# Homework Help: Comparing Lagrangians to find sets of values

1. Oct 15, 2014

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