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Covariant form of the Lagrangian for Lorentz force.

  1. Oct 25, 2008 #1
    If I use the following Lagrangian:

    [tex]
    \mathcal{L} = \frac{1}{2} m v^2 + e A \cdot v/c = \frac{1}{2} m \dot{x}_\mu \dot{x}^\mu + e A_\nu \dot{x}^\nu /c
    [/tex]

    I can arrive at the Lorentz force equation in tensor form:

    [tex]
    m \ddot{x}_\mu &= (q/c) F_{\mu\beta} \dot{x}^\beta
    [/tex]

    details offline here:
    http://www.geocities.com/peeter_joot/lut/maxwell_tensor_lagrangian.pdf


    However, reading a translation of the DeBroglie thesis he appears to use the same Lagrangian, but it differs by a factor of two in the v^2 term

    http://www.nonloco-physics.000freehosting.com/ldb_the.pdf

    (equation 2.3.5 on page 26 of the pdf)

    I haven't gotten far enough that I see what he does with this, but would like to understand why my previous calculation appears to be off by a factor of two before continuing.

    Does anybody know of the correct covariant Lagrangian to arrive at the Lorentz force equation? I started with the non-covariant form in Goldstein and got the equation above with a bit of guess work.
     
  2. jcsd
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