Phase Space

The multidimensional space in which each dimension represents one of the essential measures of a
physical system is called phase space.  Taken together, the measures uniquely describe the state of
the system.  It might seem that it should be called state space, but that name is used for several
slightly different physical and mathematical spaces.

For the two wheels in the example sculpture, four quantities are required to specify the state of the
system of wheels. The four quantities might be chosen in a variety of ways, but the most common
way is to use the angular position and angular velocity of each wheel.

If the state were graphed in the most straightforward way, I would have to have a four-dimensional
plot: one quantity graphed as the distance along each of the four axes.  Four dimensional graphs are
very hard to make in a three-dimensional world!  In my plots, I use the two dimensions of the flat
screen for displaying two quantities and ribbon color and ribbon width to substitute for the two other
dimensions my screen doesn't have.

If the sculpture is placed in a particular state—a particular value for each angle and angular velocity—
then it will move in a particular way.  The laws of classical physics, Isaac Newton's laws, determine
how it moves going forward in time.  If you grab the sculpture and place it back precisely in the same
state and let go, it will do precisely the same motion every time you try it.

As mentioned in the discussion of chaos theory, it is the preciseness that is difficult to achieve.
Copyright  2008 James W. Wiggins.  All rights reserved.