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I Brachistochrone for Specific Ratios

  1. May 15, 2017 #1
    It is commonly known that the solution to the brachistochrone problem is a cycloid.

    However, in order for a solution curve to be a cycloid, the ratio between points A on the y-axis and B on the x-axis has to be r/pi*r, since that is the ratio between the "height" of the cycloid and half of its "length". However, what is the solution curve to the brachistochrone problem if points A and B share a different ratio to each other - say, 1/2 ?

    Two possible solutions that I have considered:
    1. The curve is an affine function of the cycloid; the curve is stretched by a factor k along one of the directions.
    2. The curve is a segment of a larger cycloid.
    How would I approach this problem?
  2. jcsd
  3. May 15, 2017 #2
    Have you looked into the "Calculus of Variations"? Where you can use single variable calculus to find points on a curve corresponding to the extrema (ie. min/max), the Calculus of Variations can be used to find entire functions that are extrema for some characteristic. Taylor's Classical Mechanics text for example has a chapter that introduces this and I believe the Brachistochrone is one of the solutions presented (Morin's Mechanics text also touches on this.)

    I suspect having gone through that you would be able to learn how to introduce new constraints that produce different curves.
  4. May 15, 2017 #3
    I don't quite understand how I would go about adding such a constraint as I mentioned above
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