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Finding geodesics on a cone of infinite height

  1. Jan 27, 2013 #1

    Leb

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    1. The problem statement, all variables and given/known data
    Find the geodesics on a cone of infinite height, [itex]x^{2}+y^{2} = \tan{\alpha}^{2}z^{2}[/itex] using polar coordinates [itex](x,y,z)=(r\cos{\psi},r\sin{\psi},z) with z=r\tan(\alpha)[/itex]
    HW.png

    3. The attempt at a solution

    I am not sure with how should I expres the element [itex]dz^{2}[/itex] ? When it is a function of α (My calculus was always weak especially stuff with creating a derivative by dividing...)

    Thanks.
     
  2. jcsd
  3. Jan 27, 2013 #2

    haruspex

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    The question is wrong there. They mean ##z=r\cot(\alpha)##
    α is a constant, r2 = x2+y2. You can obtain an expression for dz in terms of x, y, z, dx and dy. Does that help?
     
  4. Jan 28, 2013 #3

    Leb

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    Thanks haruspex!

    I just tried to write them down like this:

    [itex]dx=\cos(\psi) dr - r\sin (\psi) d\psi[/itex]
    [itex]dy=\sin(\psi) dr + r\cos (\psi) d\psi[/itex]
    [itex]dz=\cot (\alpha) dr[/itex]
    and
    [itex]ds = \sqrt{(1+\cot^{2}(\alpha))dr^{2}+r^{2}d\psi^{2}}[/itex]
    taking dr^{2} out of the square root and calling the constant term as k
    [itex]ds = \sqrt{k+r^{2}\frac{d\psi^{2}}{dr^{2}}}dr[/itex]


    And now to integrate with limits from zero to infinity ? (Does not matter since we are looking for L (Lagrangian), right ?)

    Update
    OK, so I think I have found a solution, [itex]r_{0}=const=r\cos(\frac{\psi + C}{\sqrt{k}})[/itex] I now should do it with lagrange multipliers. Will I get the same answer up to a constant ?
     
    Last edited: Jan 28, 2013
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