{2}=g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2}$

And I'm stuck lol...how do I use these to find the most general values for m_1, m_2 and g?

Thank guys

Last edited: Oct 15, 2014
2. Oct 20, 2014

### Staff: Admin

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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