1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Comparing Lagrangians to find sets of values

  1. Oct 15, 2014 #1
    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:

    [itex] \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}[/itex]

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

    [itex] \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}[/itex]


    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

    [itex]\mathcal{L}(\phi',\partial_{\mu}\phi')=\mathcal{L}(\phi,\partial_{\mu}\phi)[/itex]

    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:
    [itex]\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2})=\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1})[/itex]
    [itex]\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})=\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})[/itex]
    [itex]g(\phi_{1}\phi_{2})^{2}=g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2}[/itex]

    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. jcsd
  3. Oct 20, 2014 #2
    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Comparing Lagrangians to find sets of values
  1. Finding the Lagrangian (Replies: 14)

Loading...