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Euler-Lagrange Brachistochrone Problem in rotating system

  1. Aug 9, 2011 #1

    jpo

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    1. The problem statement, all variables and given/known data
    Bead slides on a wire (no friction) shaped as [itex]r = r(\theta) [/itex] in the Oxy plane. The Oxy plane and the constraining wire rotate about Oz with [itex]\omega = const[/itex]
    [itex]r, \theta[/itex] is the rotating polar frame; [itex]r, \phi[/itex] is the stationary frame.
    Find the trajectory [itex]r = r(\phi)[/itex] in the stationary frame.


    2. Relevant equations
    Minimize the time of motion
    [itex]T = \frac{1}{\omega} \int_{\pi/2}^{\phi_1} \sqrt{\frac{r'^2+r^2}{r}} d\phi[/itex]
    in polar coordinates

    3. The attempt at a solution
    The trajectory [itex]r = r(\phi)[/itex] in the inertial frame is unknown; In the rotating frame the trajectory is known: it is [itex]r = r(\theta) [/itex]. Then the integral in the rotating frame becomes

    [itex]T = \frac{1}{\omega} \int_{0}^{t_1} \sqrt{\frac{\dot{r}^2(\theta)+r^2(\theta) (\dot{\theta}+\omega)^2}{r(\theta)}} dt[/itex],
    because
    [itex]\phi=\theta+\omega t[/itex] and
    [itex]\frac{dr}{d\phi}=\frac{\dot{r}}{\dot{\phi}}=\frac{\dot{\theta}}{\dot{\theta}+\omega}[/itex]

    I have tried to write the integral into the rotating polar frame and then enforce the constraint, by substituting [itex]r[/itex] and its derivative through [itex]\theta[/itex] and [itex]\dot{\theta}[/itex]. Then I'll minimize [itex]T[/itex]with Euler-Lagrange
    [itex]\frac{\partial J}{\partial\theta}-\frac{d}{dt}\frac{\partial J}{\partial\dot{\theta}}=0[/itex]

    Is this correct?
     
  2. jcsd
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