INTEGRABILITY OF DIFFERENTIAL EQUATIONS AND “COMPLEXITY CONVERSION”
Alexey Andreevich Sukhno, Vyacheslav Vladimirovich Gulin
Institute of Mechanics of Moscow State University, Moscow, Russia
Keywords: differential equations, integrability, laws of physics, intuition, n-body problem, limits of knowledge, qualitative methods, phase space, complexity
Abstract
The article examines the situation around differential equations as a tool of natural science. On the one hand, during the “Quiet Scientific Revolution” of the 18th century, they made it possible to overcome the limitations of human intuition and reveal the potential of analytical methods for understanding nature. On the other hand, by the end of the 19th century, the problem of integrability of differential equations, which most clearly manifested itself with regard to the “n-body problem,” showed the need to rehabilitate intuition as the most important factor in scientific knowledge. It was the appeal to intuition that determined the creation of qualitative/geometric methods for studying nonlinear systems. Based on the analysis of this situation, the article draws a conclusion about the changes in methodology when scientific knowledge meets its limits: here the choice of mathematical tools is made in such a way that the “complexity” of one of the elements of the research situation, which has become an insurmountable barrier for knowledge, is transferred to some other element. So, this “complexity conversion” allows progress in the study of nature.
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