- #1
JFuld
- 23
- 0
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 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
----------------------------
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