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Geodesic on a cylinder - have I done this correctly?

  1. Oct 20, 2009 #1
    Geodesic on a cylinder - have I done this correctly??

    1. The problem statement, all variables and given/known data

    ds[tex]^{2}[/tex] = a[tex]^{2}[/tex]d[tex]\theta^{2}[/tex] + dz[tex]^{2}[/tex]

    ds = [tex]\sqrt{a^{2}d\theta^{2} + dz^{2}}[/tex]

    [tex]\int\sqrt{a^{2} + dz'^{2}}[/tex] d[tex]\theta[/tex] = Min

    E-L equation

    df/dz - d/d[tex]\theta[/tex](df/dz') = 0

    df/dz = 0,

    d/d[tex]\theta[/tex][[tex]\frac{z'}{\sqrt{a^{2} + z'^{2}}}[/tex]] = 0

    Integrating gives:


    [tex]\frac{z'}{\sqrt{a^{2} + z'^{2}}}[/tex] = B


    z' = B[tex]\sqrt{a^{2} + z'^{2}}[/tex]

    z = B [tex]\int\sqrt{a^{2} + z'^{2}}[/tex]

    I am now stuck, I should be able to get to:

    z = b[tex]\theta[/tex] + c (i think)

    But I'm not sure how...

    Have I made any mistakes??
     
  2. jcsd
  3. Oct 20, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Re: Geodesic on a cylinder - have I done this correctly??

    You don't have to integrate it. You just have to show that z'/sqrt(a^2+z'^2)=B means z'^2 must be a constant. Which in turn means z' is a constant.
     
  4. Oct 20, 2009 #3
    Re: Geodesic on a cylinder - have I done this correctly??

    ah yes, thank you so much!!
     
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