1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Derivation of the Eqation of Motion from Fermi Lagrangian density

  1. Dec 26, 2017 #1
    1. The problem statement, all variables and given/known data
    Hello, I am trying to find the equations of motion that come from the fermi lagrangian density of the covariant formalism of Electeomagnetism.


    2. Relevant equations
    The form of the L. density is:
    $$L=-\frac{1}{2} (\partial_n A_m)(\partial^n A^m) - \frac{1}{c} J_m A^m$$

    where J is the electric current.
    The result has to be:

    $$\partial_n \partial^n A^m = \frac{1}{c} J^m$$

    3. The attempt at a solution
    Using the Euler- Lagrange equations that derive the eq. of motion I do not understant if I have to treat the fields Am and Am. In addition, the Euler Lagrange equations from a L.density of the form:
    $$L=-\frac{1}{2} (\partial_n A_m)(\partial_n A_m) - \frac{1}{c} J_m A^m$$
    giveaway the wanted result. But I could not relate the two L.densities.
     
  2. jcsd
  3. Dec 27, 2017 #2

    maajdl

    User Avatar
    Gold Member

    How did you lower the indices to get the density in §3 ?
    I think the density in §3 is not covariant and it is then incorrect.
    There must be elements of the metric tensor missing in the density of §3 .
     
  4. Dec 27, 2017 #3

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    Indeed. OP, please show your work.
     
  5. Dec 30, 2017 #4
    I did not lower any indices. And yes it is not covariant at all so it could not be a lagrangian. It was something a digged up on the internet while I was working at the solution of my problem. However, I did not notice while posting the thread that it was not covariant.

    Now, Lets focus on the first L.density.

    I tried playing with the indexes and my metric.

    Is the following correct? :

    $$L=-1/2(\partial_n A_m)( \partial^n A^m) = -\frac{1}{2} (g_{ns} \partial^s A_m)(g^{nr} \partial_r A^m) = -\frac{1}{2} \delta_{s}^{r}(\partial^s A_m)( \partial_r A^m)= -\frac{1}{2} (\partial^s A_m)(\partial_s A^m) $$

    So the partial derivative $$\frac{ \partial L}{\partial (\partial_s A^m)}$$ equals:

    $$\partial_s (\frac{ \partial L}{\partial (\partial_s A^m)})= \partial_s \partial^s A^m$$
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...