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Conserved quantity

  1. Nov 30, 2004 #1
    I have a system of equations here:
    [tex]\frac{d\theta}{dt} = u-(\frac{1}{u})cos(\theta)[/tex]
    [tex]\frac{du}{dt} = -sin(\theta)[/tex]

    It asks to show that [tex]C(\theta,u) = u^3-3ucos(\theta)[/tex]. That's fine, if it worked. From looking at it and taking the partial derivatives, it doesn't seem to be a conserved quantity. Any ideas, or am I missing something?
     
  2. jcsd
  3. Nov 30, 2004 #2
    Sorry to add it here as well, but I don't want to start a new thread. For this system above, I am supposed to sketch certain solutions in the xy plane (this system is for the motion of a glider, theta is the angle it starts at, u is its initial velocity). What's the code to do this, I can't find it anywhere?
     
  4. Dec 1, 2004 #3

    HallsofIvy

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    We can't answer your first question because you haven't told us what "C(θ,u)
    means! Without knowing that, we can't even say if it should be a conserved quantity.
     
  5. Dec 1, 2004 #4
    I wasn't told what it means either :cry:

    I assumed it was just a Hamiltonian of the system, so I tried taking the partials and it comes out close if you fudge a few numbers or variables here or there, but otherwise I get zilch.
     
  6. Dec 1, 2004 #5

    robphy

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    Given
    [tex]\frac{d\theta}{dt} = u-(\frac{1}{u})cos(\theta)[/tex]
    and
    [tex]\frac{du}{dt} = -sin(\theta)[/tex]
    the quantity defined by
    [tex]C(\theta,u) = u^3-3ucos(\theta)[/tex]
    has the property that
    [tex]\frac{dC}{dt}=0[/tex], i.e., it is unchanged as "t" varies.

    [tex]
    \begin{align*}
    0
    &\stackrel{?}{=}
    \frac{d}{dt}\left(u^3-3u \cos\theta \right)\\
    &\stackrel{?}{=}
    3u^2\dot u-3(\dot u \cos\theta - u\sin\theta\dot\theta)\\
    &\stackrel{?}{=}
    3u^2[-\sin\theta]-3([-\sin\theta]\cos\theta - u\sin\theta[u-\frac{1}{u}\cos\theta ])\\
    &\stackrel{\surd}{=}0
    \end{align*}
    [/tex]

    I was rushing when I did this... Please check.
     
  7. Dec 1, 2004 #6
    Ah, thanks very much! Much appreciated - I didn't think of it that way :smile:

    Have an ideas on the maple one? I know that really isn't as much math oriented it's just that I don't really know how to use Maple :grumpy:
     
    Last edited: Dec 1, 2004
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