Stable States of System



Alexander Liss





Sometimes it is beneficial to have a simple view of the system, a macro-model, which captures factors essential to particular study and ignores others.

For example, a set of strings fixed at both ends could be used as a musical instrument, where one makes strings oscillate and produce sound. For each string, one could describe a set of elementary types of oscillations (standing waves) and present any oscillation as a mixture of oscillations of elementary types. Those oscillations of elementary types are tones and overtones produced by a string. Now, one could ignore variability of oscillations of the same elementary type and deal only with types of oscillations. It is a simple and practical model; it allows tuning the musical instrument and playing music.

The part of Mechanics, which describes rigid bodies at rest Statics, in fact is a macro-model, where one ignores inevitable oscillations of rigid bodies. Instead of oscillating rigid body, Statics deals with types of oscillations, where each type is presented with the body, which with infinitesimally small oscillations. This aspect of Statics is often omitted, when Statics is taught, but it quickly becomes important, when one tries to apply Statics to solve practical problems. One has to analyze potential oscillations of the body and decide, if Statics is a proper model to be used.

Analysis of potential stable states of the system, and creation of a macro-model, which deals with the system of stable states, is a powerful tool of analysis of complex systems and it has numerous applications.

To support imagination, when one is searching for stable states, following provides a good example. One could see a stable state in movement of a ball in a bowl. Push the ball a little, and it rotates inside the bowl until it loses energy and settles on the bottom of the bowl. This system has only one stable state. Characteristics of the ball, as speed and position, are changing periodically oscillate.