International Encyclopedia of Systems and Cybernetics

The problem, first considered by H. POINCARÉ (In "Méthodes nouvelles de la Mécanique Céleste", 1892-99) is to compute exactly the future or past position at some instant of three celestial bodies whose initial positions and velocities are known and which are submitted to Newtonian reciprocal attraction.

POINCARÉ showed that such a prediction was not possible by using anyone of the usual functions: linear, trigonometric, exponential and their combinations. Moreover he demonstrated that a three bodies system may well be unstable.

This has been confirmed during the 1960's by KOLMOGOROV, ARNOL'D and MOSER (The so-called KAM theorem), who showed that for some values of the initial conditions, the trajectories are quasi periodic, as if they were integrable, but for other initial conditions, arbitrarily close, instabilities zones do appear (chaotic zones), where dynamics are quitemore intricated.

This has been confirmed by computing comets and asteroids orbits, which result chaotic in the long run, i.e. subject to significant discontinuous changes. Very generally the classical newtonian two-bodies problem is an arbitrary simplification, obtained by elimination of secondary perturbations. Thus chaotic dynamics is also a feature of celestial mechanics and, conversely, it appears that Newtonian mechanics may imply non absolutely deterministic consequences.

As explained by R. ROSEN, the three bodies problem has wide implications in relation to the interactions of elements within any system or for groups of interacting systems: "Yet we might argue that we could study such a threebody system by decomposing it into subsystems; a two-body system and a one-body system, both of which are solvable in closed form. This does not help us, because the very act of decomposing the original system in that fashion will irreversibly destroy the dynamical laws of the original system; in other words, the properties of the subsystems prepared in that fashion will have nothing whatever to do with the properties of the system we are interested in. The properties of the subsystems will be artifacts. Stated another way, a three-body system is simply not the algebraic sum of a two-body and a one-body system"…

"Now biological systems are typically organized in a manner like that of the three-body system; they resist physical decomposition into subsystems which would possess the same dynamical properties in isolation that they did when connected into the system" (1974, p. 172).

As expressed by J. CASTI: "The essence of the three bodies problem resides somehow in the linkages between all three bodies" (1994, p. 41 – emphasis ours). Simultaneities of causes and effects, when complex, cannot be perfectly modelized, and still less so at long term.

This is possibly the clearest indictment of reductionism applied to complex systems.