DIFFERENTIAL EQUATION: limits to its uses
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 917 ▶ |
| Object type | Epistemology, ontology or semantics, Methodology or model |
These very significant limits are thus stated by A. RAPOPORT: “Both the strength and the limitations of the mechanical outlook reside in the mathematical methods used in the construction of mechanistic theories. The fundamental tool of this method is the differential equation which is essentially a precise statement about how certain quantities and their rates of change are related” (1966, p.3).
After reminding the role of second derivatives (the rates of change of the rates of change), he notes that: “If several bodies are involved… their motions would then be described by a system of differential equations in which the relations between the positions and the accelerations are all interwoven by a network of interdependencies. Now if the differential equations comprising a system are linear, i. e., if the variables and their rates of changes appear at most in the first degree, the same general methods of solution apply regardless of how many equations are involved” (Ibid).
This unfortunately, is a very special case, to which most complex phenomena cannot be reduced, lest the model would eliminate the very nature of the modelized system.
In celestial mechanics, for example, perturbations calculus can be superposed on a basically dual system (The sun and the earth for example — a two-body system), but with three or more bodies of comparable mass, the differential equations cannot be solved anymore (POINCARÉ) and we enter the realm of deterministic chaos.