Let $(M, g)$ be a Riemannian manifold. Lately, I've grown interested in what you may call a "modified geodesic" problem. Given some smooth, non-negative scalar field $V$ on $M$ (aptly called the potential), what are the critical points of $E_V(\gamma) = \frac12\int_{a}^{b} \left[\left|\left|\dot{\gamma}(t)\right|\right|_g^2 + V(\gamma(t))\right]dt$ and how do they behave? The necessary conditions are easy to derive as $\frac{D}{\partial t} \dot{\gamma}(t) + \text{grad}V(\gamma(t)) = 0$ (using the Levi-Civita connection), and from here you can simply try to work through the standard results of geodesics, Jacobi Fields, etc. with this formalism.

To make things a little less tedious, my hope was to define a metric $\text{dist}_V: M \times M \to \mathbb{R}_+$ on $M$ by $$\text{dist}_V(p, q) := \inf\left\{\int_a^b \left[||\dot{\gamma}(t)||_g + V(\gamma(t))\right]dt: \gamma: [a,b] \to Q \ \text{ an admissible curve with } \ \gamma(a) = p, \ \gamma(b) = q\right\}$$ and recover from this metric a new Riemannian metric $g_V$ on $M$ such that the distance function induced by $g_V$ (in the usual way via the length of curves connecting two points) aligns with $\text{dist}_V$. In that sense, the modified geodesic problem could be understood purely from the theory of geodesics. We need only move to a new space where the geometry has been appropriately modified by the potential.

I've already seen from the discussion in this post that we in general cannot expect a given metric to be induced by a Riemannian structure, so I suppose that this may not be a fruitful method. Nevertheless, I'm curious if there is some class of potentials for which this can be done. For instance, disregarding the smoothness condition for a moment, it (perhaps naively) appeals to my intuition that with a potential like $V(q) = \text{dist}(q, p)$ for some fixed $p \in M$, the manifold would stretch/curve in a way that the geodesics get 'pulled' towards $p$.

However, in that same post I linked above, it is mentioned that it is necessary for $\text{dist}_V$ to be a "path-metric." If true, that could be a major obstruction here, as it seems to me that the "length" defined by $L_V(\gamma) = L(\gamma) + \int_{a}^{b} V(\gamma(t))dt$ is not well-defined unless $V$ is a constant map (as it is otherwise dependent upon the parameterization of $\gamma$).

Let $(M, g)$ be a Riemannian manifold. Lately, I've grown interested in what you may call a "modified geodesic" problem. Given some smooth, non-negative scalar field $V$ on $M$ (aptly called the potential), what are the critical points of $E_V(\gamma) = \frac12\int_{a}^{b} \left[\left|\left|\dot{\gamma}(t)\right|\right|_g^2 + V(\gamma(t))\right]dt$ and how do they behave? The necessary conditions are easy to derive as $\frac{D}{\partial t} \dot{\gamma}(t) + \text{grad}V(\gamma(t)) = 0$ (using the Levi-Civita connection), and from here you can simply try to work through the standard results of geodesics, Jacobi Fields, etc. with this formalism.

To make things a little less tedious, my hope was to define a metric $\text{dist}_V: M \times M \to \mathbb{R}_+$ on $M$ by $$\text{dist}_V(p, q) := \inf\left\{\int_a^b \left[||\dot{\gamma}(t)||_g + V(\gamma(t))\right]dt: \gamma: [a,b] \to Q \ \text{ an admissible curve with } \ \gamma(a) = p, \ \gamma(b) = q\right\}$$ and recover from this metric a new Riemannian metric $g_V$ on $M$ such that the distance function induced by $g_V$ (in the usual way via the length of curves connecting two points) aligns with $\text{dist}_V$. In that sense, the modified geodesic problem could be understood purely from the theory of geodesics. We need only move to a new space where the geometry has been appropriately modified by the potential.

I've already seen from the discussion in this post that we in general cannot expect a given metric to be induced by a Riemannian structure, so I suppose that this may not be a fruitful method. Nevertheless, I'm curious if there is some class of potentials for which this can be done. For instance, disregarding the smoothness condition for a moment, it (perhaps naively) appeals to my intuition that with a potential like $V(q) = \text{dist}(q, p)$ for some fixed $p \in M$, the manifold would stretch/curve in a way that the geodesics get 'pulled' towards $p$.

However, in that same post I linked above, it is mentioned that it is necessary for $\text{dist}_V$ to be a "path-metric." If true, that could be a major obstruction here, as the "length" function $L_V(\gamma) = L(\gamma) + \int_{a}^{b} V(\gamma(t))dt$ is dependent upon the parameterization of $\gamma$ unless $V$ is a constant map, and so I'm not sure if this could be a well-defined notion of length.