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Let $(M,g)$ be a Riemannian manifold, and let $\nabla$ denote the Levi-Civita connection. Then we say a smooth curve $\gamma:J\to M, t\mapsto\gamma(t)$ is a geodesic if $$D_t\gamma'=0.$$ We say a smooth curve $\hat{\gamma}:I\to M, s\mapsto\hat{\gamma}(s)$ is a pregeodesic if there exists a diffeomorphism $\phi:J\to I$ such that $\gamma:=\hat{\gamma}\circ\phi$ is a geodesic for some open interval $J\subseteq\mathbb{R}$.

Let's now turn to the local representation of the above, that is, suppose $(U,x^j)$ are coordinates on $M$ with Christoffel symbols $\Gamma_{ij}^k$. Then then geodesic equation reads $$\ddot{\gamma}^k+\dot{\gamma}^i\dot{\gamma}^j\Gamma_{ij}^k=0.$$ Then a fairly straightforward application of the chain rule yields the result: A curve $\hat{\gamma}:I\to U$ is a pregeodesic if and only if $$\frac{d^2\hat{\gamma}^k}{ds^2}+\frac{d\hat{\gamma}^i}{ds}\frac{d\hat{\gamma}^j}{ds}\Gamma_{ij}^k=f(s)\frac{d\hat{\gamma}^k}{ds},\qquad (*)$$ for some continuous $f:I\to\mathbb{R}$. Indeed (for the relevant direction), suppose $\gamma=\hat{\gamma}\circ\phi$ is a geodesic for some diffeomorphism $\phi:J\to I, s=\phi(t)$. Then $\hat{\gamma}$ satisfies the above system $(*)$ with $$f(\phi(t))=-\frac{\frac{d^2\phi}{dt^2}}{\left(\frac{d\phi}{dt}\right)^2}.$$

This leads to my question: I've come across in the literature that $(*)$ is equivalent to the equation $$\frac{d^2\hat{\gamma}^k}{ds^2}+\frac{d\hat{\gamma}^i}{ds}\frac{d\hat{\gamma}^j}{ds}\Gamma_{ij}^k=F(\gamma')\frac{d\hat{\gamma}^k}{ds},\qquad (**)$$ for some continuous $F:TU\to\mathbb{R}$ which is homogeneous of degree $1$ in the tangent variable.

I don't understand what this function $F$ is. Clearly, any curve $\hat{\gamma}$ satisfying $(**)$ is a pregeodesic. However, the result is saying that there exists some $F:TU\to\mathbb{R}$ with the above properties so that all pregeodesics satisfy $(**)$ with that specific $F$.

Now, the function $f(s)$ depends on the diffeomorphism $\phi$, which in turn depends on the geodesic $\gamma$ with starting point $(x,\xi)\in TU$. There is some homogeneity of geodesics when dealing with the initial condition, so this certainly seems reasonable, but I can't piece all of this together in coherent form.

I think this may actually be related to general sprays, and this $F$ is a reformulation of the Liouville vector field associated to the geodesic spray, but this is a bit outside my field (at the moment).

Any help or references would be appreciated.

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