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Proving that the free particle lagrangian is rotationally symmetric

  1. Feb 26, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that the free particle lagrangian is invariant to rotations in $$\Re^{3}$$, but I assume this means invariant up to a gauge term.
    $$L=m/2 [\dot{R^{2}} + R^{2}\dot{θ^{2}} +R^{2}Sin^{2}(θ)\dot{\phi^{2}}$$


    2. Relevant equations
    I consider an aribtrary infinitesimal rotation:
    $$ \theta(t,\epsilon)=\theta(t,0)+\epsilon \Delta \theta $$
    $$ \phi(t,\epsilon)=\phi(t,0)+\delta \Delta \phi $$


    3. The attempt at a solution
    The new angle derivitives are identical to the first, since we evaluate them by taking the time partial of the transformed coordinates.
    I am running into issues with the $$Sin^{2}(\theta)$$ term.
    $$Sin^{2}(\theta) \rightarrow Sin^{2}(\theta)+2Cos(\theta)Sin(\theta) \epsilon \Delta \theta + O(\epsilon^{2})$$

    The epsilon term is throwing me off, because I can't get it to disappear or rewrite it as a gauge term.
     
  2. jcsd
  3. Feb 26, 2013 #2

    George Jones

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    Re: Proving that the free particle lagrangian is rotationally symmetri

    Why do you have to use infinitesimal rotations and a Lagrangian written in terms of spherical coordinates?

    Whey not apply a finite rotation to

    [tex]L = \frac{1}{2} m \dot{\vec{r}} \cdot \dot{\vec{r}} ?[/tex]
     
  4. Feb 26, 2013 #3
    Re: Proving that the free particle lagrangian is rotationally symmetri

    It seemed more straightforward to apply displacements in the angular directions. To apply a general rotation matrix would be a ton more algebra, no?

    I need to use infinitesimal rotations because I am trying to prove a continuous symmetry for noether's theorem
     
    Last edited: Feb 26, 2013
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