What is phase space?
Phase space is the collection of possible states of a dynamical system. A phase space can be finite (e.g. for the ideal coin toss, we have two states heads and tails), countably infinite (e.g. state variables are integers), or uncountably infinite (e.g. state variables are real numbers). Implicit in the notion is that a particular state in phase space specifies the system completely; it is all we need to know about the system to have complete knowledge of the immediate future. Thus the phase space of the planar pendulum is two-dimensional, consisting of the position (angle) and velocity. According to Newton, specification of these two variables uniquely determines the subsequent motion of the pendulum. Note that if we have a non-autonomous system, where the map or vector field depends explicitly on time (e.g. a model for plant growth depending on solar flux), then according to our definition of phase space, we must include time as a phase space coordinate–since one must specify a spec
