1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Calculus of Variations again
  1. Calculus of variations (Replies: 8)

  2. Calculus of variations (Replies: 0)

  3. Calculus by variations (Replies: 2)

  4. Variational Calculus (Replies: 0)

  5. Calculus of variations (Replies: 3)

Loading...