What is the vector field associated with potential energy? The mere concept of a line integral is defined for a vector field, and I thus thought the following was a rigorous and general definition of potential energy:
Definition: Given a conservative force field $f:U\to\mathbb{R}^n$, where $U$ is an open subset of $\mathbb{R}^n$, we define the potential energy associated to $f$ as the function $V:U\to\mathbb{R}$ for which
$$\int_Cf = V(x_A)-V(x_B)$$
for any piecewise curve $C\subseteq U$ that starts at $x_A$ and ends at $x_B$. Alternatively, our map $V$ is that for which
$$f = -\nabla V.$$
Observe that $V$ is defined up to adding a constant.
In the above definition, potential energy is a property of a force field. However, I don't think this always needs to be the case. In the case of two point-masses $m$ and $M$, for example, one may talk about the work done in moving the mass $m$ through a certain curve even when there does not seem to be a single force field involved (if anything, at any time $t$ there is a force field $f_t$ created by the point-mass $M$) and one may even show -although I'm confused by the proof- that
$$W=\frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}$$
is the work done on the point-mass $m$ due to gravity as $m$ moves from $\vec{r}(t_1)$ to $\vec{r}(t_2)$, and where $r(t_i)$ is the distance between $m$ and $M$ at time $t_i$.


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*What is a general, rigorous definition of potential energy? If we define it by means of the formula

$$W=\int_C\vec{F}\cdot d\vec{r}=U(x_A)-U(x_B)$$
where $C$ is a curve that starts at $x_A$ and ends at $x_B$, then what is the vector field involved in the integral above? How does such vector field relate with the function $\vec{F}$?


*In the case of two (or more) point-masses, what is the vector field involved?

 A: In the general case of $N$ point-like masses in $D$ dimensions, we have a $D$-dimensional position vector ${\vec r}_i$ for each particle and a corresponding $D$-dimensional force vector ${\vec F}_i$. The work done by the force on the$i$-the particle when moving the $i$-th particle from position ${\vec r}^A_i$ to position ${\vec r}^B_i$ along a path $C$ from $A$ to $B$, keeping fixed all the other particles is
$$W_{i,AB}=\int_{C}\vec{F}_i\cdot d\vec{r}_i.
$$
If we move all the particles along a path $C'$ in the $DN$-dimensional space starting from a point $\{ \vec r_i^A \}$ and ending at a point $\{ \vec r_i^B \}$, we can consider the total work.
$$W_{AB}=\int_{C'}\sum_i\vec{F}_i\cdot d\vec{r}_i.
$$
If such a quantity is independent of the path for all the pairs of the initial and final points, the differential form in $DN$-dimensional  allows introducing a potential energy scalar function as
$$W=U(\{ \vec r_i^A \})-U(\{ \vec r_i^B \})$$.
Summarizing, the vector field involved in the work for a system of $N$ particles is the $ND$ dimensional vector field $\{ \vec F_i \}$  ($i = 1,\dots,N$).
Thinking the potential energy as a property of the field of force is correct. However, one has not to forget that the connection of the potential energy with observable quantities is through $ {\vec F}_i = - \nabla_i U  $, for all the values of $i$.
