Finding geodesics on a cone of infinite height

[Leb]

Homework Statement

Find the geodesics on a cone of infinite height, $x^{2}+y^{2} = \tan{\alpha}^{2}z^{2}$ using polar coordinates $(x,y,z)=(r\cos{\psi},r\sin{\psi},z) with z=r\tan(\alpha)$

The Attempt at a Solution

I am not sure with how should I expres the element $dz^{2}$ ? When it is a function of α (My calculus was always weak especially stuff with creating a derivative by dividing...)

Thanks.

 

[haruspex]

with $z=r\tan(\alpha)$

The question is wrong there. They mean $z=r\cot(\alpha)$
α is a constant, r² = x²+y². You can obtain an expression for dz in terms of x, y, z, dx and dy. Does that help?

 

[Leb]

Thanks haruspex!

I just tried to write them down like this:

$dx=\cos(\psi) dr - r\sin (\psi) d\psi$
$dy=\sin(\psi) dr + r\cos (\psi) d\psi$
$dz=\cot (\alpha) dr$
and
$ds = \sqrt{(1+\cot^{2}(\alpha))dr^{2}+r^{2}d\psi^{2}}$
taking dr^{2} out of the square root and calling the constant term as k
$ds = \sqrt{k+r^{2}\frac{d\psi^{2}}{dr^{2}}}dr$ And now to integrate with limits from zero to infinity ? (Does not matter since we are looking for L (Lagrangian), right ?)

Update
OK, so I think I have found a solution, $r_{0}=const=r\cos(\frac{\psi + C}{\sqrt{k}})$ I now should do it with lagrange multipliers. Will I get the same answer up to a constant ?