Free particle — phase space

The same free particle, viewed simultaneously through its position-space and velocity-space coordinates.

Phase space is the joint space of all the coordinates needed to specify the instantaneous state of a system. For a single particle moving in two dimensions, that state is four numbers: the particle’s position (x, y) and its velocity (ẋ, ẏ).More precisely, in classical mechanics, phase space is parameterized by position and momentum, not velocity. For a particle of unit mass, the distinction vanishes.

The two panels below display the same particle from two angles. The left panel is configuration space: where the particle is. The right panel is velocity space: how fast and in what direction it is moving. Drag the dot in either panel and the system updates everywhere it appears.

With no force on the particle, its velocity is constant in time — the dot in the right panel does not move once you press start. The position panel records a straight-line trajectory.This is Newton’s first law. The free particle is the simplest non-trivial dynamical system: a single integration, no coupling. Together the panels form a complete picture of the state: pick any two points (one per panel) and the entire future is determined.