n-body Problem - Three-body Problem

Three-body Problem

Not as much is known about the n-body problem for n ≥ 3 as for n = 2. The case n = 3 was most studied and for many results can be generalized to larger n. Many of the early attempts to understand the 3-body problem were quantitative, aiming at finding explicit solutions for special situations.

• In 1687 Isaac Newton published in the Principia the first steps taken in the definition and study of the problem of the movements of three bodies subject to their mutual gravitational attractions. His descriptions were verbal and geometrical, see especially Book 1, Proposition 66 and its corollaries (Newton, 1687 and 1999 (transl.), see also Tisserand, 1894).
• In 1767 Euler found collinear motions, in which three bodies of any masses move proportionately along a fixed straight line.
• In 1772 Lagrange discovered two classes of periodic solution, each for three bodies of any masses. In one class, the bodies lie on a rotating straight line. In the other class, the bodies lie at the vertices of a rotating equilateral triangle. In either case, the paths of the bodies will be conic sections. Those solutions led to the study of central configurations, for which for some constant k>0 .

Specific solutions to the three-body problem result in chaotic motion with no obvious sign of a repetitious path. A major study of the Earth-Moon-Sun system was undertaken by Charles-Eugène Delaunay, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in perturbation theory.

The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun-Earth-Moon system and many others). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point.

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Poincaré at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the restricted problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. This may be easier to visualize if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the equilibrium points maintaining the 60 degree-spacing ahead of and behind the less massive body almost in its orbit (although in reality neither of the bodies is truly stationary; they both orbit the center of mass of the whole system). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.