Remarkable Three-body Motions

The motions of planets and other celestial bodies give the most convincing observational support for the laws of classical Newtonian mechanics. In this wonderful space laboratory all phenomena are observed in their purest form, without the complication of friction and air resistance that are inevitable in an ordinary earth laboratory.

The differential equations of motion for a body under the central inverse square gravitational force (for a planet orbiting a star or a satellite orbiting the planet) have exact analytic solutions (a single-body Kepler problem). The striking mathematical simplicity of trajectories is a distinctive feature of Keplerian motion. Any possible motion in the Newtonian inverse square gravitational field occurs along one of the conic sections - curves formed by the intersection of a circular cone by a plane. Exact analytic solutions exist also for the motions of two celestial bodies attracted by mutual gravitational forces - this two-body problem mathematically may be reduced to the case of a single body which moves in an effective stationary inverse-square gravitational field.

The most fascinating phenomena of celestial mechanics are revealed in the motions of three or more bodies attracted to one another by gravitational forces. If a third body is added to a system of two interacting bodies, the three-body problem generally becomes analytically unsolvable, that is, there exist no general formulas that describe the motion and permit the calculation of positions and velocities of the bodies from arbitrary initial conditions. The lack of analytic solutions is related to the extraordinary complexity of possible motions. Some examples included in the presented collection of Java applets allow us to observe fascinating trajectories of three-body motions that delight the eye and challenge our intuition. However, among the great variety of extremely complex motions there exist a finite subset of very simple regular motions