Geodesics and a general pregeodesic equation

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.