Disclaimer: I have asked this question on Physics SE a week ago, but got no answers. I know that some MO users are interested in the $n$-body problem, so I decided to cross post here as well.
Consider the $n$-body problem where we are interested in describing the time evolution of $n$ masses interacting through a potential $U$. Let $D$ be the matrix containing all pairwise distances between our masses; i.e. $D_{ij} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2 + (z_i - z_j)^2}$ where $x_i, y_i, z_i$ are the Cartesian coordinates of the $i$'th mass. Assume that $U$ only depends on $D$, and not individual positions. [1]
I am interested in finding a differential equation that describes the time evolution of $D$, without individual positions appearing in it. Is it possible to formulate such an equation in classical mechanics?
[1] This is indeed true for Newton's Law of Gravitation, but it need not hold for a generic potential.