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A Noether's Theorem to Multi-parameter Transformations

  1. Jul 17, 2016 #1
    When you have single parameter transformations like this in Noether's Theorem
    [itex]
    \begin{array}{l}
    {\rm{ }}t' = t + \varepsilon \tau + ...{\rm{ }}\\
    {\rm{ }}{q^\mu }^\prime = {q^\mu } + \varepsilon {\psi ^\mu } + ...
    \end{array}
    [/itex]

    The applicable form of the Rund-Trautman Identity is
    [itex]
    {\rm{ }}\frac{{\partial L}}{{\partial {q^\mu }}}{\psi ^\mu } + {p_\mu }{{\dot \psi }^\mu } + \frac{{\partial L}}{{\partial t}}\tau - H\dot \tau = \frac{{dF}}{{dt}}
    [/itex]

    and the conserved quantity is
    [itex]
    {\rm{ }}{p_\mu }{\psi ^\mu } - H\tau - F.
    [/itex]


    Can someone confirm that with multi-parameter transformations like
    [itex]
    \begin{array}{l}
    {\rm{ }}t' = t + {\varepsilon _i}{\tau _i} + ...{\rm{ }}i = 1,2,...,N\\
    {\rm{ }}{q^\mu }^\prime = {q^\mu } + {\varepsilon _i}{\psi _i}^\mu + ...
    \end{array}
    [/itex]

    The Rund-Trautman Identity becomes N identities
    [itex]
    {\rm{ }}\frac{{\partial L}}{{\partial {q^\mu }}}{\psi _i}^\mu + {p_\mu }{{\dot \psi }_i}^\mu + \frac{{\partial L}}{{\partial t}}{\tau _i} - H{{\dot \tau }_i} = \frac{{d{F_i}}}{{dt}}
    [/itex]

    and the conserved quantity becomes N conserved quantities
    [itex]
    {\rm{ }}{p_\mu }{\psi _i}^\mu - H{\tau _i} – F_i.
    [/itex]
     
  2. jcsd
  3. Jul 22, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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