A critique of some mathematical models of mechanical systems with differential constraints

Abstract

Theories (including mathematical models) put forward for a particular phenomenon are unacceptable, if they do not satisfy certain criteria (see, for instance, [1, p. 14; and 2, pp. 190–196]). The objective criterion of correctness demands that the behaviour predicted by the model agrees with the observed data, to the accepted accuracy. ‡ ‡ “We are bound to require of every fundamental law of our mechanical system, that when applied to approximately correct relations it should always lead to approximately correct results, not to results which are entirely false.” [1, p. 21]. There might be several other correct theories, to the given accuracy. § § Feynman asserts “… every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics. He knows that they are all equivalent, and that nobody is ever going to be able to decide which one is right at that level…” [3, p. 168]. In practice, preference is given to the theory which the researcher deems to be the simplest. ¶ ¶ V. V. Novozhilov: “When any new problem arises, a rough model is first constructed. It is then verified by experiment (if not preceded by it) and better models are constructed, for as long as necessary” [4, p. 361]. At the very beginning of his first lecture, Kirchhoff puts forward the requirement of the simplicity of a theory [5, p. 5], thereby emphasizing the importance of this criterion in the natural sciences. Courant, in concluding his essay [6, p. 27], emphasizes a fundamental difference between the research formulations and goals of the mathematician and of the scientist solving an applied problem. For the mathematician, the only criterion of the applicability of a theory is its logical consistency. The desire for generality is incompatible with the requirement that the theory should be simple. And since the object of mathematical research is the properties of mathematical relations, even in cases where these relations have arisen in mechanics, for instance, they are usually treated as abstract generalizations, and their correctness (in the above sense) is not discussed. However, if such research claims to have an applied value, it should, of course, satisfy the criteria of correctness and simplicity. In a number of cases, it is found that the mathematical construction cannot be applied in practice. This assertion is demonstrated here on the models of Lindelöf and Kozlov.