Phase space communicates both the motion and configuration (location with shape) of a system (x1,x2, x1‘,x2‘), wheras configuration space only mentions the values of the degrees of freedom of the system, therby decribing its location and configuration (x1,x2). Clearly the number of entries in the configuration space will be equal to the number of degrees of freedom of the system. The number of entries of phase space will be twice the number as it includes velocities as well.
Here is the key observation, shouldn’t we include accelerations as well or for that matter any number of higher order derivatives (x,x’,x”,x”’…) to make the description of motion complete? No! It is complete in phase space and any addition would be redunant. How?
It is because everything moves in Newtonian mechanics (or the equivalent Lagrangian formulation) according to second order differential equations. For example: x” = -bx’-kx. Hence, when we say the system is at a partciular point in phase space, the immediate future motion is determinied by the governing second order differential equation. In other words, there can be no intersection of trajectories in phase space. When the system arrives at a particular point, how the first derivative moves forward is dictated by the value of the second order derivative x”. Moreover, x” will evaluate to the same value at that point as long as the governing law does not change. And how x changes for the future trajectory is dictated by x’, which can be read off the current value of the point on the phase plot.
This is a good test if any imaginary system is truly second order. If the phase plot contains intersections with all the DOF accounted for, with their positions and velocities forming the axes then it points to a richer underlying (higher order) dynamics that is not captured adequately by the plot. In this case the plot is merely a lower dimension shadow or projection of the full dynamics in a higher dimension space.
Cars and Physics 