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Homework Help: Calculus of Variations again

  1. Dec 3, 2005 #1
    Presume the earth is spherical, homogeneous and of radius [tex]R[/tex]. What should be the shape of a tunnel connecting two points on the surface in order to minimize the time it takes for a particle to travel between the two points.

    I have had a go at doing it in both polar and cartesian co-ordinates but am getting stuck. I'm fairly sure we are supposed to do it in cartesian but this way is proving particularly tricky.

    What i have done is this;

    Using some basic physics and energy conservation I have found that

    [tex]\displaystyle{v(r) = \sqrt{\frac{g}{R}(R^2 - r^2)}}[/tex]

    or

    [tex]\displaystyle{v = \sqrt{\frac{g}{R}(R^2 - x^2 - y^2)}}[/tex]

    [tex]ds^2 = dx^2 + dy^2[/tex]

    [tex]ds = \sqrt{1 + (y')^2} \; dx[/tex]

    So [tex]\displaystyle{\int t = \frac{ds}{v} = \sqrt{\frac{R}{g}} \int \frac{\sqrt{1 + (y')^2}}{\sqrt{R^2 - x^2 - y^2} \; dx}[/tex]

    The functional is

    [tex]\displaystyle{T[x, y, y'] = \sqrt{\frac{R}{g}} \frac{\sqrt{1 + (y')^2}}{\sqrt{R^2 - x^2 - y^2}}[/tex]

    This is the problem, I've never dealt with a functional like this before. Up until now, any functional has had a cyclic coordinate which has made it nice and simple. This fuctional depends explicitly on both [tex]x[/tex] and [tex]y[/tex].

    Is what i've done so far correct? How can I go about finishing the problem


    The question also gives the answer that should be obtained

    [tex]\displaystyle{x(\theta) = (R - r) \cos \left(\frac{R}{r}\theta \right) + r \cos \left(\frac{R - r}{R} \theta \right)}[/tex]

    and

    [tex]\displaystyle{y(\theta) = (R - r) \sin \left(\frac{R}{r}\theta \right) - r \sin \left(\frac{R - r}{R} \theta \right)}[/tex]
     
  2. jcsd
  3. Dec 4, 2005 #2
    bump... any ideas anyone?
     
  4. Dec 4, 2005 #3
    Assuming your functional is correct, why don't you proceed with calculating the Euler-Lagrange differential equation. If you do it in polar coordiantes with the correct functional, the ELDE can be integrated once giving the equation for a hypocycloid in polar coordiantes. It can then be written parametrically giving you the answer your book provided. I don't know why your professor would have a bias towards either method, generally they like to see the simplest solution.
     
  5. Dec 6, 2005 #4

    CarlB

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    Science Advisor
    Homework Helper

    So you get to neglect Coriolis effects?

    Carl
     
  6. Dec 7, 2005 #5
    I found I a cool link on a GR brachistrone problem. It becomes an interesting problem in that there are then 2 distinct frames from which to minimize the travel time.
     
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