We now specialise to the case of three bodies moving under mutual gravitational attraction. In Section 7.1 we give a detailed proof of Lagrange's theorem on homographic solutions to the three-body problem. Lagrange's argument includes, in principle, the collinear solutions found earlier by Euler. For simplicity, however, we shall prove Lagrange's theorem under the assumption that the triangle formed by the three bodies is at all times non-degenerate. Then, in Section 7.2, we give a direct derivation of Euler's solutions under some simplifying assumptions. In Section 7.3 we derive various general results about the so-called restricted three-body problem, where one of the three masses is assumed to be negligibly small compared with the other two.

Lagrange's homographic solutions

In Section 6.6 we have given a complete description of the non-collinear planar homographic solutions of the three-body problem. The following theorem of Lagrange says that all non-collinear homographic solutions of the three-body problem are in fact planar, and hence of the form described earlier.

Theorem 7.1 (Lagrange) Let (r1, r2, r3) be a non-collinear homographic solution of the three-body problem, with centre of mass fixed at the origin. Then the following statements hold.

1. (i) The plane determined by the three bodies remains fixed in space.

2. (ii) The sum of the Newtonian forces acting on each of the three bodies is directed towards the origin.

3. (iii) The three bodies form an equilateral triangle.

4. (iv) The three bodies move on conic sections that are similar to one another, each having the origin as one of its foci.

Figure 7.1 shows such a homographic solution in the case of three equal masses, where the three conic sections are congruent. Figure 7.2 shows a relative equilibrium with three different masses moving on circles.

Remark 7.2 Lagrange's solutions to the three-body problem can be observed in nature. In the plane of Jupiter's orbit around the Sun, near the two points forming an equilateral triangle with Sun and Jupiter, there are large groups of asteroids known as the Trojans.