Phase space
The 2N-dimensional space of (position, momentum) pairs in which every classical state is a single point.
Definition
Phase space is the 2N-dimensional space of (q, p) pairs for a system with N degrees of freedom. A single point in phase space completely specifies the state of the system: every position and every momentum. A trajectory is a curve in phase space; an ensemble of states is a region.
Phase space is the natural stage for Hamiltonian mechanics, statistical mechanics, and chaos theory. Liouville's theorem says phase-space volume is conserved under Hamiltonian flow. Poincaré recurrence says that a bounded system eventually returns near any previous state. The KAM theorem says that most integrable tori in phase space survive small perturbations. All three are statements about the geometry of phase space rather than about any particular system.