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!