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The Brachistochrone problem

  1. Feb 2, 2006 #1
    find the curve for which the body will follow such that the time of travel is a minimim.
    Hints Minimize [tex] t_{12} = \int_{x_{1}}^{x_{2}} dt = \int_{x_{1}}^{x_{2}} \frac{ds}{v} = \int_{x_{1}}^{x_{2}} \sqrt{\frac{1+y'^2}{2gy}} dx [/tex]
    since F does not depend on x i can use hte beltrami identity (from the previous post)
    [tex] H = \frac{-1}{\sqrt{2gy} \sqrt{1+y'^2}}[/tex]
    [tex] \frac{dy}{dx} = \frac{1}{2gyH^2} -1 [/tex]
    this is where i am stuck
    SOlving this creates an ugly mess! How can i get the parametric equations from this?
  2. jcsd
  3. Feb 3, 2006 #2


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    There's no hint, you need to set the problem right. Use conservation of total mechanical energy.

  4. Feb 3, 2006 #3
    havent i already considered that using [tex] \frac{1}{2} mv^2 = mgh ==> v = \sqrt{2gh} [/tex]??

    would be very desirable to get this in terms of the parametric equations... they are far better in recognizing the cycloid
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