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Homework Help: Lagrangian of a particle + conserved quantities

  1. May 12, 2010 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Consider the spherical pendulum. In other words a particle with mass m constrained to move over the surface of a sphere of radius R, under the gravitational acceleration [tex]\vec g[/tex].
    1)Write the Lagrangian in spherical coordinates [tex](r, \phi, \theta)[/tex] and write the cyclical coordinates and conserved quantities.
    2)Define an effective potential energy [tex]U(\phi)[/tex] and determine (at least qualitatively) the allowed regions for the motion of the particle.
    2. Relevant equations

    Lots of.

    3. The attempt at a solution
    1)As the Lagrangian, I got [tex]L=\frac{m}{2}r^2 \left [ \dot \phi ^2 + \dot \theta ^2 \sin ^2 (\phi) \right ]+mgR \left [ 1- \cos (\theta) \right ][/tex]. Notice that [tex]\dot r=0[/tex].
    I can already say that the energy is conserved since the Lagrangian doesn't depend explicitly on time.
    I want to see if the linear momentum is conserved (is it stupid? I mean it seems obvious that no...)
    So [tex]\vec P=\frac{\partial L}{\partial \vec \dot q}[/tex]. For [tex]\dot q = \dot r[/tex], I get [tex]\vec P=\vec 0[/tex]. What does that mean? It's a constant. So the "r" component of the linear momentum is conserved? It doesn't make sense to me to say it under these words.
    I also reach [tex]\frac{\partial L}{\partial \dot \phi}=mR^2 \dot \phi[/tex]. And [tex]\frac{\partial L}{\partial \dot \theta}=m \dot \theta \sin ^2 (\phi)[/tex].
    Since the energy is conserved, I can apply Euler-Lagrange's equation. I get that [tex]\dot \phi =0[/tex] which makes sense to me. Thus [tex]\frac{\partial L}{\partial \dot \phi}=0[/tex] and so the [tex]\phi[/tex] component of the linear momentum is conserved. (Does this make sense to talk like this? What should I say instead? Maybe the angular momentum with respect to [tex]\phi[/tex]?).
    Lastly I found out that [tex]\frac{\partial L}{\partial \dot \theta}[/tex] is not necessarily constant (it is constant if the pendulum becomes the coplanar pendulum I believe) and thus the [tex]\theta[/tex] component of the linear momentum is not conserved.

    I don't know what are the cyclical coordinates. I'll look into this further.
    2)I just opened Landau's book to check out the effective potential energy and it is [tex]\frac{M^2}{2m R^2}[/tex] which seems to assume 2 particles... in my exercise there's only a massless sphere and a particle. I'm no clue what to do here.
     
  2. jcsd
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