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What are the 'closed geodesic' ?

  1. Sep 14, 2008 #1
    i have been reading about 'Selberg Trace formula'

    i know what a Laplacian is but i do not know what is the author referring to when he talks about 'Closed Geodesic' i know what the Geodesic of a surface is

    [tex] l(\gamma)=\int_\gamma \sqrt{ g(\dot\gamma(t),\dot\gamma(t)) }\,dt\ ,[/tex]

    but i do not know what means 'closed' or why the geodesic of a torus would have the lenght (?) [tex] l_n =na [/tex] a=radius ??
     
  2. jcsd
  3. Sep 14, 2008 #2

    tiny-tim

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    Hi mhill! :smile:

    "closed" simply means that the geodesic meets itself, so it has a finite length.

    If the tangent of the angle to one axis is a rational multiple of the ratio of the axes, then it will meet itself. If irrational, it will go on for ever. :smile:

    n is the number of turns "through the hole" before the geodesic joins up. The more turns, the longer the geodesic (though I must admit, I don't see why it's proportional :redface:).
     
  4. Sep 16, 2008 #3
    thanks tiny-tim, then you mean that a geodesic will be closed if for example x(a)=x(b) , so there is a point where the geodesic intersect itself.

    and for the 'Selberg Trace' is there a pedestrian proof or a proof that a profane non-mathematician could understand ??
     
  5. Sep 16, 2008 #4

    tiny-tim

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    Hi mhill! :smile:

    "a point" is rather an understatement … it intersects itself everywhere, an infinite number of times. :wink:
    Sorry … I've no idea what a Selberg trace is. :redface:
     
  6. Sep 16, 2008 #5
    ' Selberg Trace' is related to the fact that you can express a sum [tex] f(E_n ) [/tex]

    with [tex] -\nabla f(x)= E_n f(x) [/tex] using the 'length' of the closed geodesics

    http://mathworld.wolfram.com/SelbergZetaFunction.html
     
  7. Sep 16, 2008 #6

    HallsofIvy

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    For a cylinder the closed geodesics are the circles perpendicular to the axis of the cylinder.

    For a sphere the closed geodesics are the great circles.
     
  8. Sep 17, 2008 #7

    tiny-tim

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    torus

    Hi HallsofIvy! :smile:

    You missed the torus :wink::
     
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