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Brachistochrone Problem/Calculus of Variations

  1. Oct 11, 2006 #1
    I'm working on the Brachistochrone probem and I've gotten to the equation:

    (dy/dx)^2 = (c^2*y)/(1-c^2*y)

    the 'hint' given is to use y = sin^2(theta)/c^2 to solve the integral. I haven't done any math for 5 months and i haven't been in a pure math class for over a year, so I'm drawing a complete blank on how to solve this. If someone could point me in the right direction that would help.

    Also, I'm having trouble approaching a problem in which I'm asked to find the geodesics on a sphere of unit radius using calculus of variations. I'm asked to express phi as a function of theta but that isn't making sense to me...again, some direction would help me get on track.
    (^ is supposed to denote a power)

    Thanks
     
    Last edited: Oct 11, 2006
  2. jcsd
  3. Oct 11, 2006 #2
    If you make the substitution
    [tex]y = \sin^2(\theta)/c^2[/tex]
    then you also need to substitute for [itex]dy[/itex] the value
    [tex]dy = 2/c^2 \sin(\theta) \cos(\theta) d\theta[/tex].
    Grouping [itex]\theta[/itex] terms and x terms on opposite sides of the equation should give you something you can integrate.

    Look familiar? Think back to your integral calculus class :)

    For the geodesics on a sphere you need to consider the differential line element in spherical coordinates
    [tex]ds^2 = dr^2 + r^2 d\theta^2 + r^2\sin^2(\theta) d\phi^2[/tex]
    and remember that for a geodesic the quantity you are trying to minimize is
    [tex]\int ds[/tex].

    On the surface of a sphere [itex]r[/itex] is constant, so that simplifies things a bit, and you can then factor out the [itex]d\theta^2[/itex] and follow the standard calculus of variations method using the Euler-Lagrange equation.
     
    Last edited: Oct 11, 2006
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