Eulers equation in cylindrical coordinates

[JFuld]

find the geodesics on a cylinder, where R^2 = x^2 + y^2

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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 isn't a function of theta, nor is theta a function of z, so I don't really know how to apply the euler eq

 

[phyzguy]

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.