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Thread: Calculus of Variations Rund Trautman Electromagnetism Problem

  1. #1
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    Calculus of Variations Rund Trautman Electromagnetism Problem

    I have been having trouble with a bunch of examples to do with the Rund Trauman Identity. I have also posted this query on Physics help forum but I generally don't have much luck on that forum.


    I have the Rund-Trautman identity in this form/notation:


    $\displaystyle \frac {\partial L}{\partial q^ \mu}\zeta ^\mu+p_\mu \dot \zeta^\mu+\frac{\partial L}{\partial t}\tau-H \dot \tau=\frac{dF}{dt}$


    Now for the electromagnitism problem I have the Lagrangian:


    $\displaystyle L=\frac12 m \underline{\dot r}^2+q \underline v \cdot \underline A -qV$


    I wish to test for invariance under the transformation:


    $\displaystyle t'=t(1+\epsilon)$ $\displaystyle x^{\mu'}=x^{\mu}(1+\frac12 \epsilon)$


    Now the electric scalar potential and the magnetic scalar potential are the bits that are giving me trouble. If I consider just the scalar potential and I use spherical coordinates then I get:


    $\displaystyle \frac {\partial qV}{\partial r}\frac r2+\frac{\partial qV}{\partial t}t=$


    From here I am stuck. Perhaps I need to show that this is equal to the derivative of a function of time, and my calculus just isn't good enough. Or maybe I need to add the terms for the magnetic vector potential and then (somehow) use one of Maxwells equations to show that the result is zero?


    In any case I have tried many of these things and am stuck. Can someone suggest an approach?
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  2. #2
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    Re: Calculus of Variations Rund Trautman Electromagnetism Problem

    Zeta and Tau come from

    $\displaystyle t'=t(1+\epsilon)$ and $\displaystyle x^{\mu'}=x^{\mu}(1+\frac12 \epsilon)$

    giving

    $\displaystyle \zeta ^{\mu}=\frac {x^{\mu}}{2}$ and $\displaystyle \tau = t$

    They come from

    $\displaystyle t'=T(t,q^{\mu},\epsilon)$

    $\displaystyle \tau = (dT/d \epsilon)_{\epsilon = 0}$ and a similar equation for zeta.
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