Let $Q^n$ be a closed manifold and $M = TQ$ be its tangent bundle.
In [1], it is worked out in Equation 5.31 that there is a kinetic energy Riemannian metric $g$ on $Q$ with Levi-Civita connection $\nabla_g$ such that for any potential function $V$ and time-dependent forcing vector field $F$, the equation of motion is ${\nabla_g}_{\dot{\gamma}}\dot{\gamma} = -\text{grad}V + F$. This can be extended to include a Raleigh dissipative forces function $R$ and equation of motion ${\nabla_g}_{\dot{\gamma}}\dot{\gamma} = -\text{grad}_qV + \text{grad}_{\dot{q}}R + F$. In the absence of a time-dependent forcing vector field $F$, this becomes ${\nabla_g}_{\dot{\gamma}}\dot{\gamma} = -\text{grad}_qV + \text{grad}_{\dot{q}}R$. EDIT: Note that when $V = R = 0$ and $F = 0$, the equation of motion becomes ${\nabla_g}_{\dot{\gamma}}\dot{\gamma} = 0$, and the system simply follows its "kinetic energy geodesics".
My question is, is it worked out anywhere that if one has a time-independent, complete vector field $\xi$ on $M$, there is a Levi-Civita connection $\nabla_{\xi}$ on $M$ with the flow lines $\Phi_t$ of $\xi$ being geodesics of $\nabla_{\xi}$?
This seems a natural generalization of the ideas of [1], but I'm unable to find a reference. I can likely work it out myself if necessary, but I'm hoping it's already done so I won't have to "re-invent the wheel".
[1] Bullo, Francesco; Lewis, Andrew D., Geometric control of mechanical systems. Modeling, analysis, and design for simple mechanical control systems., Texts in Applied Mathematics 49. Berlin: Springer (ISBN 0-387-22195-6/hbk). xxiv, 726 p. (2005). ZBL1066.70002.
Best Answer
As stated the answer is negative: the velocity of geodesics has constant length, but an integral curve for a general vector field does not have to.
The question becomes interesting if you formulate it by asking that the support of the curve is the support of a geodesic. But even in this case the answer is negative. Indeed, consider a vector field over a compact manifold, vanishing at some points $p_-$ and $p_+$, $p_-\neq p_+$, and suppose that there is a flowline $\gamma$ limiting to $p_-$ and $p_+$ at $-\infty$ and $+\infty$ respectively. The support of $\gamma$ cannot be the support of a geodesic, as a geodesic on a compact manifold is a curve that has either infinite length or is diffeomorphic to $\mathbb S^1$.