To find Geodesics on a cone I used the cylindial coordinates
$x=rcos\theta$
$y=rsin\theta$
$z=z$
Is this parameterization correct.How can I know how to parameterize?
Then arc length $ds^2=r^2 d^2\theta+dz^2$
$dz=\sqrt{r^2 +({dz\over d\theta})^2}d\theta$
Lagrangian = $\sqrt{r^2 +({dz\over d\theta})^2}=L(\theta,z,z')$
Since Lagrangian does not depend on $\theta $ I used the Theorem on Hamiltonian function as
$H(u,u')=L(u,u')-u'{\partial L \over\partial u'}$ is a first integral for the Euler Lagrange equation meaning that it is constant on each solution.
Then I end up with $H(z,z')={r^2 \over \sqrt{r^2+(z')^2}}$.Thus ${r^2 \over \sqrt{r^2+(z')^2}}=k$,where k is a constant.
And integrating $z={r\theta\over k}\sqrt{r^2-k^2}+m$, m is a constant of integration.
I want to know if what I have done is correct. And is solving for $z $ alone is enough