Clarifying meaning of geodesic equations
In the line element for the hyperbolic plane (this trig chart covers all but one point) which you wrote down,
<br />
ds^2 = dr^2 + \sinh(r)^2 \, d\phi^2, \;<br />
0 < r < \infty, \; -\pi < \phi < \pi<br />
(I changed to a more suggestive notation), s is playing the role of arc length. The standard method of obtaining the geodesic equations is to read off what MTW call the "dynamical Lagrangian"
L = \dot{r}^2 + \sinh(r)^2 \, \dot{\phi}^2
Then apply the Euler-Lagrange equations:
0 = \frac{\partial L}{\partial r} - \frac{d}{d s} \; \frac{\partial L}{\partial \dot{r}}, \; \;<br />
0 = \frac{\partial L}{\partial \phi} - \frac{d}{d s} \; \frac{\partial L}{\partial \dot{\phi}}
Simplify the result so that \ddot{r},\ddot{\phi} are monic and collect terms in the first order derivatives. The result is the geodesic equations in their standard form and the coefficients are the Christoffel coefficients:
<br />
\ddot{r} + \sinh(r) \, \cosh(r) \, \dot{\phi}^2 = 0, \;<br />
\ddot{\phi} + 2 \, \coth(r) \, \dot{r} \, \dot{\phi} = 0<br />
where "dot" means differentiation wrt the parameter s. Here, s is playing the role of an arc length parameter, so the solutions of these equations give the geodesics as arc length parameterized curves. The second equation has a first integral
\dot{\phi} = \frac{L}{\sinh(r)^2}
which is obviously similar to the analogous first integral for a radial chart for the Euclidean plane (substitute \sinh(r) \mapsto r and consider what happens for small r).
Exercise: consider a simple semi-Riemannian analogue:
<br />
ds^2 = -dt^2 + \cosh(t)^2 \, d\phi^2, \;<br />
0 < t < \infty, \; -\pi < \phi < \pi<br />
(This is the upper half of H^{1,1}, the hyperboloid of one sheet in E^{1,2}, the signature 1,1 manifold of constant curvature, aka "two-dimensional tachyonic momentum space".) Why do the resulting equations give all geodesics, including null geodesics, as affine-parameterized curves? (This point recently confused someone in another thread.)
