The Role of Singular Solutions in the Study of Physical Systems

TL;DR

This paper emphasizes the importance of singular solutions in physical systems, demonstrating their role in chaos and sensitivity, and proposing methods to identify and analyze them, even in classical problems like Kepler's.

Contribution

It introduces the significance of singular solutions in physical systems, showing their impact on chaos, sensitivity, and providing criteria for their detection using invariant measures.

Findings

Singular solutions often lead to chaotic regimes.

Considering singular solutions can reveal unusual dynamics in classical problems.

Invariant measures offer a simple criterion for identifying singular solutions.

Abstract

The necessity and benefit of singular solutions in the study of physical systems is shown. By singular solutions we mean solutions that are not contained in the general solution of the system of equations that describes the dynamic system under study. In addition, at the points of singular solutions the conditions of the uniqueness theorem are violated. It is shown that the presence of singular solutions, first of all, leads to the emergence of chaotic regimes. The dynamics of the system under study in the presence of singular solutions can differ radically from the dynamics in which singular solutions are not considered. It is shown that in many cases, in the presence of singular solutions, the system under study may turn out to be anomalously sensitive to small perturbations. Typically, singular solutions are not considered at analysing the dynamics of physical systems. It is shown that taking such solutions into account even in well-studied physical problems (for example, in the Kepler problem) can lead to unusual chaotic regimes. Examples of physical systems in which considering singular solutions turned out to be useful are given. It is shown that the use of an invariant measure provides a simple criterion for the appearance of singular solutions. It is shown that any regular function can be represented as a set of random functions.