When one has a system with n degrees of freedom, one can take these n 'measurements' and plot these in an n-dimensional space producing a 'phase plot'. This space is called the phase space.

Phase space has the property that each point in the space corresponds to a unique state of the system. This is then useful in analysing how the system changes over time. By joining together subsequent points as the system evolves over time, one obtains a phase path.

An example of a simple system which appears 'nice' in phase space is that of an ideal pendulum. The state of the pendulum can be described completely by two pieces of information - the current angle from vertical and the current angular velocity, and so it can be represented in a two dimensional phase space.

Plotting the angle of the pendulum against its angular velocity at each point in time, one gets a closed curve*.

Adding some friction to the system, one sees this curve spiral in to the centre as energy is lost (and so it swings less far each time, and with reduced maximum speed.

* for a pendulum of mass m, of length a, one expression of this curve is:

theta = arccos( (a/2g)*(d theta/dt)² +c) where g is gravitational field strength, c is some constant (depending on the initial conditions of the pendulum - i.e. from how high it was dropped)