Stable States
of System
Introduction
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.