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Lagrange/Hamilton system

  1. Mar 3, 2008 #1
    [SOLVED] Lagrange/Hamilton system

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

    There is a circle without mass, radius r. On the edge of the circle there is a mouse, being forced to move around the circle.

    The angle the mouse makes with respect to the centre of the circle is called [tex]\theta(t)[/tex]

    At the same time, the circle is hold at it's place at a point Q on the edge and is rotated around this point Q with constant angular velocity [tex]\omega[/tex]. Forget about friction.

    write down the equation of motion of the mouse.

    3. The attempt at a solution

    the number of degrees of freedom for this system is 2, [tex]\omega[/tex] and [tex]\theta[/tex]

    The Lagrangian L=K-V
    K=\frac{m v^2}{2} [/tex]

    where [tex] v=v_c+v_\theta[/tex]
    and [tex]v_c[/tex] is the velocity of the centre of the circle, and [tex]v_\theta[/tex] is the velocity of the mouse around the circle.

    [tex]v_c=\omega l [/tex]
    [tex]v_\theta=\theta' l[/tex]

    so [tex] \delta J = \delta {\int_{t_1}}^{t_2} \frac{m (\omega l + \theta' l)^2}{2} dt = \delta {\int_{t_1}}^{t_2} \frac{m (\omega^2 l^2 + 2 \omega \theta' l^2 + \theta'^2 l^2)}{2} dt = 0[/tex]

    Now I have two problems.

    1) is this a good way to write down the Lagrangian?
    2) how do I get the [tex] "\delta" [/tex] inside the integral in a good way?

    I suppose after that it's integration by parts and setting the integrand equal to zero.

    EDIT: I solved the problem myselve today =)

    thanks for anyone who was willing to look at it.
    Last edited: Mar 4, 2008
  2. jcsd
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