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Eulers equation in cylindrical coordinates

  1. Feb 18, 2012 #1
    find the geodesics on a cylinder, where R^2 = x^2 + y^2


    so the goal is to find a function F, that gives the minimum distance between any two points on the cylinder.

    in cylindrical coordinates, dl = sqrt( ds^2 +(sdθ)^2 +dz^2 )

    since we are on the surface, s=R, and ds=0

    then dl = sqrt ( R^2 dθ^2 + dz^2 ) = F

    so I want to minimize F.

    We have been using eulers eq. for finding geodesics; eulers equation in polar coordinates is

    d/dr(dF/dθ') -dF/dθ = 0 , where F = F(r,θ,θ'), & θ'=dθ/dr

    but z isnt a function of theta, nor is theta a function of z, so I dont really know how to apply the euler eq
  2. jcsd
  3. Feb 19, 2012 #2


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    Science Advisor

    In polar coordinates, your two coordinates are r and θ. On the cylinder, r is no longer a coordinate, it is a constant which describes the radius of the cylinder. So your two coordinates are θ and z. So replace r with z in your Euler-Lagrange equation.
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