I'm stuck on the following problem;
a) Calculate the Christoffel symbols for the metric
$dr^2 + G(r,\phi)d\phi^2$.
b) Write down the geodesic equations and verify the identity
$(r')^2 + G(\phi')^2 = constant$
is satisfied by virtue of these equations.
c) Show that the curves $\phi = constant$ are geodesics.
So for a) I have
$\Gamma^r_{rr} = 0, \Gamma^\phi_{rr} = 0, \Gamma^r_{r\phi} = 0, \Gamma^\phi_{r\phi} = G_r/2G, \Gamma^r_{\phi \phi} = -GG_r/2G = -G_r/2, \Gamma^\phi_{\phi\phi} = G_\phi/2G$
and for b), I get for the geodesic equations
$-\frac{G_r}{2}(\phi')^2 + r'' = 0$
$\frac{G_r}{G}r'\phi' + \frac{G_\phi}{2G}(\phi')^2 + \phi'' = 0$.
But after this Im stuck. The hint is that you differentiate the identity with respect to $t$ and sub in $r''$ and $\phi''$ from the geodesic equations. So I try that:
$\frac{d}{dt}\left(\frac{dr}{dt}\right)^2 + \frac{d}{dt}G\left(\frac{d\phi}{dt}\right)^2 = 0$
to get
$2r'r'' + 2G\phi'\phi'' = 0$
but when I substitute in $r''$ and $\phi''$ I get
$-G_rr'(\phi')^2 - G_\phi(\phi')^3$
and I don't see how this can equal zero? Have I calculated the Christoffel symbols correctly? Also Im not sure I differentiated the identity correctly - in that sense is $G$ a functions of $t$ through $r$ and $\phi$? If anyone can help me it would be greatly appreciated.