Parameterize a geodesic using one of the coordinates

[PLuz]

I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so please ignore that simplification (I shall indicate it when it comes up).

Given a Riemannian manifold with metric
$$\tag{1} ds^{2}=\frac{dr^2}{a+r^2}+r^2\left(d \theta^2+\sin^2(\theta) \, d\phi^{2}\right),$$
consider a curve $c(\lambda)=(r(\lambda),\phi(\lambda),\pi/2)$ whose tangent vector is $c'(\lambda)=(r'(\lambda),\phi'(\lambda),0)$. The geodesic equations (if I didn't mess up) are given by:
\begin{cases}
\tag{2} r''-\frac{r'^{2}}{2(a+r)}-r(a+r)\phi'^{2}=0,\\
\phi''+2\frac{r'}{r}\phi'=0.
\end{cases}

The second equation of Eq.$(2)$ can be integrated, such that:
$$\tag{3} \phi'(\lambda)=\frac{C}{r^{2}(\lambda)},$$
where $C$ is a constant of integration.

Now I introduce a new equation:
$$\tag{4}r'^{2}\left(\frac{1}{a+r^{2}}\right)+\phi'^{2}r^{2}=1,$$
which is basically impose unit speed. Substituting Eq.$(3)$ in Eq.$(4)$ we have that:
$$\tag{5} r'=\sqrt{(a+r^{2})\left(1-\frac{C^{2}}{r^{2}}\right)}.$$
(Here is the difference from my case since Eq.$(5)$ can be integrated analytically and in my case, the congener equation can't).

Dividing Eq.$(3)$ by Eq.$(5)$ we have that
$$\tag{6} \frac{d\phi}{dr}=\frac{C}{r^{2}(\lambda)}\frac{1}{\sqrt{(a+r^{2})\left(1-\frac{C^{2}}{r^{2}}\right)}}.$$

So my question is:Does Eq.$(6)$ allow me to write the curve $c$ as $c(r)=(r,\phi(r))$, where $\phi(r)$ is given by $(6)$ and $c$ is a geodesic?

Note: This treatment is based on the Clairaut parametrization but I'm not sure if I can do it in this kind of problem where the $g_{rr}$ component of the metric varies...

 

[epr1990]

I'm not sure if you can simply introduce a new equation when its not given. Have you tried to find an expression which yields the r equation after substitution? Such as an integrating factor so you could put it into sturm-liouville form? I don't have a pencil and paper right now but it just looking at it, seems that there should be some expression like maybe [(a+r)^2*r']' yields the rest of the equation when dividing by some other expression of a+r it seems like it would resemble bessel's equation... If you don't think so, then you are probably right. I'm just diving into Tensors and Diff Geometry. But, that is what I see when looking at that equation because it seems to have some kind of symmetry.

However, I don't know the exact context of the problem, but from what is given to you, I would think that the method of characteristics would be the best approach. I found a really good demonstration of the method on google last year when I needed to learn it for a mathematical modeling class, since they didn't teach it in PDE's at my school.