# Covariant form of the Lagrangian for Lorentz force.

1. Oct 25, 2008

### Peeter

If I use the following Lagrangian:

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

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

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

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.