[NMSU, N. Vogt]

We consider the problem of two gravitationally interacting particles, and elect to change coordinates from r₁ and r₂ to R (r₁, r₂), the position of the center of mass, and r (r₁, r₂) = r₁ - r₂.

We can now find the equation of motion for r, using our conserved quantities.

We have already show that angular momentum L is conserved.

and we can evaluate the angular momentum per unit mass, l.

We will make use of these facts, in conjunction with the following vector identities.

Evaluating the angular momentum per unit mass, we find the following.

We finish by re-evaluating the second time derivative of r in terms of l.