restricted three body problem equations of motion using particle distances and one angle variable

Question

If we solve numerically a three (or $N$) body planar problem, it's easy to calculate the distances of the bodies as function of time. Conversely if we know the interparticle distances as functions of time and the angle of one distance, we can calculate the positions of all three (or $N$) bodies up to a reflection of the configuration (and of course translation). This is known as the distance geometric problem.

• So is it possible to write differential equations for the distances $r_{12}, r_{13}$ and $r_{23}$ and say an angle $a$ in case of three bodies or generally $N$ bodies?

In the case of two bodies (for example in the Kepler problem) that of course is possible and well known.

• Moreover, might the equations for the distances be of the form
  $$ \frac{\mathrm{d}^2r_{12}}{\mathrm{d}t^2} = f(r_{12}, r'_{12};r_{12},r'_{12},r_{23},r'_{23}) $$ (written for distance between bodies 1 and 2) that is, independent of the angle variable a like is the case with 2-body case?

Answer 1

The equations of motion cannot be written as $$ \ddot{r}_{ij} = f_{ij}(r_{12},\ldots, r_{n-1,n}, \dot{r}_{12},\ldots, \dot{r}_{n-1,n}) $$ Consider $n$ equally massive particles arranged equidistantly around a circle. If they are all initially stationary, the system will collapse to the centre of the circle. But if they are rotating at the correct speed, the system will be in equilibrium with all distances remaining constant. At time $0$ both systems have the same $r_{ij}$ and $\dot{r}_{ij}=0$.