Find range of parameters for which a given curve is a geodesic I am working on the following problem;
Given  parameterised surface $X(u,v)=(u \cdot \cos v,u \cdot 
\sin v,v)$ determine for which values $\alpha$ the curve $\gamma_{\alpha}=(t \cdot \cos (\alpha t),t \cdot \sin(\alpha t),\alpha t)$ is a geodesic.
According to a theorem a curve $\gamma = X \circ \beta $ is a geodesic if and only if $\beta(t)=(u(t),v(t))$ satisfy
$\frac{d}{dt}(E\dot{u}+F\dot{v})=\frac{1}{2}(E_{u}(\dot{u})^{2}+2F_{u}\dot{u}\dot{v} +G_{u}(\dot{v})^{2})$
$\frac{d}{dt}(F\dot{u}+G\dot{v})=\frac{1}{2}(E_{v}(\dot{u})^{2}+2F_{v}\dot{u}\dot{v} +G_{v}
(\dot{v})^{2})$.
Given the data in this problem we have,
$X_{u}=(\cos v,\sin v,0)$
$X_{v}=(-u \cdot \sin v, u \cdot \cos v, 1 )$
Hence
$<X_{u},X_{u}>=E=1$
$<X_{u},X_{v}>=F=0$ 
$<X_{v},X_{v}>=G=u(t)^2+1$
Therefore we get the system
$0=\alpha^2\cdot t$
$\alpha\cdot 2t=0$
Which means that $\alpha$ must be zero. This however dosn't seem quite right. Can anyone see where I go wrong and what the right answer should be?

My calculations seems to be correct.
