Geodesic on a cylinder - have I done this correctly?

[martinhiggs]

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

Homework Statement

ds^{2} = a^{2}d\theta^{2} + dz^{2}

ds = \sqrt{a^{2}d\theta^{2} + dz^{2}}

\int\sqrt{a^{2} + dz'^{2}} d\theta = Min

E-L equation

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

df/dz = 0,

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

Integrating gives:


\frac{z'}{\sqrt{a^{2} + z'^{2}}} = B


z' = B\sqrt{a^{2} + z'^{2}}

z = B \int\sqrt{a^{2} + z'^{2}}

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

z = b\theta + c (i think)

But I'm not sure how...

Have I made any mistakes??

 

[Dick]

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.

 

[martinhiggs]

ah yes, thank you so much!