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Homework Help: Conservation of ang. momentum using Euler-Lag. Eqns (help! ;-)

  1. Feb 26, 2008 #1

    T-7

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    Hi,

    I'm stuck on what I suspect is a really easy proof. I'd appreciate some rapid help!

    1. The problem statement, all variables and given/known data

    For a trajectory with given initial position and velocity, we can always rotate the system of coordinates such that initially [tex]d\theta / dt = 0 [/tex] and [tex]\theta= \pi/2[/tex] say, t = 0.

    Using the Euler-Lagrange equations, show that [tex]\theta[/tex] remains at [tex]\pi/2[/tex] for all times t.

    3. The attempt at a solution

    Having obtained the Lagrangian (L=mv^2/2 + U, rewriting v as [tex]\dot{r}^{2}+r^{2}(\dot{\phi}^{2}sin^{2}\theta + \dot{\theta}^{2})[/tex] in spherical coordinates, and taken partial derivatives with respect to [tex] \dot{\theta} [/tex] and [tex] \theta [/tex], and [tex]\phi[/tex] and [tex]\dot{\phi}[/tex], I have found the following relations (I think they're correct?)

    [tex]d/dt [mr^2\dot{\theta}] = mr^2\dot{\phi^2}cos(\theta)sin(\theta)[/tex]

    [tex]d/dt [mr^{2} \dot{\phi} sin^{2} (\theta)] = 0[/tex]

    (There is a third expression, from the partial derivatives wrt. [tex]r[/tex] and [tex]\dot{r}[/tex], but I doubt it's relevant here.)

    I'd be tempted to integrate both sides of the first to get an expression for [tex]\dot{\theta}[/tex], but that's where I become unstuck just now... I'm not sure how to integrate the RHS with respect to time, given what I know about the components. But presumably integration is going to be involved here, or the initial conditions would never come into play (?). I can deduce from the second that [tex]mr^{2} \dot{\phi} sin^{2} (\theta) = const = c[/tex], and so at t = 0 [tex]mr^{2} \dot{\phi} = c[/tex], but nothing more of interest seems to be following at the moment...

    Incidentally, I take the geometrical meaning to be that the motion is confined to a plane orthogonal to L, ang. momentum is conserved.

    Advice gratefully received (as soon as possible!) Many thanks :-)
     
    Last edited: Feb 27, 2008
  2. jcsd
  3. Feb 28, 2008 #2
    The statement you are trying to prove is not true for any potencial energy U. Does the problem include something like U=U(r)?
     
  4. Feb 28, 2008 #3

    T-7

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    Yep. :-)
     
  5. Feb 28, 2008 #4
    If you put theta(t)=pi/2, then the equation is always true. I think this proves that theta must be constant, since a physical system can not have more than one solution.
     
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