State Spaces

Life is usually easier if we imagine the states of a dynamical system as points in space.

Points on the cylinder can be used to represent all possible states of our pendulum, for example.

The "lengthwise" position of the point represents the speed and direction of the motion of the pendulum.

The position of the point in the "circular" direction represents the angle of the pendulum.

This lets us imagine the changing state of the system as a point moving through space as time goes by.

We usually trace a line along the path followed by the moving point (as in the picture) to make it it is easier to follow what is happening.

Of course, the way in which the point moves will be determined by the rules which govern the system.

QUESTION: Can you work out how the penulum would have to move for its state to be changing as shown in the animation? Does this seem possible?

The rules governing a system often keep the point within a particular region of space.

State spaces may come in many shapes and forms. For our pendulum the state space is just a cylinder.

Visualising the changing states of a system as a point moving through space helps our intuition about what is possible, even though there are often too many measurements to allow a realistic picture to be drawn.

This is called the "state space" or "phase space" of the system.