Results 1 to 2 of 2

Math Help - Calculus of Variations Rund Trautman Electromagnetism Problem

  1. #1
    Member
    Joined
    May 2008
    From
    Melbourne Australia
    Posts
    216
    Thanks
    29

    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:


    \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:


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


    I wish to test for invariance under the transformation:


    t'=t(1+\epsilon) 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:


    \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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    May 2008
    From
    Melbourne Australia
    Posts
    216
    Thanks
    29

    Re: Calculus of Variations Rund Trautman Electromagnetism Problem

    Zeta and Tau come from

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

    giving

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

    They come from

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

    \tau = (dT/d \epsilon)_{\epsilon = 0} and a similar equation for zeta.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. problem of calculus of variations
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 22nd 2011, 08:04 AM
  2. [SOLVED] Calculus of variations
    Posted in the Calculus Forum
    Replies: 0
    Last Post: August 31st 2010, 04:30 AM
  3. Calculus of Variations
    Posted in the Calculus Forum
    Replies: 4
    Last Post: August 7th 2010, 06:16 PM
  4. Calculus of variations
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 16th 2010, 06:55 AM
  5. Calculus of Variations problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 14th 2010, 10:08 PM

Search Tags


/mathhelpforum @mathhelpforum