If I use the following Lagrangian:(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Covariant form of the Lagrangian for Lorentz force.

Can you offer guidance or do you also need help?

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