Phase space

Phase space is the space of all possible states of a physical system, particularly in classical mechanics, where each point represents a complete microstate specified by both positions and momenta of all particles. Unlike the state space of a general dynamical system, phase space carries a symplectic structure that preserves volume under Hamiltonian flow — a constraint with deep consequences for the system's long-term behavior.

The phase space formulation reveals that deterministic systems can exhibit chaotic dynamics even when their phase space volume is conserved. The Liouville theorem states that Hamiltonian systems preserve phase space volume, yet trajectories can stretch and fold into fractal structures, producing the appearance of irreversibility from perfectly reversible dynamics. This is the foundation of statistical mechanics: the macroscopic arrow of time emerges from the geometry of phase space, not from any asymmetry in the microscopic laws.

Phase space is also the natural setting for the study of observability in physical systems: not all coordinates of phase space are accessible to measurement, and the problem of reconstructing the full state from partial observations is the central challenge of experimental physics.