.
 Phase SpaceThe 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. 