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Lagrangian where time is a dependent coordinate

  1. Oct 24, 2013 #1
    1. The problem statement, all variables and given/known data

    I don't know why I'm having trouble here, but I want to show that, if we let [itex] t = t(\theta) [/itex] and [itex] q(t(\theta)) = q(\theta) [/itex] so that both are now dependent coordinates on the parameter [itex] \theta [/itex], then

    [tex] L_{\theta}(q,q',t,t',\theta) = t'L(q,q'/t',t) [/tex]

    where [itex] t' = \frac{dt}{d\theta}, q' = \frac{dq}{d\theta} [/itex]

    3. The attempt at a solution

    Writing [itex] L = \frac{m}{2} \dot{q}^2 - V(q) [/itex], we let [itex] \frac{d}{dt} \to \frac{d\theta}{dt}\frac{d}{d\theta} [/itex] and then,

    [tex] L = \frac{m}{2} \frac{q'^2}{t'^2} - V(q) [/tex]

    Which clearly doesn't agree with what I need to show... where am I going wrong here?
  2. jcsd
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