Conserved quantities for geodesics

Brewer

1. The problem statement, all variables and given/known data
In comoving coordinates, a one dimensional expanding flat universe has a metric [tex]ds^2 = -c^2dt^2 + at(t)^2dr^2[/tex]. Derive an expression for a conserved quantity for geodesics in terms of [tex]a, \tau[/tex] and [tex]r[/tex], where [tex]\tau[/tex] is the time measured in the rest frame of the freely falling particle.


2. Relevant equations



3. The attempt at a solution
I have the answer to the question in front of me, I just don't follow one of the steps, so I just wondered if anyone could explain it to me.

After writing a Lagrangian as [tex]L = c^2(\frac{dt}{d\tau})^2 - a(t)^2(\frac{dr}{d\tau})^2[/tex]

it can be seen that since r does not appear explicitly that it has something to do with it a constant.

The next line in the answer goes onto say
[tex]\frac{dL}{d\frac{dr}{d\tau}}[/tex] is a constant, but I don't know why this is.

At a guess I would say this is because the [tex]\frac{dL}{dr}[/tex] term in the Euler Lagrange equations is zero (because of the lack of dependence on r), and as such you get [tex]\frac{d}{d\tau}\frac{dL}{d\frac{dr}{d\tau}} = 0[/tex], and so by integrating with respect to [tex]\tau[/tex] you'll get a constant on the right hand side right? Or am I scratching at the wrong tree?

Any help would be appreciated.